The Twelve-Mile Circle is an approximately circular arc which forms most of the boundary between the Commonwealth of Pennsylvania and the State of Delaware in the United States. It is not actually a circle, but rather a combination of different circular arcs that have been feathered together.  
It is nominally a circle with a supposed—yet in fact only approximate and variable—12-mile (19 km) radius,  centered in the town of New Castle, Delaware. In 1750, the center of the circle was fixed at the cupola of the courthouse in New Castle. The Twelve-Mile Circle continues into the Delaware River. A small portion of the circle, known as the "Arc Line," also forms part of the Mason-Dixon line, separating Delaware and Maryland. Two other small portions, although not actually demarcated until 1934, form parts of the boundary between the states of Delaware and New Jersey.  Although the Twelve-Mile Circle is often claimed to be the only territorial boundary in the United States that is a true arc, the Mexican boundary with Texas includes several arcs,  and many cities in the South (such as Plains, Georgia ) also have circular boundaries.
Its existence dates back to 1681, when Charles II granted a deed to William Penn north of the already chartered Maryland. Charles created an exception, consisting of 12 miles around the town of New Castle and extending down the peninsula, as these lands were held by the Duke of York, who had won them in conquest from the Dutch colonists. Later, on August 24, 1682, the Duke granted these lands to Penn as well, giving him:
all that the Towne of Newcastle otherwise called Delaware and All that Tract of Land lying within the Compass or Circle of 12 Miles about the same scituate lying and being upon the River Delaware in America And all Islands in the same River Delaware and the said River and Soyle thereof lying North of the Southermost part of the said Circle of 12 Miles about the said Towne.
The boundaries of the circle were the focal point of the eighty-year Penn–Calvert Boundary Dispute.
The fact that the circle extends into the Delaware River makes for an unusual territorial possession. Most territorial boundaries that follow watercourses split the water course between the two territories by one of two methods, either by the median line of the watercourse (the Grotian Method, after Hugo Grotius) or, more often, the center of the main flow channel, or thalweg (the lowest point in the stream channel). However, due to the text of the deed, within the Twelve-Mile Circle, all the Delaware River to the low-tide mark on the east ( New Jersey) side is territory of the state of Delaware.
New Jersey has often disputed this claim, as the rest of its territorial boundaries along the Delaware River are determined by the midline and thalweg methods. In 1813, Delaware's legislature passed an act deeding Pea Patch Island to the United States government, and in 1820 New Jersey disputed that they owned the island since it was primarily on the New Jersey side of the river. Attorney General William Wirt sided with Delaware.  In the 1840s, the Pea Patch Island disagreement led to two conflicting circuit court decisions—the circuit of Delaware ruling that the entirety of the river (and its islands) belonged to Delaware, and the circuit in New Jersey ruling that the island had belonged to New Jersey, which had deeded it to Dr. Henry Gale, a citizen of New Jersey. At the recommendation of President James K. Polk, the parties agreed to arbitration, which resulted in a confirmation of Delaware's claim. 
The arbitration did not ultimately resolve the dispute, and it has been brought to the Supreme Court of the United States on several occasions (all titled New Jersey v. Delaware), most notably in 1877, 1934,  1935,  and 2007. The court's opinion for the 1934 case contained an extensive history of the claims to this territory, and the 1935 opinion memorably  enjoined New Jersey and Delaware from ever disputing their jurisdictions again.
Regardless of the Supreme Court's admonition to the two states against further litigation on this subject, they were back before the court as late as November 2005, when New Jersey's desire to approve plans by BP to build a liquefied natural gas terminal along the New Jersey shore of the Delaware River fell afoul of Delaware's Coastal Zone Act.  The court on January 23, 2006, appointed a special master to study the border dispute,  and on March 21, 2008, it upheld his report, which largely supported Delaware's authority.  Meanwhile, the Delaware House of Representatives considered a (symbolic) bill to call out the National Guard to safeguard the state's interests, while New Jersey legislators made comments about the battleship New Jersey, moored upriver from the site. 
There are persistent rumors that David Rittenhouse, the famous astronomer and instrument-maker of Pennsylvania, surveyed the circle around New Castle, but this is likely not correct. The circle was first laid out by surveyors named Taylor and Pierson in 1701.  The 1813 "Memoirs of the life of David Rittenhouse" by William Barton surmises that Rittenhouse was involved in such a survey in 1763, based on a letter in which Rittenhouse mentions being paid "for my attendance at New Castle," but there is no clear record of what, exactly, Rittenhouse was paid for. His biographer, Brooke Hindle, guessed that Rittenhouse assisted with latitude or longitude calculations. 
- Penn–Calvert Boundary Dispute
- The Wedge
- Transpeninsular Line
- Killcohook National Wildlife Refuge
- Mason-Dixon line
- Schenck, William S. (2007). "Delaware's State Boundaries". Delaware Geological Survey. Archived from the original on 2008-12-05. Retrieved 2008-11-17.
- [Perry L., 1934, The circular boundary of Delaware, Civil Engineering: v. 4, no. 11, p. 576-580.]
- Frank Jacobs, "Enigma, Georgia: Mystery of the South's Circular Towns", Strange Maps, 30 April 2014.
- United States Congress. "American State Papers: Documents, Legislative and Executive, of the Congress of the United States, Volume 33". Retrieved February 28, 2015.
- West Publishing Company. "The Federal Cases: Comprising Cases Argued and Determined in the Circuit and District Courts of the United States from the Earliest Times to the Beginning of the Federal Reporter, Arranged Alphabetically by the Titles of the Cases and Numbered Consecutively, Book 30". Retrieved February 28, 2015.
- State of New Jersey v. State of Delaware, 291 US 361
- State of New Jersey v. State of Delaware, 295 US 694
- The state of Delaware, [is] perpetually enjoined from disputing the sovereignty, jurisdiction, and dominion of the state of New Jersey over the territory adjudged to the state of New Jersey by this decree; and the state of New Jersey, [is] perpetually enjoined from disputing the sovereignty, jurisdiction, and dominion of the state of Delaware over the territory adjudged to the state of Delaware by this decree.
- "NJ to file lawsuit against Delaware over Delaware River jurisdiction". Philadelphia Business Journal. July 28, 2005.
- "DE Receives Favorable Recommendation in U.S. Supreme Court Dispute with NJ over Regulation of Wharves on the Delaware River". Connolly Bove Lodge & Hutz LLP. April 2007. Archived from the original on 2008-08-20.
- "New Jersey v. Delaware". Cornell University Law School. March 2008.
- Search archives of DelawareOnline.com for "BP gas terminal dealt setback", January 24, 2006
- Thomas D. Cope, "Some Local Scholars Who Counselled the Proprietors of Pennsylvania and their Commissioners during the Boundary Surveys of the 1760s," Proceedings of the American Philosophical Society, vol. 99, no. 4 (August 1955), pp. 268–276, p. 275.
- Brooke Hindle, David Rittenhouse, Princeton University Press, 1964, pp. 20–21.