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Astroscopus guttatus

Astroscopus guttatus (northern stargazer) is a fish that can reach lengths of 22 in and is found on the Atlantic shores between the states of North Carolina and New York in the United States. The northern stargazer can be found up to depths of 120 ft. Stargazers have a flat forehead with a lot of body mass up front near the mouth.

Description
The northern stargazer has a blackish brown body with white spots that are of the same size all over its head and back. It has three dark horizontal stripes on its (white) tail. The mouth of the stargazer faces up so that it can ambush prey while hiding in the sandy bottoms of coastal bodies of water. The top of the stargazer has electric organs in the orbitae which can generate and transmit an electric shock. The electric apparatus is composed of two organs, which form two vertical columns roughly oval in horizontal section, and placed behind and somewhat under each eye. It is composed of about 200 thin layers of electric tissue.

Ecology
Northern stargazers live primarily along the eastern seaboard of the United States. They bury themselves in the sand and wait for prey (usually smaller fish) to happen by. Their eyes are situated on top of the head and poke up through the sand, hence the name stargazer. The stargazer's scientific name is Astroscopus guttatus where Astroscopus means "one who aims at the stars" and guttatus translating into "speckled" – referring to the white spots on the fish's back.

Life cycle
Stargazers lay small, transparent eggs on the bottoms of bays. These eggs float to the surface after they are released. They hatch into larvae which grow up to 6 - 7 mm. They slowly grow a dark coloring and develop the electrical organs from eye muscles when they are 12 - 15 mm. After this they swim to the bottom and grow into adults.

Taxonomy
The northern stargazer was first described by Charles Conrad Abbott in 1860.

title: Astroscopus guttatus


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Talk:Sports Interactive

National Eastside Hockey
Someone added a statement that Sports Interactive develop "National Eastside Hockey" but I cant find any reference to such a game on their website. Is this correct???--Ukdan999 18:35, 22 November 2005 (UTC)

 * No. As far as I know, Sports Interactive develops only NHL Eastside Hockey Manager (link to the official game site), not other hockey game. I will remove that statement, as there is no evidence that National Eastside Hockey exists. Regards, Carioca 20:11, 22 November 2005 (UTC)

 * Some confusion may have come from the heritage of the game - it used to be developed independently as shareware (I think) but SI brought it in-house by employing the developer (Risto Remes). Also, the NHL title was not conferred immediately, but was gained after a licensing process. Either way, the name is wrong and should be changed. DanGravell

AFC Wimbledon
Someone has stated that this is shown at start up on Championship Manager 01-02. This is not correct as AFC Wimbledon were not founded until May 2002, this game was released in October 2001,however it may be present on a later seasons game.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Sports Interactive


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Reginald Stuart Poole

Reginald Stuart Poole (27 January 1832 – 8 February 1895), known as Stuart Poole, was an English archaeologist, numismatist and Orientalist. Poole was from a famous Orientalist family; his mother Sophia Lane Poole, his uncle Edward William Lane and his nephew Stanley Lane-Poole were all famous for their work in this field. His other uncle was Richard James Lane, a distinguished Victorian lithographer and engraver.

Life
Born in London, Poole was the son of the Rev. Edward Poole, a well-known bibliophile. His parents became estranged during his early childhood, and his mother, Sophia Lane Poole, took her sons to Egypt to live with her brother, the Orientalist Edward William Lane. During their seven-year residence in Cairo from 1842 to 1849, Lane Poole wrote The Englishwoman in Egypt, while her son was imbibing an early taste for Egyptian antiquities.

In 1852 he became an assistant in the British Museum and was assigned to the Department of Coins and Medals, of which in 1870 he became keeper. In that capacity he worked as a writer, teacher, and administrator. He was largely responsible for founding the Egypt Exploration Fund in 1882 and for starting the Society of English Medallists in 1884. He was Yates Professor of Archaeology at University College, London from 1889, and also a lecturer at the Royal Academy. In 1883 he received an honorary degree from Cambridge University. He received the Medal of the Royal Numismatic Society in 1892.

On 6 August 1861, he had married Eliza Christina Forlonge, daughter of William Forlonge, with whom he had four children, including Sir Reginald Ward Poole.

Poole retired in 1893 and died in 1895.

Works
Some of Poole's best work was done in his articles for the 9th edition of the Encyclopædia Britannica, on Egypt, Hieroglyphics and Numismatics; he also wrote for Smith's Dictionary of the Bible and published volumes dealing with his special subjects.

Poole was one of the strong defenders in England of the work of Champollion when he was criticized harshly by Sir George Lewis even as late as 1862. In reply to Lewis critique, Poole defended Champollion's method describing it as "the method of interpreting Hieroglyphics originated by Dr. Young and developed by Champollion".

Selected publications
[NC = Numismatic Chronicle]
 * 1860 Cairo, Sinai, Jerusalem, and The Pyramids of Egypt: A Series of Sixty Photographic Viewsby Francis Frith, Sophie Lane Poole and Reginald Stuart Poole.
 * 1860 The Genesis of the Earth and of Man: a critical examination of passages in the Hebrew and Greek Scriptures, chiefly with a view to the solution of the question, Whether the varieties of the Human Species be of more than one origin
 * 1861 On a coin of Mallus in Cilicia [read 26 Jan 1859], NC, New Series, vol. 1 (1861), part 2, pp. 87–90, text fig.
 * 1861 On a copper coin of the class struck after the death of Alexander the Great, by his generals, before they assumed regal titles [read 25 April 1861], NC, New Series, vol. 1 (1861), part 3, pp.; 137–0139, text fig.
 * 1861 On two Cretan coins in the British Museum [read 22 March 1860], NC, New Series, vol. 1 (1861), part 3, pp. 168–74, pl.
 * 1861 On a coin from the Cyrenaica, presented to the British Museum by the late F.H. Crowe, Esq., H.M. Consul at Cairo [read 24 Oct 1861], NC, New Series, vol. 1 (1861), part 4, pp. 201–3, text fig.
 * 1862 On a new coin of ancient Italy [read 20 Nov 1862], NC, New Series, vol. 2 (1862), part 8, pp. 300–1, text fig.
 * 1863 The late Colonel Leake's Collection of Greek Coins [offered to Cambridge University for £5000], NC, New Series, Vol. 3 (1863), part 12, pp. 266–67. [accepted, see 1864, 75]
 * 1864 The coins of the Ptolemies (to be continued), NC, New Series, vol. 4 (1864), part 13, pp. 7–16, pl., 2 tables [see 159, 231 below; 1865, 126, 321; 1866, 1; 1867, 161]
 * 1864 Coins of the Ptolemies, continued, NC, New Series, vol. 4 (1864), part 15, pp. 159–73, 2 pl.
 * 1864 Coins of the Ptolemies, continued, NC, New Series, vol. 4 91864), part 16, pp. 231–35, 2 pl.
 * 1864 On Greek coins as illustrating Greek art [delivered 27 May 1864], p. 236-47, pl.
 * 1865 Coins of the Ptolemies, continued, NC, New Series, vol. 5 (1865), part 18, pp. 126–60, pl, 2 charts.
 * 1865 Coins of the Ptolemies, continued, NC, New Series, vol. 5 (1865), part 20, pp. 321–36, 2 pl, 2 charts.
 * 1866 Coins of the Ptolemies, continued, NC, New Series, vol. 6 (1866), part 21, pp. 1–20, 3 text figs.
 * 1867 Coins of the Ptolemies, concluded, NC, New Series, vol. 7 (1867), part 27, pp. 161–202.
 * 1875–1890 Catalogue of Oriental coins in the British Museum, by Stanley Lane-Poole and Reginald Stuart Poole.
 * 1878 A descriptive catalogue of Swiss coins in the South Kensington Museum, by Reginald Stuart Poole, Chauncy Hare Townshend, and Victoria and Albert Museum.
 * 1885 Coins and medals : their place in history and art by Reginald Stuart Poole and Stanley Lane-Poole.
 * 1892 Catalogue of the coins of Alexandria and the Nomes / by Reginald Stuart Poole (in the series: Catalogue of the Greek coins in the British Museum, ed. by Reginald Stuart Poole).
 * 1854/85 On the coinage of the dynasties called the Benee-Tooloon and the Ikhsheedeeyeh, ruling in Egypt, NC 17 (1854–55), pp. 116–26, pl.
 * 1876 Catalogue of Greek Coins in the British Museum, ed. by R.S. Poole, with B.V. Head and P. Gardner, London, 1876. (announced in NC, New Series, vol. 16 (1876), part 63, p. 276.
 * 1877 Catalogue of Oriental Coins in the British Museum: vol. II, Coins of the Mohammedan Dynasties, Classes III-X, by Stanley Lane Poole, ed. by R.S. Poole, London, 1876 (announced in NC, New Series, vol. 17 (1877), part 65, p. 78.
 * 1883 Athenian coin-engravers in Italy (coins of Terina) [read 21 Feb 1884], NC, Third Series, vol. 3 (1883), part 12, pp. 269–77, 2 pl.
 * 1889 Catalogue of the Greek coins in the British Museum – Pontus, Paphlagonia, Bithynia and the Kingdom of the Bosporus, by Warwick Wroth, ed. by R.S. Poole, London, 1889 (announced by B.V. Head in NC, Third Series, vol. 38 (1890), pp. 173–74.

Obituaries

 * The Times, 9 February 1895; pg. 5; Issue 34496; col F
 * The Advertiser (Adelaide, Australia), 11 February 1895, p. 5.
 * Australian Town and Country Journal (NSW, Australia), 16 February 1895, p. 12.
 * The Mercury (Hobart, Tasmania), 13 February 1895, p. 3.

title: Reginald Stuart Poole


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Fort Michilimackinac

Fort Michilimackinac was an 18th-century French, and later British, fort and trading post at the Straits of Mackinac; it was built on the northern tip of the lower peninsula of the present-day state of Michigan in the United States. Built around 1715, and abandoned in 1783, it was located along the Straits, which connect Lake Huron and Lake Michigan of the Great Lakes of North America.

The present-day village of Mackinaw City developed around the site of the fort, which has been designated as a National Historic Landmark. It is preserved as an open-air historical museum, with several reconstructed wooden buildings and palisade, and is now part of Fort Michilimackinac State Park.

