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First class constraint

In physics, a first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated.

First and second class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.

The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.

Poisson brackets
Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional).

Suppose we have some constraints
 * $$ f_i(x)=0, $$

for n smooth functions
 * $$\{ f_i \}_{i= 1}^n$$

These will only be defined chartwise in general. Suppose that everywhere on the constrained set, the n derivatives of the n functions are all linearly independent and also that the Poisson brackets
 * $$\{f_i,f_j\}$$

and
 * $$\{f_i,H\}$$

all vanish on the constrained subspace.

This means we can write
 * $$\{f_i,f_j\}=\sum_k c_{ij}^k f_k$$

for some smooth functions $$c_{ij}^k$$   −−there is a theorem showing this; and
 * $$\{f_i,H\}=\sum_j v_i^j f_j$$

for some smooth functions $$v_i^j$$.

This can be done globally, using a partition of unity. Then, we say we have an irreducible first-class constraint (irreducible here is in a different sense from that used in representation theory).

Geometric theory
For a more elegant way, suppose given a vector bundle over $$\mathcal M$$, with $$n$$-dimensional fiber $$V$$. Equip this vector bundle with a connection. Suppose too we have a smooth section $f$ of this bundle.

Then the covariant derivative of $f$ with respect to the connection is a smooth linear map $$\nabla f$$ from the tangent bundle $$T\mathcal M$$ to $$V$$, which preserves the base point. Assume this linear map is right invertible (i.e. there exists a linear map $$g$$ such that $$(\Delta f)g$$ is the identity map) for all the fibers at the zeros of $f$. Then, according to the implicit function theorem, the subspace of zeros of $f$ is a submanifold.

The ordinary Poisson bracket is only defined over $$C^{\infty}(M)$$, the space of smooth functions over M. However, using the connection, we can extend it to the space of smooth sections of $f$ if we work with the algebra bundle with the graded algebra of V-tensors as fibers.

Assume also that under this Poisson bracket, $$\{f,f\}=0$$ (note that it's not true that $$\{g,g\}=0$$ in general for this "extended Poisson bracket" anymore) and $$\{f,H\}=0$$ on the submanifold of zeros of $f$ (If these brackets also happen to be zero everywhere, then we say the constraints close off shell). It turns out the right invertibility condition and the commutativity of flows conditions are independent of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.

Intuitive meaning
What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other on the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.

Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical observable, should only be defined on that subspace. Equivalently, we can look at the equivalence class of smooth functions over the symplectic manifold, which agree on the constrained subspace (the quotient algebra by the ideal generated by the  $f$  's, in other words).

The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.

Look at the orbits of the constrained subspace under the action of the symplectic flows generated by the $f$ 's. This gives a local foliation of the subspace because it satisfies integrability conditions (Frobenius theorem). It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions A1 and B1, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A1,f}={B1,f}=0 over the constrained subspace)and another two A2 and B2, which are also constant over orbits such that A1 and B1 agrees with A2 and B2 respectively over the restrained subspace, then their Poisson brackets {A1, B1} and {A2, B2} are also constant over orbits and agree over the constrained subspace.

In general, one cannot rule out "ergodic" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have self-intersecting orbits.

For most "practical" applications of first-class constraints, we do not see such complications: the quotient space of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a differentiable manifold, which can be turned into a symplectic manifold by projecting the symplectic form of M onto it (this can be shown to be well defined). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.

In general, the quotient space is a bit difficult to work with when doing concrete calculations (not to mention nonlocal when working with diffeomorphism constraints), so what is usually done instead is something similar. Note that the restricted submanifold is a bundle (but not a fiber bundle in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a section of the bundle instead. This is called gauge fixing.

The major problem is this bundle might not have a global section in general. This is where the "problem" of global anomalies comes in, for example. A global anomaly is different from the Gribov ambiguity, which is when a gauge fixing doesn't work to fix a gauge uniquely, in a global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum gauge theory discovered by Witten in 1980.

What have been described are irreducible first-class constraints. Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the cotetrad formulation of general relativity, at the subspace of configurations where the cotetrad field and the connection form happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.

One way to get around this is this: For reducible constraints, we relax the condition on the right invertibility of Δf into this one: Any smooth function that vanishes at the zeros of f is the fiberwise contraction of f with (a non-unique) smooth section of a $$\bar{V}$$-vector bundle where $$\bar{V}$$ is the dual vector space to the constraint vector space V. This is called the regularity condition.