History
The primary purpose of the fort was as part of the French-Canadian trading post system, which stretched from the Atlantic Coast and the St. Lawrence River to the Great Lakes, and south to the Mississippi River through the Illinois Country. The fort served as a supply depot for traders in the western Great Lakes.

The French had first established a presence in the Straits of Mackinac in 1671 when Father Marquette established the Jesuit St. Ignace Mission at present-day St. Ignace in the Upper Peninsula of Michigan. In 1683, they augmented the mission with Fort de Buade. In 1701, Sieur de Cadillac moved the French garrison to Fort Detroit and closed the mission.

By 1713, however, the French decided to re-establish a presence along the Straits of Mackinac, and built the wooden Fort Michilimackinac on the northern tip of the lower peninsula. They sent Constant le Marchand de Lignery with a contingent of soldiers and workmen in 1715 to accomplish the job. Over the decades, they made several modifications and expansions to the palisade walls. Chevalier Jacques Testard de Montigny, who was a Lt. and a Knight of the Order of St. Louis, was appointed in 1730 and served for three years as commandant of the fort. He was previously commandant of Fort La Baye (Green Bay, Wisconsin). Many of his relatives settled in Michigan.

The French relinquished the fort, along with their territory in Canada, to the British in 1761 following their defeat in the French and Indian War, the North American front of the Seven Years' War. The British continued to operate the fort as a major trading post, but most residents were French and Métis (Ojibwe-French), who spoke predominantly French and worshipped at Sainte Anne Church in a small log structure. Other civilian residents included British fur traders, some of whom resided within the fort in the southeastern row house.

The Ojibwe in the region soon became dissatisfied with British policies, particularly their cancellation of the annual policy of distributing gifts to the Indians. On June 2, 1763, as part of the larger conflict known as Pontiac's War, a group of Ojibwe staged a game of baaga'adowe (a forerunner of modern lacrosse) outside the fort as a ruse to gain entrance. After entering the fort, they killed most of the British inhabitants. They held the fort for a year before the British regained control, promising to offer more and better gifts to the native inhabitants of the area.

The British eventually determined that the wooden fort on the mainland was too vulnerable. In 1781 they built a limestone fort on nearby Mackinac Island. Now known as Fort Mackinac, it was initially named Fort Michilimackinac. The British then moved related buildings to the island by dismantling them and moving them across the water in the summer and over ice in winter to the island during the next two years. Ste. Anne's Church was also moved. Patrick Sinclair, the lieutenant governor of Michilimackinac, ordered the remains of the original Fort Michilimackinac to be burned after the move.

Today
In 1960 the fort grounds were designated a National Historic Landmark. The grounds were then restored, largely through archaeologically informed reconstruction, as a tourist attraction. The fort and grounds operate, as of 2022, as part of Colonial Michilimackinac State Park in Mackinaw City, a major component of the Mackinac State Historic Parks. Interpreters, both paid and volunteer, help bring the history to life with music, live demonstrations and reenactments, including musket and cannon firing demonstrations. The site has numerous reconstructed historical wooden structures based on archeological excavations. This is considered one of the most extensively excavated early colonial French archaeological sites in the United States.

The State Park grounds, adjacent to the fort, also contain other resources. The grounds feature the foot of the Mackinac Bridge, the Old Mackinac Point Light (a 1892 lighthouse), a day-use park with a view of the Mackinac Bridge and Mackinac Island, and a visitor center with gift shop.

title: Fort Michilimackinac


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Talk:Afonso IV of Portugal

Untitled
So did he die on May 12 or May 28? --Naddy 18:46, 12 May 2004 (UTC)

Choice of portrait of Afonso IV
, which portrait is most suitable to the article is debatable, but the style of the portrait you've restored is much more primitive and less painterly than the one you reverted, which is admittedly less flattering. That, however, should not be the criterion by which we judge the artistry of the portrait. Contrary to your subjective opinion of which is the better image, his rendering of the armor alone is enough to demonstrate the superior skill of the painter of the earlier portrait. Carlstak (talk) 15:24, 19 July 2015 (UTC)
 * Oh I agree that whether the image is flattering to the subject or not is completely irrelevant and that the painting itself is of a lesser caliber than the original, but I think the other's actual image quality (pixels, colouration, etc.) is just horrendous, therefor I believe the better quality one (in terms of image quality and not subject quality) would be better suited - though obviously both are not perfectly ideal. Do you see what I mean? Cristiano Tomás (talk) 21:24, 19 July 2015 (UTC)
 * I see what you mean, although personally, I would accept the degraded image quality to have what I consider an image more expressive of Afonso's character. It's not a big issue, so I can live with your choice. Carlstak (talk) 23:04, 19 July 2015 (UTC)

title: Talk:Afonso IV of Portugal


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Bubba Ray Dudley

Mark LoMonaco (born July 14, 1971), known by the ring names Bubba Ray Dudley (also spelled Buh-Buh Ray Dudley) and Bully Ray, is an American professional wrestler. He is best known for being one-half of The Dudley Boyz tag team, where he competed alongside his partner D-Von Dudley, during his tenures in Extreme Championship Wrestling (ECW) and World Wrestling Entertainment (WWE), the latter of which they are also signed to under "Legends" contracts.

Debuting in 1991, LoMonaco came to prominence when he joined ECW as a member of The Dudley Brothers in 1995. He formed a tag team with D-Von Dudley called The Dudley Boyz, who became eight-time ECW World Tag Team Champions. In 1999, they moved to the WWF (now WWE) and became eight-time World Tag Team Champions, as well as one-time WWE Tag Team Champions and WCW World Tag Team Champions. They are considered one of the major teams that revived tag team wrestling during the Attitude Era. Dudley also held the WWE Hardcore Championship 8 times before both men left in 2005. They joined TNA later that year, becoming one-time NWA World Tag Team Champions and two-time TNA World Tag Team Champions. After the team disbanded in 2010, LoMonaco competed as a singles wrestler under the name Bully Ray and went on to hold the TNA World Heavyweight Championship twice. He left TNA in 2015, and The Dudley Boyz returned to WWE for two years before joining Ring of Honor (ROH) until 2020.

The Dudley Boyz are recognized as one of the most prolific tag teams in professional wrestling history, with 23 world tag team championships between ECW, WWE, Impact, and NJPW. They were inducted into the TNA Hall of Fame in 2014 and the WWE Hall of Fame in 2018. LoMonaco himself has won 36 titles between ECW, WWE, Impact, NJPW and ROH, factoring in the aforementioned tag team championships, eight WWE Hardcore Championships, two TNA World Championships, and one ROH World Six-Man Tag Team Championship (held with The Briscoe Brothers).

Early life
Mark LoMonaco was born in the Queens borough of New York City on July 14, 1971. He moved to Dix Hills, New York, in his youth and graduated from Half Hollow Hills East High School, where he was a member of the track and football team.

Early career (1991–1995)
LoMonaco trained under Johnny Rodz, and debuted in 1991. He used the ring name Mongo Vyle with the gimmick of a biker billed from Hell's Kitchen who was inspired by The Nasty Boys, The Road Warriors, and Maxx Payne.

Extreme Championship Wrestling (1995–1999)
LoMonaco received a tryout with Extreme Championship Wrestling in September 1995, appearing as Mongo, the bodyguard of Bill Alfonso. After impressing ECW owner Paul Heyman by taking a chokeslam from wrestler 911, LoMonaco was hired. He was recast as Buh Buh Ray Dudley, the stuttering, dancing, overweight, hillbilly member of the sprawling Dudley family. The unusual spelling of "Bubba" was a result of his stutter. In saying his name, Buh-Buh would stutter, causing him to say "my name is Buh-buh-buh-buh" until Big Dick Dudley struck him in the chest, at which point he would shout "my name is Bubba Ray Dudley". The fans embraced the comical Dudley boys and during Buh Buh's matches would chant "Buh Buh Buh Buh". The spelling eventually stuck.

On April 13, 1996, at Massacre on Queens Boulevard, D-Von Dudley debuted in ECW and began feuding with the other members of the Dudley Family (his kayfabe half-brothers), claiming that their comedic antics were not the way true Dudleys should act. D-Von eliminated Dances With Dudley, Dudley Dudley, and Chubby Dudley before joining forces with Buh Buh Ray, Big Dick, Sign Guy Dudley, and Joel Gertner. Known collectively as Dudley Boyz, Buh Buh Ray and D-Von dominated the ECW tag team division, winning the ECW World Tag Team Championship a record eight times and defeating the four major tag teams in ECW: The Public Enemy, The Eliminators, The Gangstas, and Sabu and Rob Van Dam. Buh Buh Ray, D-Von, and Gertner all achieved a degree of infamy for their vitriolic interviews, which antagonized audiences to a point of near riot.

The Dudley Boyz (1999–2002)
In 1999, Buh Buh Ray and D-Von left ECW and joined the World Wrestling Federation, where the "Buh Buh" stutter-spelling of "Bubba Ray Dudley" was rescinded. Throughout 2000 and 2001, the Dudley Boyz engaged in a three-way feud for the WWF Tag Team Championship with The Hardy Boyz and Edge & Christian. The feud incorporated three critically acclaimed Tables, Ladders, and Chairs matches, the first at WrestleMania 2000, second at SummerSlam (2000), and the third at WrestleMania X-Seven. While they were initially heels, one of the things Bubba Ray was known for was driving women through tables including WWE Divas Terri, Mae Young, Lita, Trish Stratus, Tori, Torrie Wilson, Stacy Keibler and Jazz. The Dudley Boyz were soon cheered by the crowd and were turned face in early 2000. On the December 28, 2000 episode of SmackDown!, Bubba Ray received a title shot against WWF Champion Kurt Angle, which he lost. In early 2001, The Dudleys were joined in the WWF by Spike Dudley.

In mid-2001, the Dudley Boyz turned heel once again by joining The Alliance, a large stable of former ECW and World Championship Wrestling (WCW) wrestlers led by Shane McMahon and Stephanie McMahon-Helmsley who attempted to take over the WWF (Spike, along with other ECW alumni, did not join The Alliance). The ECW-WCW invasion ended at the Survivor Series, when five WWF wrestlers defeated five Alliance wrestlers in a match to determine the ownership of the WWF. The Alliance disbanded and its members left the WWF, but the Dudley Boyz retained their jobs due to their possession of the WWF Tag Team Championship, which they had unified with the WCW Tag Team Championship that night.

Hardcore Champion and Dudley Boyz reunion (2002–2005)
Following WrestleMania X8, the WWF was renamed "World Wrestling Entertainment" (WWE) in May 2002 and the roster was divided into two brands: Raw and SmackDown!. The Dudley Boyz were separated when Bubba Ray was drafted to the Raw brand and D-Von was drafted to the SmackDown! brand. On Raw, Bubba Ray found moderate success as he competed for the WWE Hardcore Championship, formed a tag team with Spike, and even competed in a World Heavyweight Championship match against Triple H. D-Von's new gimmick on SmackDown! failed to get over, however, and he was reunited with Bubba Ray and Spike at Survivor Series on November 17, when he moved to the Raw brand to help Bubba Ray and Spike defeat Rico and 3-Minute Warning (Rosey and Jamal) in a Six Man elimination tables match.

The Dudley Boyz became a force to be reckoned with in the Raw tag team division over the next sixteen months, feuding with teams such as 3-Minute Warning, La Résistance, and various combinations of The Un-Americans. They held the World Tag Team Championship several more times before being traded (along with Booker T) to the SmackDown! brand on March 22, 2004, in exchange for Triple H. Shortly after arriving on SmackDown!, the Dudley Boyz turned heel once again, siding with Paul Heyman and feuding with Rob Van Dam and Rey Mysterio. On the May 27 episode of SmackDown!, the Dudley Boyz kidnapped Paul Bearer, the manager of Heyman's enemy The Undertaker. On June 27 at The Great American Bash, The Undertaker defeated the Dudley Boyz in a handicap match. In July, the Dudley Boyz reunited with Spike. For the remainder of the year, they assisted Spike in his matches for the WWE Cruiserweight Championship. They also won the WWE Tag Team Championship one more time. In early 2005, the Dudley Boyz were removed from WWE television and sent to Ohio Valley Wrestling while the WWE creative team attempted to devise an angle for them.

The Dudley Boyz returned to WWE television in June 2005 in order to promote ECW One Night Stand, an ECW reunion show. In the weeks preceding ECW One Night Stand they, along with several other ECW alumni, vied with General Manager Eric Bischoff and his "anti-hardcore crusaders". At ECW One Night Stand on June 12, the Dudley Boyz defeated Tommy Dreamer and The Sandman in the main event of the night after driving Dreamer through a flaming table. On July 5, 2005, WWE announced that it had opted not to continue contract renewal negotiations with the Dudley Boyz. In addition, fifteen other wrestlers were released by WWE, which was decreasing its spending as a result of a fall in projected revenue. In August 2005, all three former Dudleys were issued with legal notices instructing them not to use the (WWE trademarked) name "Dudley". This led to a degree of acrimony between the former Dudleys and their erstwhile employers, as they had used the names since 1996, several years before all ECW intellectual property was acquired by WWE as a result of bankruptcy proceedings. Soon after, LoMonaco and Hughes (D-Von) announced that they intended to pursue legal action against WWE.

Team 3D (2005–2010)
No longer able to use the Bubba Ray Dudley name, LoMonaco adopted (and trademarked) the ring name Brother Ray Deadly, while Hughes became Brother Devon Deadly. The team also trademarked the terms "Deadly Brothers" and "Deadly Death Drop". In August and September 2005, Ray and Devon made several appearances on the independent circuit, most notably Hardcore Homecoming, an unofficial ECW reunion show organized by ECW alumnus Shane Douglas, a frequent critic of WWE and Chairman Vince McMahon. On September 21, 2005, it was announced that they had signed multi-year contracts with Total Nonstop Action Wrestling.

Ray and Devon debuted in TNA on the October 1, 2005 episode of TNA Impact! – the first episode of Impact! to be aired on Spike TV and in a prime time slot. They were identified as "Brother Ray" and "Brother Devon" respectively (with the suffix "Deadly" apparently having been dropped) and as Team 3D collectively. Team 3D quickly established themselves as faces by confronting the heel NWA World Heavyweight Champion Jeff Jarrett and his allies, the NWA World Tag Team Champions, America's Most Wanted (AMW). Team 3D defeated AMW at pay-per-view events in November and December 2005, but failed to defeat them in a title match at Final Resolution on January 15, 2006, due to the interference of Team Canada. Ray and Devon continued to feud with America's Most Wanted and Team Canada over the following months. On the April 13, 2006 episode of Impact!, an attempted ambush by Team Canada was foiled by the debuting Spike Dudley, identified as "Brother Runt". They had many battles with The Latin American Xchange (LAX) until winning the NWA World Tag Team Championship from LAX at Lockdown in an "electrified" steel cage match.

Upon the disintegration of TNA's contract with the NWA, Ray and D-Von were stripped of the NWA Tag Team Championships and awarded the TNA World Tag Team Championship as compensation. They defended these belts against the team of Road Warrior Animal and Rick Steiner at Slammiversary. Following a match on Impact! to determine opponents, Ray would headline Victory Road alongside his partner in a match of champions encounter against TNA World Champion Kurt Angle and X Division Champion Samoa Joe. At Victory Road, Ray and D-Von lost their match, and the TNA tag titles, to Angle and Joe, though only Joe gained complete control of the titles as he scored the pinfall and could select a partner of his choosing.

The next feud had Team 3D along with the X-Division traitor Johnny Devine against The Motor City Machine Guns (Alex Shelley and Chris Sabin) and the entire TNA X Division. Team 3D and Devine lost at Against All Odds 2008 meaning that they have to be under two-hundred seventy-five pounds in order to compete in the X Division. However, the angle was dropped. Brother Ray and Devon aligned themselves with Kurt Angle during his feud with A.J. Styles. This led to them feuding with Christian Cage and Rhino as well. At Hard Justice, they lost to Cage and Rhino in a New Jersey Street Fight. They then feuded with Matt Morgan and Abyss, losing to them at No Surrender. At Bound for Glory IV, they participated in a Monster's Ball match for the Tag Team Championship, but came up short. Team 3D appeared to join The Main Event Mafia but turned face by double crossing the stable and jumping them along with The TNA Front Line leaders Samoa Joe, A.J. Styles, and Rhino. They also attempted to put former ally Kurt Angle through a table.

At Lockdown Team 3D won the TNA World Tag Team Championship and retained the IWGP Tag Team Championship against Beer Money, Inc. (Robert Roode and James Storm). At Slammiversary they lost the TNA tag titles to Beer Money, Inc. and on the July 30 episode of Impact! the IWGP tag titles to The British Invasion of Brutus Magnus and Doug Williams. On October 18, 2009, at Bound for Glory, Team 3D captured their 23rd tag team championship when they defeated Beer Money, British Invasion, and The Main Event Mafia's Scott Steiner and Booker T in a Full Metal Mayhem Tag Team match.

With both the IWGP and TNA Tag tiles on the line, they captured the IWGP titles which were formerly held by The British Invasion. During the following weeks Team 3D turned heel and aligned themselves with Rhino in a battle against the younger talent of the company. On the November 19 episode of Impact! Team 3D Academy of Professional Wrestling and Sports Entertainment graduate Jesse Neal joined Team 3D and Rhino and two weeks later Suicide joined Morgan, Hernandez and Dinero to level the playing field. At Final Resolution Morgan, Hernandez, Dinero and Suicide defeated Neal, Team 3D and Rhino in an eight-man elimination tag team match. On January 4, 2010, at Wrestle Kingdom IV in Tokyo Dome Team 3D lost the IWGP Tag Team Title to No Limit (Tetsuya Naito and Yujiro) in a three-way hardcore match, which also included Bad Intentions (Giant Bernard and Karl Anderson). When Hulk Hogan and Eric Bischoff took over TNA at the beginning of 2010, Team 3D's angle with Rhino and Neal was discontinued and they reverted to being faces, while starting a feud with The Nasty Boys (Brian Knobs and Jerry Sags), who were a part of the new wave of wrestlers brought in by Hogan and Bischoff. At Against All Odds The Nasty Boys defeated Team 3D in a tag team match, when Jimmy Hart made his return to the company and interfered in the match on the Nasty Boys' behalf.

Team 3D avenged their loss on the February 25 episode of Impact!, when Jesse Neal helped them defeat the Nasty Boys in a tables match. On the March 15 episode of Impact! Team 3D and Neal were scheduled to face the Nasty Boys and Hart in a six-man tag team match, but prior to the match the Nasty Boys attacked Neal backstage and put him through a table. Team 3D found Neal a replacement in the returning Brother Runt, but were still defeated in the match by the Nastys and Hart. However, after the match Neal made the save for Team 3D and helped them put Sags through a table. On May 16 at Sacrifice Ray turned heel by attacking Jesse Neal and costing him and Shannon Moore their match for the TNA World Tag Team Championship, after feeling he had been disrespected by Neal. The following month at Slammiversary VIII Neal defeated Ray in a singles match after a distraction from the debuting Tommy Dreamer. The following month at Victory Road Ray faced Jesse Neal and Devon in a three-way match. During the match the members of Team 3D attacked each other, before Neal accidentally speared Devon and was then pinned by Ray. On the following episode of Impact! Devon joined fellow ECW alumni Mick Foley, Tommy Dreamer, Raven, Stevie Richards, Rhino, Pat Kenney and Al Snow and TNA World Heavyweight Champion Rob Van Dam in attacking Brother Ray, Abyss and the rest of the TNA locker room. The following week, Ray declined Devon's offer to join the ECW alumni, before TNA president Dixie Carter agreed to give them their own reunion pay-per-view event, Hardcore Justice: The Last Stand, as a celebration of hardcore wrestling and a final farewell to the company. However, on the July 29 episode of Impact! Ray decided to join the ECW alumni and seemingly buried the hatchet with his brother, becoming a face again in the process. On August 8 at Hardcore Justice Team 3D, accompanied by Joel Gertner, defeated Axl Rotten and Balls Mahoney, billed as Kahoneys, in a South Philadelphia Street Fight. After the match Ray and Devon were assaulted by the Gangstas. On the following episode of Impact!, the ECW alumni, known collectively as Extreme, Version 2.0 (EV 2.0), were assaulted by A.J. Styles, Kazarian, Robert Roode, James Storm, Douglas Williams and Matt Morgan of Ric Flair's Fourtune stable, who thought they did not deserve to be in TNA. This would mark Team 3D's final appearance on their old TNA contracts, which expired shortly thereafter. The two then entered negotiations over new contracts. Team 3D returned to TNA television two month later on the October 7 live episode of Impact!, promising a major announcement at Bound for Glory. At the pay-per-view Team 3D announced their retirement, but asked for a one final match against the TNA World Tag Team Champions, the Motor City Machine Guns. Their new multi-year contracts with TNA were confirmed shortly thereafter. At Turning Point the Motor City Machine Guns defeated Team 3D to retain the TNA World Tag Team Championship.