Constrained Hamiltonian dynamics from a Lagrangian gauge theory
First of all, we will assume the action is the integral of a local Lagrangian that only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are on shell and off shell configurations. The constraints that hold off shell are called primary constraints while those that only hold on shell are called secondary constraints.

Examples
Consider the dynamics of a single point particle of mass $m$ with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold  $S$ with metric g. Assume also that the parameter $τ$ describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance). Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form $ω$.

If we coordinatize T * S by its position $x$ in the base manifold $S$ and its position within the cotangent space p, then we have a constraint
 * f = m2 −g(x)−1(p,p) = 0.

The Hamiltonian $H$ is, surprisingly enough, $H$ = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian $H$ '= $f$ instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See Hamiltonian constraint for more details.

Consider now the case of a Yang–Mills theory for a real simple Lie algebra $L$ (with a negative definite Killing form  $η$) minimally coupled to a real scalar field  $σ$, which transforms as an orthogonal representation  $ρ$ with the underlying vector space  $V$ under  $L$ in ( $d$ − 1) + 1 Minkowski spacetime. For $l$ in  $L$, we write

as

for simplicity. Let A be the $L$-valued connection form of the theory. Note that the A here differs from the A used by physicists by a factor of $i$ and  $g$. This agrees with the mathematician's convention.

The action $S$ is given by
 * $$S[\mathbf{A},\sigma]=\int d^dx \frac{1}{4g^2}\eta((\mathbf{g}^{-1}\otimes \mathbf{g}^{-1})(\mathbf{F},\mathbf{F}))+\frac{1}{2}\alpha(\mathbf{g}^{-1}(D\sigma,D\sigma))$$

where g is the Minkowski metric, F is the curvature form
 * $$d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}$$

(no $i$s or  $g$s!) where the second term is a formal shorthand for pretending the Lie bracket is a commutator,  $D$ is the covariant derivative
 * Dσ = dσ − A[σ]

and $α$ is the orthogonal form for  $ρ$.

What is the Hamiltonian version of this model? Well, first, we have to split A noncovariantly into a time component $φ$ and a spatial part  $A$. Then, the resulting symplectic space has the conjugate variables $σ$, $ρ(l)[σ]$ (taking values in the underlying vector space of $$\bar{\rho}$$, the dual rep of  $ρ$),  $A$,    $π$A,  φ and πφ. For each spatial point, we have the constraints, πφ=0 and the Gaussian constraint
 * $$\vec{D}\cdot\vec{\pi}_A-\rho'(\pi_\sigma,\sigma)=0$$

where since $ρ$ is an intertwiner
 * $$\rho:L\otimes V\rightarrow V$$,

$ρ$ ' is the dualized intertwiner
 * $$\rho':\bar{V}\otimes V\rightarrow L$$

( $L$ is self-dual via $η$). The Hamiltonian,
 * $$H_f=\int d^{d-1}x \frac{1}{2}\alpha^{-1}(\pi_\sigma,\pi_\sigma)+\frac{1}{2}\alpha(\vec{D}\sigma\cdot\vec{D}\sigma)-\frac{g^2}{2}\eta(\vec{\pi}_A,\vec{\pi}_A)-\frac{1}{2g^2}\eta(\mathbf{B}\cdot \mathbf{B})-\eta(\pi_\phi,f)-<\pi_\sigma,\phi[\sigma]>-\eta(\phi,\vec{D}\cdot\vec{\pi}_A).$$

The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by $f$. In fact, since the last three terms vanish for the constrained states, we may drop them.

Second class constraints
In a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second class constraint.

See Dirac brackets for diverse illustrations.

An example: a particle confined to a sphere
Before going on to the general theory, consider a specific example step by step to motivate the general analysis.

Start with the action describing a Newtonian particle of mass $m$ constrained to a spherical surface of radius $R$ within a uniform gravitational field $g$. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.

In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.

The action is given by
 * $$S=\int dt L=\int dt \left[\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mgz+\frac{\lambda}{2}(x^2+y^2+z^2-R^2)\right]$$

where the last term is the Lagrange multiplier term enforcing the constraint.

Of course, as indicated, we could have just used different, non-redundant, spherical coordinates and written it as
 * $$S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]$$

instead, without extra constraints; but we are considering the former coordinatization to illustrate constraints.