Immortal (2010–2012)
During Team 3D's retirement ceremony on the following episode of Impact!, Ray turned heel by attacking Devon, reigniting the feud between the two. The following week Ray claimed that Devon had gotten weak, since Chris Sabin had managed to kick out of a 3D at Turning Point, calling him the Marty Jannetty and himself the Shawn Michaels of the team. On the December 16 episode of Impact! LoMonaco renamed himself Bully Ray as he defeated Amazing Red in a squash match. On January 9, 2011, at Genesis, Ray defeated Devon via disqualification, after Devon hit his former partner with his own chain. In the following weeks Devon's sons Terrence and Terrell began making appearances on Impact! and were regularly abused by Ray. At Against All Odds Ray faced his former tag team partner in a Street Fight. After Devon's sons interfered in the match, Ray low-blowed him and pinned one of his sons for the win. On March 13 at Victory Road, Devon and his sons interfered in a Falls Count Anywhere match between Ray and Tommy Dreamer, costing Ray the match. On the following episode of Impact!, Ray's feud with Devon was abruptly ended as he aligned himself with Immortal and moved on to feuding with Fortune. At the show's end, Ray powerbombed A.J. Styles off the entrance stage through a table, sidelining him with a storyline injury. On April 17 at Lockdown, Immortal, represented by Ray, Abyss, Matt Hardy and Ric Flair, was defeated by Fortune members James Storm, Kazarian and Robert Roode and Christopher Daniels, who replaced A.J. Styles, in a Lethal Lockdown match. During the match Styles made his return, attacking Ray as he was beating Daniels. On the April 28 episode of Impact!, Ray challenged Sting for the TNA World Heavyweight Championship, but was defeated following interference from Styles. Ray then recruited Tommy Dreamer as his partner and helped him defeat Styles at Sacrifice. On the May 26 episode of Impact Wrestling, Styles and Daniels defeated Ray and Dreamer in a No Disqualification Street Fight. On June 12 at Slammiversary IX, Ray defeated Styles in a Last Man Standing match to win the feud. After the match Ray, though still retaining his status as a heel, put over Styles' resilience.

On the July 28 episode of Impact Wrestling, Ray began having problems with fellow Immortal member Mr. Anderson, when he cost him his steel cage match against Kurt Angle. The tension between the two stablemates eventually led to a match on August 7 at Hardcore Justice, where Ray defeated Anderson with a low blow. From June to September, Ray was one of the twelve participants in the Bound for Glory Series to determine the number one contender to the TNA World Heavyweight Championship. When the group stage of the tournament concluded, Ray finished in the top four and thus advanced to the finals at No Surrender along with James Storm, Robert Roode and his Immortal stablemate, Gunner. On September 11 at No Surrender, Ray defeated Storm via disqualification, moving him up to 52 points. However, later in the event Roode managed to defeat Gunner via submission, which meant that he tied Ray's score to set up a tiebreaker match between the two. In the tiebreaker match, Ray was defeated by Roode via pinfall. On October 16 at Bound for Glory, Ray was defeated by former Immortal stablemate Mr. Anderson in a Falls Count Anywhere match. On November 13 at Turning Point, Ray and Scott Steiner were defeated by Anderson and Abyss in a tag team match. On January 8, 2012, at Genesis, Ray was defeated by Abyss in a Monster's Ball match. Ray later aligned himself with Bobby Roode, interfering in his matches especially ones relating to Roode's title. On February 12 at Against All Odds, Ray was unable to capture the TNA World Heavyweight Championship from Roode in a four-way match, which also included James Storm and Jeff Hardy. On the February 16 episode of Impact Wrestling, Ray was defeated by James Storm in a number one contender's match. On March 18 at Victory Road, Ray was defeated by Storm in a rematch to retain his spot as the number one contender. On the following episode of Impact Wrestling, Ray started a rivalry with X Division Champion Austin Aries, after interrupting an X Division Championship match between Aries, Anthony Nese, Kid Kash and Zema Ion. The rivalry between Ray and Aries continued on April 15 at Lockdown, where the two were on opposing teams in the annual Lethal Lockdown match. On the April 12, 2012 episode of Impact, Ray with Eric Bischoff, Ric Flair, Kazarian, Christopher Daniels and Gunner defeated A.J. Styles with Garett Bischoff, Austin Aries and Mr. Anderson). Aries' team, led by Garett Bischoff, ended up defeating Ray's team, led by Eric Bischoff. On May 13 at Sacrifice, Ray was defeated by Aries in a singles match. Afterwards, Ray returned to his rivalry with Abyss, when his supposed brother Joseph Park accused him of Abyss' disappearance. On June 10 at Slammiversary, Park defeated Ray in an Anything Goes match, following interference from Abyss. On the following episode of Impact Wrestling, Ray entered the 2012 Bound for Glory Series, taking part in the opening gauntlet match, from which he was eliminated by an interfering Abyss. Ray and Park had a rematch on the July 12 episode of Impact Wrestling in an Anything Goes match, which Ray won after hitting his opponent with a chain. After the match, Park attacked Ray and hit him with Abyss' signature Black Hole Slam.

Aces & Eights (2012–2013)
In August, Ray came together with the rest of the TNA locker room to battle the villainous Aces & Eights stable, who had been attacking random TNA wrestlers during the past months, in the process making a gradual face turn. In September, Ray, unlike his longtime tag team partner Devon, signed a two-year contract extension with TNA. On the September 6 episode of Impact Wrestling, Ray finished his Bound for Glory Series group stage with a win over Rob Van Dam, thus finishing third and advancing to the semifinals of the tournament. On September 9 at No Surrender, Ray defeated James Storm in his semifinal match, following interference from Bobby Roode, before losing to Jeff Hardy in the finals of the 2012 Bound for Glory Series, despite Hardy being attacked by the Aces & Eights before the match. On the following episode of Impact Wrestling, Ray was granted another opportunity to become the number one contender to the TNA World Heavyweight Championship, but was again defeated by Hardy. On the September 20 episode of Impact Wrestling, Ray defeated World Heavyweight Champion Austin Aries in a non-title match, after hitting him with a steel chain. On the October 4 episode of Impact Wrestling, Ray defeated both Aries and his number one contender Jeff Hardy in a three-way match, after which Hulk Hogan agreed to let him become Sting's partner for the match against the Aces & Eights stable at Bound for Glory. On October 14 at Bound for Glory, Ray and Sting were defeated by the Aces & Eights, who in the process earned full access to TNA. Afterwards, the returning Devon was revealed as one of the members of the Aces & Eights. On November 1, Ray and Devon faced off in the main event of Impact Wrestling, however, the match ended in a brawl between the Aces & Eights and the TNA locker room.

Over the next few weeks, tension would be teased between Ray and Hulk Hogan after Austin Aries revealed a secret relationship between Ray and the general manager's daughter Brooke. On December 9 at Final Resolution, Ray was defeated by Austin Aries in a singles match, following a distraction from both Brooke and Hulk Hogan and a low blow from Aries. On the January 3, 2013 episode of Impact Wrestling, Hogan suspended both Ray and Brooke after seeing them kissing in the parking garage two weeks before. The following week, Ray proposed to Brooke after saving her from the Aces & Eights, which she accepted. The ceremony took place the next week on Impact Wrestling, where Ray's groomsman Taz interrupted and revealed himself a member of Aces & Eights, leading to the group attacking Ray, Hulk Hogan, and the rest of the groomsmen. On the January 31 episode of Impact Wrestling, Hogan reinstated Ray and pitted him and Sting against Aces & Eights (Devon and DOC) in a Tables match the following week, where Ray won the match by putting Devon through a table.

On the February 21 episode of Impact Wrestling, Hogan named Ray the number one contender to the TNA World Heavyweight Championship. Ray received his shot in a steel cage match on March 10 at Lockdown, where he defeated Jeff Hardy, following interference from Aces & Eights, to win his first World Heavyweight Championship. Afterwards, Ray revealed himself as the president of Aces & Eights, turning heel again in the process, and reuniting with Devon. Ray made his first televised title defense on the April 11 episode of Impact Wrestling, defeating Jeff Hardy in a Full Metal Mayhem match to retain his title. On June 2 at Slammiversary XI, Ray successfully defended the World Heavyweight Championship against Sting in a No Holds Barred Match after an interference by Aces & Eights, with the added stipulation, that Sting would never get another World title opportunity again. Following Slammiversary, Ray and Aces & Eights would continue feuding with Sting and his New Main Event Mafia (Kurt Angle, Magnus, Samoa Joe, and Rampage Jackson). On July 18, during the Destination X episode of Impact Wrestling, Ray lost the TNA World Heavyweight Championship to Chris Sabin, ending his reign at 130 days. Ray received his rematch in a steel cage on August 15 at Impact Wrestling: Hardcore Justice, where he defeated Sabin, following interference from Mr. Anderson and Tito Ortiz, to regain the TNA World Heavyweight Championship. On the August 22 episode of Impact Wrestling, Ray revealed his marriage with Brooke Hogan fell apart as he broke up with her, and announced his relationship with Brooke Tessmacher. Later that night, Devon was forced to leave TNA after being pinned by A.J. Styles in a five-on-five tag team match against the Main Event Mafia. The following week, tension began to be teased between Bully Ray and Aces & Eights Vice-President Mr. Anderson, with Anderson disagreeing with Ray claiming he never needed Devon and bring Tito Ortiz into the stable without a group vote. During Ray's match with Sting on the September 5 episode of Impact Wrestling, Anderson cost Ray by refusing to help him. Afterwards, Anderson attacked Ray and announced that he would be his challenger at No Surrender. The following week, Ray successfully defended his title against Anderson in a Last Man Standing match. Afterwards, as Anderson was being stretchered away, Ray attacked him and piledrived him onto the stage. While beginning a feud with Bound for Glory Series winner A.J. Styles, Ray was also dealing with the implosion of Aces & Eights. On October 20 at Bound for Glory, Ray lost the TNA World Heavyweight Championship to Styles in a No Disqualification match, ending his reign at 66 days.

On the following episode of Impact Wrestling, Ray was attacked by the returning Mr. Anderson, who later in the evening, cost Ray his title rematch with Styles. Two weeks later, after teasing a break-up with Aces & Eights, the group ambushed Anderson in retaliation. On the November 21 episode of Impact Wrestling titled "Turning Point", Mr. Anderson and Bully Ray faced off in a match where it was Anderson's career vs. the Aces & Eights stable. Anderson won, forcing the end of the Aces & Eights.

On February 13, 2014, Bully Ray lost to Mr. Anderson in a Casket match, ending the feud.

Feud with the Carters and TNA Hall of Fame (2014–2015)
On March 9, Bully Ray was introduced by Dixie Carter to be the special guest referee for the Eight-Man Lethal Lockdown match at Lockdown. During the match, he attacked Team Dixie's captain, Bobby Roode, allowing Team MVP to be victorious, turning face once again. Ray then feuded with Roode, culminating in a match at Sacrifice; Carter interfered and cost Ray the match. Over the following weeks, Bully Ray invaded TNA Headquarters and Dixie's home, threatening to put her through a table. On May 22, Dixie and Ethan Carter III came out to confront MVP and his alliance for their actions. But, when Bully Ray interrupted the confrontation, he was ultimately laid out by MVP's alliance and put through a table by EC3. On June 5, MVP along with EC3 and Rockstar Spud challenged Ray, Austin Aries and Samoa Joe to a First Blood Match, which Bully and his team won after Bully hit EC3 with a steel chain. At Slammiversary, Bully was defeated by EC3 after Bully went through a table from the top rope. Later that night it was announced that Bully, along with Devon, would be inducted into the TNA Hall of Fame as Team 3D at Bound for Glory. On August 7, Bully Ray finally succeeded in putting Dixie Carter through a table. At Bound for Glory, Team 3D defeated Tommy Dreamer and Abyss. This event was the final contracted date for each member of Team 3D.

On the May 8, 2015 episode of Impact Wrestling, Ray made a one-night appearance as the special guest referee for the TNA World Heavyweight Championship match between Kurt Angle and Eric Young. On the July 15 episode of Impact Wrestling, Bully made his return to TNA, where he was revealed by Dixie Carter as the new general manager. However, it was announced on July 24 that Bully Ray had departed from the company due to being unhappy with the "current state of affairs within the company."

Independent circuit (2013–2015)
Starting in late 2013, Bully Ray and Devon have been something of a fixture within Tommy Dreamer's House of Hardcore promotion. On November 9, 2013, at HoH 3, Bully Ray attacked Dreamer and Terry Funk following the main event, claiming that Dreamer offended him by not inviting him to the show, before challenging Dreamer to a falls-count-anywhere street fight at the TNA One Night Only pay-per-view event Old School. Spike Dudley came out to make the save only to be beaten by Bully as well, whereas Devon appeared to confront Bully—only to turn around and aid him as the two hit Dreamer with a 3D, then fended off an attempted save by the Sandman.

Team 3D were also scheduled to officially team together against Dreamer and Abyss, then against Bad Influence (Christopher Daniels and Frankie Kazarian), on June 6 and 7, 2014 at House of Hardcore 4 and 5 respectively. However, Bully Ray would be pulled out of both events by TNA President Dixie Carter, turning the on-screen feud between Bully and Carter in TNA into a crossover storyline involving talent associated with both promotions. On July 13, 2014, Team 3D defeated Kevin Steen and Jason Axe to win the 2CW Tag Team Championship. At House of Hardcore VII, Team 3D defeated Killer Elite Squad.

On February 7, 2015, Bully Ray defeated Dom Vitalli to win the Destiny Wrestling Organization (DWO) Heavyweight Championship.

Return to WWE (2015–2018, 2024)
LoMonaco made a one night appearance with WWE at the 2015 Royal Rumble entering as a surprise entrant at number 3, reprising his Bubba Ray Dudley character. He eliminated The Miz and R-Truth before being eliminated by Bray Wyatt.

On the August 24, 2015 episode of Raw, he and D-Von Dudley made a surprise return to the WWE, reviving their Dudley Boyz tag team persona. They attacked the WWE Tag Team Champions, The New Day, and gave a 3D through a table to Xavier Woods. On the August 27 episode of SmackDown, they defeated The Ascension and after the match they put Viktor through a table. On the August 31 episode of Raw, they wrestled their first match on the show in 10 years against The New Day in a winning effort. Following the match, they would attempt to 3D Big E through a table, but Big E would be saved by Woods and Kofi Kingston. On the September 3 episode of SmackDown, The Dudley Boyz defeated The Prime Time Players. At both Night of Champions, and Live from Madison Square Garden, the Dudley Boyz challenged The New Day for the WWE Tag Team Championship, but would win both matches by disqualification, which allowed New Day to retain their titles. At Hell in a Cell, the Dudley Boyz would challenge for the titles again, where they would be defeated by New Day. At Survivor Series, the Dudley Boyz would team with Goldust, Neville and Titus O'Neil to defeat Stardust, The Ascension, The Miz and Bo Dallas in a 5-on-5 traditional Survivor Series elimination tag team match.

From the November 23 episode of Raw, the Dudley Boyz began a short rivalry with The Wyatt Family, and later aligned themselves with the returning Tommy Dreamer and Rhyno in the feud. At TLC: Tables, Ladders and Chairs, The Wyatt Family defeated The ECW Originals in an eight-man elimination tables match.

On the December 31 episode of SmackDown, the Dudley Boyz teamed with Kalisto in a winning effort against The New Day in six-man tag team match. They would face The New Day in a rematch on the January 4, 2016, episode of Raw in a losing effort. At Royal Rumble, the Dudley Boyz would compete in a Fatal 4-Way tag team match on the pre-show for a spot in the Royal Rumble match itself, with the match being won by Mark Henry and Jack Swagger.

On the February 8 episode of Raw, the Dudley Boyz attacked The Usos after a match involving them against The New Day and Mark Henry, thus turning heel in the process. Explaining that they did not come back to WWE to be a "nostalgia act" and reminding the fans that they are "the baddest tag team on the planet", the Dudley Boyz abandoned using tables. In the following weeks, the Dudley Boyz and The Usos began a feud, setting up a tag team match between the two teams at WrestleMania 32, where the Dudley Boyz lost to The Usos. The following night on Raw, the Dudley Boyz defeated The Usos in a tables match, breaking their vow to never again use tables and officially ending the feud. Afterwards, the duo were interrupted by the debuting Enzo Amore and Colin Cassady, who proceeded to insult them and ignite a new feud. In a tournament set up by Shane McMahon to determine the new number 1 contenders to the WWE Tag Team Championship, they would defeat The Lucha Dragons to advance to the semi-finals, where they clashed with Enzo Amore and Colin Cassady in a losing effort. On July 19 at the 2016 WWE draft, the Dudley Boyz were drafted to Raw. At SummerSlam, they would lose to Sami Zayn and Neville.

The Day after SummerSlam, the Dudley Boyz announced that they were announcing their departure from WWE on that night's Raw. During their farewell segment, they were confronted by The Shining Stars. After receiving insults from The Shining Stars, the Dudley Boyz attacked them, turning face once again in the process. As they set up Primo for a 3D through a table, Luke Gallows and Karl Anderson attacked both Bubba Ray and D-Von, sending D-Von through the table.

On January 22, 2018, the Dudley Boyz appeared on the Raw 25 Years episode when they interrupted a fight between the team of Heath Slater & Rhyno and Titus Worldwide (Apollo Crews and Titus O'Neil), and gave a Dudley Death Drop to Heath Slater through a table.

The Dudley Boyz were inducted into the WWE Hall of Fame Class of 2018.

At Wrestlemania XL on April 7, 2024, he made a guest appearance as the special guest referee for the Philadelphia Street Fight between The Final Testament's Karrion Kross, Akam and Rezar, and Bobby Lashley and the Street Profits. After numerous spots collaborating with the latter, Lashley and the Street Profits defeated The Final Testament.

Return to the independent circuit (2016–2017)
On August 28, 2016, it was announced that Team 3D would be competing at Insane Championship Wrestling's Fear and Loathing IX event on November 20, 2016.

On September 17, 2016, at House of Hardcore 18, LoMonaco reprised his Bully Ray character as he saved Tommy Dreamer from "Broken" Matt Hardy. He later threatened to Matt that he'll put him on the table like what he did to Dixie Carter.

At Delete WCPW, LoMonaco defeated his long time rival Matt Hardy in a No Disqualification Match with the help from Adam Pacitti. He was defeated by Drew Galloway in a No Disqualification Match for the WCPW Championship.

Storyline with the Briscoes (2017–2018)
At Manhattan Mayhem VI, LoMonaco made his ROH debut by saving The Briscoe Brothers (Jay Briscoe and Mark Briscoe) and Bobby Fish from Bullet Club by putting Adam Cole through a table. On March 10, at ROH 15th Anniversary Show Ray would team with The Briscoes as they defeated War Machine (Hanson and Raymond Rowe) and Davey Boy Smith Jr. On March 11, Ray and The Briscoes (Jay Briscoe and Mark Briscoe) defeated The Kingdom to win the ROH World Six-Man Tag Team Championship. On April 1, 2017, at Supercard of Honor XI Ray and The Briscoe Brothers successfully defended the ROH World Six-Man Tag Team Championship against Bullet Club (Hangman Page, Tama Tonga and Tanga Loa). They lost the title to Dalton Castle and The Boys on June 23. Bully Ray and the Briscoes challenged for the title again on September 22 at Death Before Dishonor XV, but they were defeated by Hangman Page and The Young Bucks, when Jay turned on Ray. Afterwards, Ray started teasing his retirement from professional wrestling. On October 20, Mark Briscoe also turned on Ray, joining Jay in attacking him and Tommy Dreamer when Velvet Sky came to Bully Ray's aid during his attack. Bully Ray's last match took place at ROH Final Battle, teaming with Tommy Dreamer against The Briscoes, losing in a New York Street Fight. Bully Ray confirmed he is retired on February 10, 2018, on his official Twitter page. However, he still makes appearances for Ring of Honor as an enforcer.