The conjugate momenta are given by
 * $$p_x=m\dot{x}$$, $$p_y=m\dot{y}$$, $$p_z=m\dot{z}$$, $$p_\lambda=0$$.

Note that we can't determine $l[σ]$  from the momenta.

The Hamiltonian is given by
 * $$H= \vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)$$.

We cannot eliminate $• λ$  at this stage yet. We are here treating $• λ$  as a shorthand for a function of the symplectic space which we have yet to determine and not as an independent variable. For notational consistency, define $π_{σ}$  from now on. The above Hamiltonian with the $• λ$ term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, $u_{1} = • λ$.

We have the primary constraint

We require, on the grounds of consistency, that the Poisson bracket of all the constraints with the Hamiltonian vanish at the constrained subspace. In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion.

From this consistency condition, we immediately get the secondary constraint

$$\begin{align} 0&=\{H,p_\lambda\}_\text{PB}\\ &=\sum_{i}\frac{\partial H}{\partial q_i}\frac{\partial p_\lambda}{\partial p_i}-\frac{\partial H}{\partial p_i}\frac{\partial p_\lambda}{\partial q_i}\\ &=\frac{\partial H}{\partial \lambda}\\ &=\frac{1}{2}(r^2-R^2)\\ &\Downarrow\\ 0&=r^2-R^2 \end{align}$$

This constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient $u$2, enlarging the Hamiltonian to

H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) ~. $$

Similarly, from this secondary constraint, we find the tertiary constraint

$$\begin{align} 0&=\{H,r^2-R^2\}_{PB}\\ &=\{H,x^2\}_{PB}+\{H,y^2\}_{PB}+\{H,z^2\}_{PB}\\ &=\frac{\partial H}{\partial p_x}2x+\frac{\partial H}{\partial p_y}2y+\frac{\partial H}{\partial p_z}2z\\ &=\frac{2}{m}(p_xx+p_yy+p_zz)\\ &\Downarrow\\ 0&=\vec p\cdot\vec r \end{align}$$

Again, one should add this constraint into the Hamiltonian, since, on-shell, no one can tell the difference. Therefore, so far, the Hamiltonian looks like

H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) + u_3 \vec{p}\cdot\vec{r}~, $$ where $u$1, $u$2, and  $u$3 are still completely undetermined.

Note that, frequently, all constraints that are found from consistency conditions are referred to as secondary constraints and secondary, tertiary, quaternary, etc., constraints are not distinguished.

We keep turning the crank, demanding this new constraint have vanishing Poisson bracket

0=\{\vec{p}\cdot\vec{r},\, H\}_{PB} = \frac{p^2}{m} - mgz+ \lambda r^2 -2 u_2 r^2. $$

We might despair and think that there is no end to this, but because one of the new Lagrange multipliers has shown up, this is not a new constraint, but a condition that fixes the Lagrange multiplier:

u_2 = \frac{\lambda}{2} + \frac{1}{r^2}\left(\frac{p^2}{2m}-\frac{1}{2}mgz \right). $$

Plugging this into our Hamiltonian gives us (after a little algebra)

$$ H = \frac{p^2}{2m}(2-\frac{R^2}{r^2}) + \frac{1}{2}mgz(1+\frac{R^2}{r^2})+u_1p_\lambda+u_3\vec p \cdot\vec r $$

Now that there are new terms in the Hamiltonian, one should go back and check the consistency conditions for the primary and secondary constraints. The secondary constraint's consistency condition gives

\frac{2}{m}\vec{r}\cdot\vec{p} + 2 u_3 r^2 = 0. $$ Again, this is not a new constraint; it only determines that

u_3 = -\frac{\vec{r}\cdot\vec{p}}{m r^2}~. $$ At this point there are no more constraints or consistency conditions to check!

Putting it all together,
 * $$H=\left(2-\frac{R^2}{r^2}\right)\frac{p^2}{2m} + \frac{1}{2}\left(1+\frac{R^2}{r^2}\right)mgz - \frac{(\vec{r}\cdot\vec{p})^2}{mr^2} + u_1 p_\lambda$$.