Final storylines and departure (2018–2020)
On April 7, 2018, at Supercard of Honor XII Bully Ray turned heel by powerbombing Cheeseburger. On the May 12, 2018 edition of ROH television, Bully Ray was confronted by Cheeseburger, they exchanged a quick staredown, and then Bully Ray chokeslammed Cheeseburger. The ROH COO Joe Koff came out and told Bully Ray that he was done being an enforcer. Bully Ray told Joe Koff that he unretired. On December 14, 2018, at Final Battle, Bully Ray was defeated by Flip Gordon in an "I Quit" match.

On April 1, 2020, Ray left Ring of Honor after his contract expired.

National Wrestling Alliance (2022)
On June 11, 2022, Ray was the special guest commentator for the NWA World Women's Championship at Alwayz Ready. On the June 14 episode of NWA Powerrr, Bully Ray, special master of ceremonies for the night, was confronted by Mike Knox and VSK of the Cardona Family. Knox initially acted in kind to Ray, due to their past as members of Aces & Eights in Impact Wrestling. However, after accusing Ray of "leaving him high and dry" in Impact, Knox and VSK would attack him before putting him through a table. At NWA 74th Anniversary Show he defeated Knox in a Tables match. On the following episode of Powerrr, Ray started a rivalry with Matt Cardona, after Cardona interrupted his interview with Kyle Davis. The rivalry between Ray and Cardona continued on February 11, 2023, at Nuff Said, Ray helped NWA Worlds Heavyweight Champion Tyrus, who agreed to give Ray a title shot anytime he wanted.

On April 7, 2023, at NWA 312, Ray defeated NWA World Television Champion Thom Latimer by disqualification, after Latimer would not stop attacking Ray in the corner. Prior to this, Ray went and grabbed Latimer's Television title, grabbed a microphone, and started talking about Kamille.

Return to Impact/TNA (2022–present)
On October 7, 2022, Ray returned to Impact Wrestling at Bound for Glory where he would win the Intergender Call Your Shot Gauntlet for a championship match of his choosing. On November 18 at Over Drive, Ray defeated Moose in a Tables match. Later in the show, Ray would confront Impact World Champion Josh Alexander and invoked his opportunity at Hard To Kill, which Alexander accepted. Ray attacked Alexander and threatened to piledrive his wife Jen, turning heel in the process. On January 13, 2023, at Hard To Kill, Ray was unsuccessful at winning the Impact World Championship from Alexander. After Hard To Kill, Ray would enter a feud with former friend Tommy Dreamer, with Dreamer challenging Ray to match for Sacrifice. On March 24 at the event, Ray defeated Dreamer in Busted Open match, after The Good Hands (Jason Hotch and John Skyler) made Dreamer bleed, while the referee was knocked out. On April 16 at Rebellion, Ray competed in a 10-wrestler Hardcore War teaming with Brian Myers, Kenny King, Masha Slamovich and Moose, facing Dreamer, Bhupinder Gujjar, Frankie Kazarian, Killer Kelly and Yuya Uemura, in a losing effort.

On May 26 at Under Siege, Ray came from behind and strangled President of Impact Wrestling Scott D'Amore with a cable. Ray and Impact World Champion Steve Maclin putting D'Amore through a flaming table. On June 9 at Against All Odds, Ray competed in the 8-4-1 match, when D'Amore struck Ray with a steel chair to incapacitate him. On the June 15 episode of Impact!, Ray cut a promo that revealed he filed a complaint with the Anthem Sports & Entertainment Board of Directors about D'Amore's involvement in his business. D'Amore confronted Ray soon after, saying that the board advised that he take a leave of absence from his duties as president. D'Amore also said he agreed with the board, though only because it allowed him to attack Ray right then and there. Maclin would soon come out and assist Ray once again before PCO appeared and took out both men. On July 15 at Slammiversary, Ray teamed with Deaner losing to D'Amore and his mystery partner which was Eric Young.

On September 14 at Impact 1000, Ray teamed up with Brother Devon reuniting Team 3D, winning over The Desi Hit Squad (Rohit Raju and Champagne Singh). After the match, Team 3D put Singh through a table turning Ray face in the process. On October 21, 2023, at Bound for Glory, Ray pushed Steve Maclin off the top rope through a barbed wire table. Later in the night, Ray was a participant in the 20-person Intergender Call Your Shot Gauntlet, eliminating KiLynn King before being eliminated by Jordynne Grace.

Other media
LoMonaco appeared on the second WWE edition of the game show The Weakest Link, losing in the sudden death final to Kane, and was the strongest link in the majority of the rounds. At the start of the game, he stated that he was playing for the American Cancer Society and his "mommy".

The Wrestling Figure Checklist records LoMonaco as having 26 different action figures produced from the 1990s to 2010s by Toymakers, Jakks Pacific, and Mattel.

In 2018, LoMonaco began co-hosting the professional wrestling-themed talk show Busted Open on Sirius XM Radio.

Alongside Hughes, LoMonaco has appeared in several wrestling video games, the most recent of which being WWE 2K24 as downloadable content.

Personal life
LoMonaco is of Italian descent, particularly having Sicilian roots.

LoMonaco dated Fawnfeather "Fawn" Carr, an esthetician, from 1999 to 2013. From May 2016 to November 2021, he was in a relationship with former TNA wrestler Jamie Szantyr, known professionally as Velvet Sky.

LoMonaco, along with longtime tag team partner Devon Hughes, operate a wrestling school called the Team 3D Academy of Professional Wrestling and Sports Entertainment, in Kissimmee, Florida.

Championships and accomplishments

 * All Japan Pro Wrestling
 * World's Strongest Tag Determination League (2005) – with Brother Devon
 * The Baltimore Sun
 * Tag Team of the Year (2007) – with Brother Devon
 * Cauliflower Alley Club
 * Other honoree (1997) – with Brother Devon
 * Destiny Wrestling Organization
 * DWO Heavyweight Championship (1 time)
 * Extreme Championship Wrestling
 * ECW World Tag Team Championship (8 times) – with D-Von Dudley
 * Hustle
 * Hustle Super Tag Team Championship (1 time) – with Brother Devon
 * New Japan Pro-Wrestling
 * IWGP Tag Team Championship (2 times) – with Brother Devon
 * Pennsylvania Premier Wrestling
 * PPW Heavyweight Championship (1 time)
 * Pro Wrestling Illustrated
 * Match of the Year (2000) with D-Von Dudley vs. Edge and Christian and The Hardy Boyz in a Triangle Ladder match at WrestleMania 2000
 * Match of the Year (2001) with D-Von Dudley vs. Edge and Christian and The Hardy Boyz in a Tables, Ladders, and Chairs match at WrestleMania X-Seven
 * Tag Team of the Year (2001, 2009) – with D-Von Dudley
 * Tag Team of the Decade (2000–2009) - with D-Von Dudley
 * Ranked No. 4 of the top 500 singles wrestlers in the PWI 500 in 2013
 * Ranked No. 354 of the top 500 singles wrestlers of the "PWI Years" in 2003
 * Ring of Honor
 * ROH World Six-Man Tag Team Championship (1 time) – with Jay Briscoe and Mark Briscoe
 * Squared Circle Wrestling
 * 2CW Tag Team Championship (1 time) – with Brother Devon
 * Total Nonstop Action Wrestling/Impact Wrestling
 * TNA World Heavyweight Championship (2 times)
 * TNA World Tag Team Championship (2 times, inaugural) – with Brother Devon
 * NWA World Tag Team Championship (1 time) – with Brother Devon
 * Tag Team Tournament (2013) – with Brother Devon
 * TNA World Cup of Wrestling (2014) – with Eric Young, Gunner, Eddie Edwards, and ODB
 * Call Your Shot Gauntlet (2022)
 * TNA Hall of Fame (2014) – as a member of Team 3D
 * World Wrestling Federation/Entertainment
 * WWF/E Hardcore Championship (8 times)
 * WWF/World Tag Team Championship (8 times) – with D-Von Dudley
 * WWE Tag Team Championship (1 time) – with D-Von Dudley
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Spherical trigonometry

Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook Spherical trigonometry for the use of colleges and Schools. Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.

Spherical polygons
A spherical polygon is a polygon on the surface of the sphere. Its sides are arcs of great circles—the spherical geometry equivalent of line segments in plane geometry.

Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also called digons or bi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.

One spherical polygon with interesting properties is the pentagramma mirificum, a 5-sided spherical star polygon with a right angle at every vertex.

From this point in the article, discussion will be restricted to spherical triangles, referred to simply as triangles.

Notation
\pi < A + B + C < 3\pi $$(Todhunter, Art.22,32). In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians. $$(Todhunter, Art.22,32).
 * Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters $A$, $B$, and $C$.
 * Sides are denoted by lower-case letters: $a$, $b$, and $c$. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length).
 * The angle $A$ (respectively, $B$ and $C$) may be regarded either as the angle between the two planes that intersect the sphere at the vertex $A$, or, equivalently, as the angle between the tangents of the great circle arcs where they meet at the vertex.
 * Angles are expressed in radians. The angles of proper spherical triangles are (by convention) less than $\pi$, so that $$
 * Sides are also expressed in radians. A side (regarded as a great circle arc) is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention, the sides of proper spherical triangles are less than π, so that $$0 < a + b + c < 2\pi
 * The sphere's radius is taken as unity. For specific practical problems on a sphere of radius $R$ the measured lengths of the sides must be divided by $R$ before using the identities given below. Likewise, after a calculation on the unit sphere the sides $a$, $b$, and $c$ must be multiplied by $R$.

Polar triangles
The polar triangle associated with a triangle $△A'B'C'$ is defined as follows. Consider the great circle that contains the side $BC$. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as $A$ is (conventionally) termed the pole of $A$ and it is denoted by $A'$. The points $B'$ and $C'$ are defined similarly.

The triangle $△ABC$ is the polar triangle corresponding to triangle $△A'B'C'$. A very important theorem (Todhunter, Art.27) proves that the angles and sides of the polar triangle are given by $$\begin{alignat}{3} A' &= \pi - a, &\qquad B' &= \pi - b, &\qquad C' &= \pi - c, \\ a' &= \pi - A, & b' &= \pi - B, & c' &= \pi - C. \end{alignat}$$ Therefore, if any identity is proved for $△ABC$ then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.