When finding the equations of motion, one should use the above Hamiltonian, and as long as one is careful to never use constraints before taking derivatives in the Poisson bracket then one gets the correct equations of motion. That is, the equations of motion are given by

\dot{\vec{r}} = \{\vec{r}, \, H\}_{PB}, \quad \dot{\vec{p}} = \{ \vec{p},\, H\}_{PB}, \quad \dot{\lambda} = \{ \lambda,\, H\}_{PB}, \quad \dot{p}_\lambda = \{ p_\lambda, H\}_{PB}. $$

Before analyzing the Hamiltonian, consider the three constraints,

\varphi_1 = p_\lambda, \quad \varphi_2 = r^2-R^2, \quad \varphi_3 = \vec{p}\cdot\vec{r}. $$ Note the nontrivial Poisson bracket structure of the constraints. In particular,

\{\varphi_2, \varphi_3\} = 2 r^2 \neq 0. $$ The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but even on-shell it is nonzero. Therefore, $p_{λ}$ and $• λ = u_{1}$  are second class constraints while  $p_{λ}=0$  is a first class constraint. Note that these constraints satisfy the regularity condition.

Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, Dirac noticed that we can turn the underlying differential manifold of the symplectic space into a Poisson manifold using his eponymous modified bracket, called the Dirac bracket, such that this Dirac bracket of any (smooth) function with any of the second class constraints always vanishes.

Effectively, these brackets (illustrated for this spherical surface in the Dirac bracket article)  project the system back onto the constraints surface. If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets, not the canonical Poisson brackets to commutation relations.

Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that, on-shell when the constraints are satisfied, the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that $λ$  dropped out of the extended Hamiltonian. Since $φ_{2}$   is a first class primary constraint, it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose $λ$, which has ceased to have any effect on the particle's dynamics. Therefore, that $λ$  dropped out of the Hamiltonian, that   $u$1 is undetermined, and that  $φ_{3}$ = pλ is first class, are all closely interrelated.

Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take $φ_{1}$ as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous $λ$ dynamical quantity. However, the example is more edifying in its current form.

Example: Proca action
Another example we will use is the Proca action. The fields are $$A^\mu = (\vec{A},\phi)$$ and the action is
 * $$S = \int d^dx dt \left[ \frac{1}{2}E^2 - \frac{1}{4}B_{ij}B_{ij} - \frac{m^2}{2}A^2 + \frac{m^2}{2}\phi^2\right]$$

where
 * $$\vec{E} \equiv -\nabla\phi - \dot{\vec{A}}$$

and
 * $$B_{ij} \equiv \frac{\partial A_j}{\partial x_i} - \frac{\partial A_i}{\partial x_j}$$.

$$(\vec{A},-\vec{E})$$ and $$(\phi,\pi)$$ are canonical variables. The second class constraints are
 * $$\pi \approx 0$$

and
 * $$\nabla\cdot\vec{E} + m^2 \phi \approx 0$$.

The Hamiltonian is given by
 * $$H = \int d^dx \left[ \frac{1}{2}E^2 + \frac{1}{4}B_{ij}B_{ij} - \pi\nabla\cdot\vec{A} + \vec{E}\cdot\nabla\phi + \frac{m^2}{2}A^2 - \frac{m^2}{2}\phi^2\right]$$.

title: First class constraint


text:
David L. Payne

David Lewis Payne (December 30, 1836 – November 28, 1884) was an American soldier and pioneer. Payne is considered by some to be the "Father of Oklahoma" for his work in opening the state to settlement.

He organized, trained, and led the "Boomer Army" on its forays into the Unassigned Lands. Payne's excursions into the Unassigned Lands, trips led by others, and lobbying of Congress by railroad interests eventually convinced the government to open the Unassigned Lands for settlement in 1889, about four years after Payne's death. The formation of Indian Territory and Oklahoma Territory followed, and Oklahoma became a state in 1907.

Early life and education
Payne was born in 1836 in Grant County, Indiana, on a farm near Fairmount. He grew up working on his father's farm. During winters he attended the local rural school.

In the spring of 1858, Payne and his brother left home intending to join in the Utah War. Their interest evidently waned by the time they crossed the Missouri River, as they stopped in Doniphan County, Kansas. There, in Burr Oak Township, Payne acquired some land and built a sawmill. It soon failed and Payne fell to hunting to support himself. Eventually private parties and then the Federal government hired him to scout for their various expeditions. These activities led to his exploration of what would later become Oklahoma.

Career
At the opening of the Civil War, Payne enlisted in the 4th Kansas Volunteer Infantry. In April 1862, his regiment and the 3rd Kansas Infantry and 5th Kansas Infantry were consolidated to form the 10th Kansas Infantry. He served from August 1861 to August 1864 as a private in Company F. During his service the unit saw action in Kansas, Missouri, Arkansas, and the Cherokee Nation, including the Battle of Prairie Grove in 1862.