Cosine rules
The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule: $$\begin{align} \cos a &= \cos b \cos c + \sin b \sin c \cos A, \\[2pt] \cos b &= \cos c \cos a + \sin c \sin a \cos B, \\[2pt] \cos c &= \cos a \cos b + \sin a \sin b \cos C. \end{align}$$

These identities generalize the cosine rule of plane trigonometry, to which they are asymptotically equivalent in the limit of small interior angles. (On the unit sphere, if $$a, b, c \rightarrow 0$$ set $$ \sin a \approx a $$ and $$ (\cos a - \cos b)^2 \approx 0$$ etc.; see Spherical law of cosines.)

Sine rules
The spherical law of sines is given by the formula $$\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.$$ These identities approximate the sine rule of plane trigonometry when the sides are much smaller than the radius of the sphere.

Derivation of the cosine rule
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler vector methods. (These methods are also discussed at Spherical law of cosines.)

Consider three unit vectors $△ABC$ drawn from the origin to the vertices of the triangle (on the unit sphere). The arc $\overarc{BC}$ subtends an angle of magnitude $a$ at the centre and therefore $OA, OB, OC$. Introduce a Cartesian basis with $OA$ along the $z$-axis and $OB$ in the $xz$-plane making an angle $c$ with the $z$-axis. The vector $OC$ projects to $ON$ in the $xy$-plane and the angle between $ON$ and the $x$-axis is $A$. Therefore, the three vectors have components:

$$\begin{align} \vec{OA}: &\quad (0,\,0,\,1) \\ \vec{OB}: &\quad (\sin c,\,0,\,\cos c) \\ \vec{OC}: &\quad (\sin b\cos A,\,\sin b\sin A,\,\cos b). \end{align}$$

The scalar product $OB · OC$ in terms of the components is $$\vec{OB} \cdot \vec{OC} =\sin c \sin b \cos A +  \cos c \cos b.$$ Equating the two expressions for the scalar product gives $$\cos a =  \cos b \cos c  +  \sin b \sin c \cos A.$$ This equation can be re-arranged to give explicit expressions for the angle in terms of the sides: $$\cos A = \frac{\cos a-\cos b\cos c}{\sin b \sin c}.$$

The other cosine rules are obtained by cyclic permutations.

Derivation of the sine rule
This derivation is given in Todhunter, (Art.40). From the identity $$\sin^2 A=1-\cos^2 A$$ and the explicit expression for $OB · OC = cos a$ given immediately above $$ \begin{align} \sin^2 A &= 1 - \left(\frac{\cos a - \cos b \cos c}{\sin b \sin c}\right)^2 \\[5pt] &= \frac{(1-\cos^2 b)(1-\cos^2 c)-(\cos a - \cos b\cos c)^2}{\sin^2\!b \,\sin^2\!c} \\[5pt] \frac{\sin A}{\sin a} &= \frac{\sqrt{1-\cos^2\!a-\cos^2\!b-\cos^2\!c + 2\cos a\cos b\cos c}}{\sin a\sin b\sin c}. \end{align}$$ Since the right hand side is invariant under a cyclic permutation of $a$, $b$, and $c$ the spherical sine rule follows immediately.

Alternative derivations
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy and spherical astronomy give different proofs and the online resources of MathWorld provide yet more. There are even more exotic derivations, such as that of Banerjee who derives the formulae using the linear algebra of projection matrices and also quotes methods in differential geometry and the group theory of rotations.

The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule. The scalar triple product, $cos A$ evaluates to $OA · (OB × OC)$ in the basis shown. Similarly, in a basis oriented with the $z$-axis along $OB$, the triple product $sin b sin c sin A$, evaluates to $OB · (OC × OA)$. Therefore, the invariance of the triple product under cyclic permutations gives $sin c sin a sin B$ which is the first of the sine rules. See curved variations of the law of sines to see details of this derivation.

Supplemental cosine rules
Applying the cosine rules to the polar triangle gives (Todhunter, Art.47), i.e. replacing $A$ by $sin b sin A = sin a sin B$, $a$ by $\pi – a$ etc., $$\begin{align} \cos A &= -\cos B \, \cos C + \sin B \, \sin C \, \cos a, \\ \cos B &= -\cos C \, \cos A + \sin C \, \sin A \, \cos b, \\ \cos C &= -\cos A \, \cos B + \sin A \, \sin B \, \cos c. \end{align}$$

Cotangent four-part formulae
The six parts of a triangle may be written in cyclic order as ($aCbAcB$). The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example ($aCbA$) or $BaCb$). In such a set there are inner and outer parts: for example in the set ($BaCb$) the inner angle is $C$, the inner side is $a$, the outer angle is $B$, the outer side is $b$. The cotangent rule may be written as (Todhunter, Art.44) $$ \cos\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{side}\end{smallmatrix}\Bigr)  \cos\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{angle}\end{smallmatrix}\Bigr) =  \cot\!\Bigl(\begin{smallmatrix}\text{outer} \\ \text{side}\end{smallmatrix}\Bigr)  \sin\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{side}\end{smallmatrix}\Bigr) -  \cot\!\Bigl(\begin{smallmatrix}\text{outer} \\ \text{angle}\end{smallmatrix}\Bigr)  \sin\!\Bigl(\begin{smallmatrix}\text{inner} \\ \text{angle}\end{smallmatrix}\Bigr), $$ and the six possible equations are (with the relevant set shown at right): $$\begin{alignat}{5} \text{(CT1)}&& \qquad \cos b\,\cos C &= \cot a\,\sin b - \cot A \,\sin C \qquad&&(aCbA)\\[0ex] \text{(CT2)}&& \cos b\,\cos A &= \cot c\,\sin b - \cot C \,\sin A &&(CbAc)\\[0ex] \text{(CT3)}&& \cos c\,\cos A &= \cot b\,\sin c - \cot B \,\sin A &&(bAcB)\\[0ex] \text{(CT4)}&& \cos c\,\cos B &= \cot a\,\sin c - \cot A \,\sin B &&(AcBa)\\[0ex] \text{(CT5)}&& \cos a\,\cos B &= \cot c\,\sin a - \cot C \,\sin B &&(cBaC)\\[0ex] \text{(CT6)}&& \cos a\,\cos C &= \cot b\,\sin a - \cot B \,\sin C &&(BaCb) \end{alignat}$$ To prove the first formula start from the first cosine rule and on the right-hand side substitute for $\pi – A$ from the third cosine rule: $$\begin{align} \cos a &= \cos b \cos c + \sin b \sin c \cos A \\ &= \cos b\ (\cos a \cos b + \sin a \sin b \cos C) + \sin b \sin C \sin a \cot A \\ \cos a \sin^2 b &= \cos b \sin a \sin b \cos C + \sin b \sin C \sin a \cot A. \end{align}$$ The result follows on dividing by $cos c$. Similar techniques with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.

Half-angle and half-side formulae
With $$2s=(a+b+c)$$ and $$2S=(A+B+C),$$ $$ \begin{alignat}{5} \sin{\tfrac{1}{2}}A &= \sqrt{\frac{\sin(s-b)\sin(s-c)}{\sin b\sin c}} &\qquad\qquad \sin{\tfrac{1}{2}}a &= \sqrt{\frac{-\cos S\cos (S-A)}{\sin B\sin C}} \\[2ex] \cos{\tfrac{1}{2}}A &= \sqrt{\frac{\sin s\sin(s-a)}{\sin b\sin c}} & \cos{\tfrac{1}{2}}a &= \sqrt{\frac{\cos (S-B)\cos (S-C)}{\sin B\sin C}} \\[2ex] \tan{\tfrac{1}{2}}A &= \sqrt{\frac{\sin(s-b)\sin(s-c)}{\sin s\sin(s-a)}} & \tan{\tfrac{1}{2}}a &= \sqrt{\frac{-\cos S\cos (S-A)}{\cos (S-B)\cos(S-C)}} \end{alignat} $$

Another twelve identities follow by cyclic permutation.

The proof (Todhunter, Art.49) of the first formula starts from the identity $$2\sin^2\!\tfrac{A}{2} = 1 - \cos A,$$ using the cosine rule to express $A$ in terms of the sides and replacing the sum of two cosines by a product. (See sum-to-product identities.) The second formula starts from the identity $$2\cos^2\!\tfrac{A}{2} = 1 + \cos A,$$ the third is a quotient and the remainder follow by applying the results to the polar triangle.

Delambre analogies
The Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807–1809.

$$ \begin{align} \frac{\sin{\tfrac{1}{2}}(A+B)} {\cos{\tfrac{1}{2}}C} =\frac{\cos{\tfrac{1}{2}}(a-b)} {\cos{\tfrac{1}{2}}c} &\qquad\qquad & \frac{\sin{\tfrac{1}{2}}(A-B)} {\cos{\tfrac{1}{2}}C} =\frac{\sin{\tfrac{1}{2}}(a-b)} {\sin{\tfrac{1}{2}}c} \\[2ex] \frac{\cos{\tfrac{1}{2}}(A+B)} {\sin{\tfrac{1}{2}}C} =\frac{\cos{\tfrac{1}{2}}(a+b)} {\cos{\tfrac{1}{2}}c} &\qquad & \frac{\cos{\tfrac{1}{2}}(A-B)} {\sin{\tfrac{1}{2}}C} =\frac{\sin{\tfrac{1}{2}}(a+b)} {\sin{\tfrac{1}{2}}c} \end{align} $$ Another eight identities follow by cyclic permutation.

Proved by expanding the numerators and using the half angle formulae. (Todhunter, Art.54 and Delambre )

Napier's analogies
$$\begin{align} \tan\tfrac{1}{2}(A+B) = \frac{\cos\tfrac{1}{2}(a-b)}{\cos\tfrac{1}{2}(a+b)} \cot\tfrac{1}{2}C &\qquad& \tan\tfrac{1}{2}(a+b) = \frac{\cos\tfrac{1}{2}(A-B)}{\cos\tfrac{1}{2}(A+B)}\tan\tfrac{1}{2}c \\[2ex] \tan\tfrac{1}{2}(A-B) = \frac{\sin\tfrac{1}{2}(a-b)}{\sin\tfrac{1}{2}(a+b)} \cot\tfrac{1}{2}C &\qquad& \tan\tfrac{1}{2}(a-b) =\frac{\sin\tfrac{1}{2}(A-B)}{\sin\tfrac{1}{2}(A+B)} \tan\tfrac{1}{2}c \end{align}$$

Another eight identities follow by cyclic permutation.

These identities follow by division of the Delambre formulae. (Todhunter, Art.52)

Taking quotients of these yields the law of tangents, first stated by Persian mathematician Nasir al-Din al-Tusi (1201–1274),

$$ \frac{\tan\tfrac12(A-B)}{\tan\tfrac12(A+B)} = \frac{\tan\tfrac12(a-b)}{\tan\tfrac12(a+b)} $$

Napier's rules for right spherical triangles

When one of the angles, say $sin a sin b$, of a spherical triangle is equal to π/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set $a$, $b$, $c$, $A$, and $B$.

Napier provided an elegant mnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).

First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with $a$ gives $aCbAcB$. Next replace the parts that are not adjacent to $C$ (that is $A$, $c$, and $B$) by their complements and then delete the angle $C$ from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the middle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine. For an example, starting with the sector containing $a$ we have: $$\begin{align} \sin a &= \tan(\tfrac{\pi}{2} - B)\,\tan b \\[2pt] &= \cos(\tfrac{\pi}{2} - c)\, \cos(\tfrac{\pi}{2} - A) \\[2pt] &= \cot B\,\tan b \\[4pt] &= \sin c\,\sin A. \end{align}$$ The full set of rules for the right spherical triangle is (Todhunter, Art.62) $$\begin{alignat}{4} &\text{(R1)}&\qquad \cos c&=\cos a\,\cos b, &\qquad\qquad &\text{(R6)}&\qquad \tan b&=\cos A\,\tan c,\\ &\text{(R2)}& \sin a &= \sin A\,\sin c, &&\text{(R7)}& \tan a &= \cos B\,\tan c,\\ &\text{(R3)}& \sin b &= \sin B\,\sin c, &&\text{(R8)}& \cos A &= \sin B\,\cos a,\\ &\text{(R4)}& \tan a &= \tan A\,\sin b, &&\text{(R9)}& \cos B &= \sin A\,\cos b,\\ &\text{(R5)}& \tan b &= \tan B\,\sin a, &&\text{(R10)}& \cos c &= \cot A\,\cot B. \end{alignat}$$
 * sine of the middle part = the product of the tangents of the adjacent parts
 * sine of the middle part = the product of the cosines of the opposite parts

Napier's rules for quadrantal triangles

A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π/2 radians at the centre of the sphere: on the unit sphere the side has length π/2. In the case that the side $c$ has length π/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle $C$ with sides $a', b', c'$ such that $△A'B'C'$, $A' = \pi − a$ etc. The results are: $$\begin{alignat}{4} &\text{(Q1)}&\qquad \cos C &= -\cos A\,\cos B, &\qquad\qquad &\text{(Q6)}&\qquad \tan B &= -\cos a\,\tan C,\\ &\text{(Q2)}& \sin A &= \sin a\,\sin C, &&\text{(Q7)}& \tan A &= -\cos b\,\tan C,\\ &\text{(Q3)}& \sin B &= \sin b\,\sin C, &&\text{(Q8)}& \cos a &= \sin b\,\cos A,\\ &\text{(Q4)}& \tan A &= \tan a\,\sin B, &&\text{(Q9)}& \cos b &= \sin a\,\cos B,\\ &\text{(Q5)}& \tan B &= \tan b\,\sin A, &&\text{(Q10)}& \cos C &= -\cot a\,\cot b. \end{alignat}$$

Five-part rules
Substituting the second cosine rule into the first and simplifying gives: $$\begin{align} \cos a &= (\cos a \,\cos c + \sin a \,\sin c \,\cos B) \cos c + \sin b \,\sin c \,\cos A \\[4pt] \cos a \,\sin^2 c &= \sin a \,\cos c \,\sin c \,\cos B + \sin b \,\sin c \,\cos A \end{align}$$ Cancelling the factor of $a' = \pi − A$ gives $$\cos a \sin c = \sin a \,\cos c \,\cos B + \sin b \,\cos A$$

Similar substitutions in the other cosine and supplementary cosine formulae give a large variety of 5-part rules. They are rarely used.

Cagnoli's Equation
Multiplying the first cosine rule by $sin c$  gives $$\cos a \cos A = \cos b \,\cos c \,\cos A + \sin b \,\sin c - \sin b \,\sin c \,\sin^2 A.$$ Similarly multiplying the first supplementary cosine rule by $cos A$  yields $$\cos a \cos A = -\cos B \,\cos C \,\cos a + \sin B \,\sin C - \sin B \,\sin C \,\sin^2 a.$$ Subtracting the two and noting that it follows from the sine rules that $$ \sin b \,\sin c \,\sin^2 A = \sin B \,\sin C \,\sin^2 a $$  produces Cagnoli's equation $$\sin b \,\sin c + \cos b \,\cos c \,\cos A = \sin B \,\sin C - \cos B \,\cos C \,\cos a$$ which is a relation between the six parts of the spherical triangle.

Oblique triangles
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.

 * Case 1: three sides given (SSS). The cosine rule may be used to give the angles $A$, $B$, and $C$ but, to avoid ambiguities, the half angle formulae are preferred.
 * Case 2: two sides and an included angle given (SAS). The cosine rule gives $a$ and then we are back to Case 1.
 * Case 3: two sides and an opposite angle given (SSA). The sine rule gives $C$ and then we have Case 7. There are either one or two solutions.
 * Case 4: two angles and an included side given (ASA). The four-part cotangent formulae for sets ($cBaC$) and ($BaCb$) give $c$ and $b$, then $A$ follows from the sine rule.
 * Case 5: two angles and an opposite side given (AAS). The sine rule gives $b$ and then we have Case 7 (rotated). There are either one or two solutions.
 * Case 6: three angles given (AAA). The supplemental cosine rule may be used to give the sides $a$, $b$, and $c$ but, to avoid ambiguities, the half-side formulae are preferred.
 * Case 7: two angles and two opposite sides given (SSAA). Use Napier's analogies for $a$ and $A$; or, use Case 3 (SSA) or case 5 (AAS).

The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because half-angles will be less than π/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article Solution of triangles presents variants on these methods with a slightly different notation.

There is a full discussion of the solution of oblique triangles in Todhunter. See also the discussion in Ross. Nasir al-Din al-Tusi was the first to list the six distinct cases (2-7 in the diagram) of a right triangle in spherical trigonometry.

Solution by right-angled triangles
Another approach is to split the triangle into two right-angled triangles. For example, take the Case 3 example where $b$, $c$, and $B$ are given. Construct the great circle from $A$ that is normal to the side $BC$ at the point $D$. Use Napier's rules to solve the triangle $cos a$: use $c$ and $B$ to find the sides $AD$ and $BD$ and the angle $△ABD$. Then use Napier's rules to solve the triangle $∠BAD$: that is use $AD$ and $b$ to find the side $DC$ and the angles $C$ and $△ACD$. The angle $A$ and side $a$ follow by addition.

Numerical considerations
Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or π. Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle.

Area and spherical excess

Consider an $AB$-sided spherical polygon and let $C$ denote the $A$-th interior angle. The area of such a polygon is given by (Todhunter, Art.99) $${\text{Area of polygon} \atop \text{(on the unit sphere)}} \equiv E_N = \left(\sum_{n=1}^{N} A_{n}\right) - (N-2)\pi.$$

For the case of a spherical triangle with angles $B$, $N$, and $A_{n}$ this reduces to Girard's theorem $$ {\text{Area of triangle} \atop \text{(on the unit sphere)}} \equiv E = E_3 = A+B+C -\pi,$$ where $n$ is the amount by which the sum of the angles exceeds π radians, called the spherical excess of the triangle. This theorem is named after its author, Albert Girard. An earlier proof was derived, but not published, by the English mathematician Thomas Harriot. On a sphere of radius $A$ both of the above area expressions are multiplied by $∠DAC$. The definition of the excess is independent of the radius of the sphere.

The converse result may be written as

$$ A+B+C = \pi + \frac{4\pi \times \text{Area of triangle}}{\text{Area of the sphere}}.$$

Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5π (3π for proper angles). For example, an octant of a sphere is a spherical triangle with three right angles, so that the excess is π/2. In practical applications it is often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc. On the Earth the excess of an equilateral triangle with sides 21.3 km (and area 393 km2) is approximately 1 arc second.

There are many formulae for the excess. For example, Todhunter, (Art.101—103) gives ten examples including that of L'Huilier: $$\tan\tfrac{1}{4}E = \sqrt{\tan\tfrac{1}{2}s\, \tan\tfrac{1}{2}(s-a)\, \tan\tfrac{1}{2}(s-b)\,\tan\tfrac{1}{2}(s-c)}$$ where $$s = \tfrac{1}{2}(a+b+c)$$

Because some triangles are badly characterized by their edges (e.g., if $a = b \approx \frac12c$ ), it is often better to use the formula for the excess in terms of two edges and their included angle $$\tan\tfrac12 E = \frac {\tan\frac12a\tan\frac12b\sin C}{1 + \tan\frac12a\tan\frac12b\cos C}.$$

When triangle $R^{2}$ is a right triangle with right angle at $B$, then $△ABC$ and $cos C = 0$, so this reduces to $$\tan\tfrac12 E = \tan\tfrac12a\tan\tfrac12b.$$

Angle deficit is defined similarly for hyperbolic geometry.

From latitude and longitude
The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes $$\lambda_1$$ and $$\lambda_2,$$ and the great-circle arc between two points with longitude and latitude $$(\lambda_1, \varphi_1)$$ and $$(\lambda_2, \varphi_2)$$ is $$ \tan\tfrac12 E_4 = \frac {\sin\tfrac12(\varphi_2 + \varphi_1)}{\cos\tfrac12(\varphi_2 - \varphi_1)} \tan\tfrac12(\lambda_2 - \lambda_1). $$

This result is obtained from one of Napier's analogies. In the limit where $$\varphi_1, \varphi_2, \lambda_2 - \lambda_1$$ are all small, this reduces to the familiar trapezoidal area, $E_4 \approx \frac12 (\varphi_2 + \varphi_1) (\lambda_2 - \lambda_1)$.

The area of a polygon can be calculated from individual quadrangles of the above type, from (analogously) individual triangle bounded by a segment of the polygon and two meridians, by a line integral with Green's theorem, or via an equal-area projection as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using a great-circle distance formula.

title: Spherical trigonometry