At the end of his three-year service Payne returned to Doniphan County and was elected to the Kansas House of Representatives, serving in the 1864, 1865, and 1872 sessions.

In March 1865, Payne enlisted for one year in the 15th Kansas Cavalry as a private assigned to Company H. The unit had been activated in response to a January state legislative resolution which called for the organization of a regiment of veteran volunteer cavalry to protect western Kansas from Indian activity.

In July 1867, Kansas Governor Samuel Johnson Crawford issued a proclamation calling for volunteers to protect Kansans from Indian attacks in the west. The 18th Kansas Cavalry was brought up for four months as a result. Payne enlisted and was mustered in as the captain of Company D. This battalion replaced the Seventh Cavalry which had been transferred to the Platte River for the summer. In October 1868, Payne mustered in as a lieutenant in Company H of the 19th Kansas Cavalry, which served in a winter campaign against Indians on the western Great Plains. During this campaign, Payne served as a scout for General Philip Sheridan.

In 1870 Payne moved to Sedgwick County, near Wichita, and the following year he was again elected to the Kansas House of Representatives. This led to appointments as Postmaster at Fort Leavenworth in 1867 and as Sergeant-at-arms for two terms of the Kansas Senate. In 1875 and 1879, Payne served as assistant to the Doorkeeper of the United States House of Representatives.

During the intervals between political engagements and military service, Payne supported himself by hunting, scouting, and guiding wagon trains.

Boomer
In 1866, shortly after the Civil War, the Federal government forced many tribes in the Indian Territory into making concessions. The government accused the various tribes of abrogating the standing treaties by joining the Confederacy. As a result, some two million acres (8,000 km2) of land in the center of Indian Territory were ceded to the United States and thought by many to be public domain land.

Elias Boudinot, a Cherokee citizen working as a lobbyist in Washington, D.C., published an article about the public land issue in the February 17, 1879, edition of the Chicago Times. Dr. Morrison Munford of the Kansas City Times began referring to this tract as the "Unassigned Lands" or "Oklahoma" and to the people agitating for its settlement as Boomers. Munford is the first person to use the terms "boom" and "boomer" to describe the movement of white settlers into these lands. To prevent settlement of the land, President Rutherford B. Hayes issued a proclamation in April 1879 forbidding unlawful entry into Indian Territory.

Inspired by Boudinot, Payne began his efforts to enter and settle the public domain lands as allowed by existing law. He returned from his job in Washington and returned to Wichita in 1879. On his first attempt to enter Indian Territory, in April 1880, Payne and his party laid out a town they named "Ewing" on the present-day site of Oklahoma City. The Fourth Cavalry arrested them, took them to Fort Reno, then escorted them back to Kansas. Payne was furious, as public law (specifically the Posse Comitatus Act) prohibited the military from interfering in civil matters. Payne and his party were freed, which effectively denied them access to federal and state courts.

Anxious to prove his case in court, Payne and a larger group returned to Ewing in July 1879. The Army again arrested the party, escorted them back to Kansas, and freed them. This time, however, Payne was charged under the Indian Intercourse Act and brought to trial in Fort Smith, Arkansas, in March 1881. Judge Isaac C. Parker ruled against Payne and fined him the maximum amount of US$1,000, but since Payne had no money and no property, the fine could not be collected. The ruling settled nothing about the public lands, however, and Payne continued his activities unabated. He organized and led several more expeditions into the territory, even forming a newspaper called the Oklahoma War Chief, the first in the Cherokee Outlet, which he published in several sites along the Kansas-Indian Territory border.

During one of his last ventures, Payne insisted on a public trial, so he and his group were hauled several hundred miles to Fort Smith. The trip began at Fort Reno and proceeded to Fort Sill, Henrietta, Texas, Texarkana, Arkansas, Little Rock, Arkansas, and then finally to Fort Smith. During another trip in July 1884, the army seized his Oklahoma War Chief press, burned his buildings, and took Payne and his group through the Cherokee Nation after their arrest. The party was paraded through the streets of Tahlequah, past the Indian people who hated him for his attempts to seize part of their lands.

Public sentiment grew so great over his mistreatment at the hands of the military that the government finally granted his trial. Payne was turned over to the United States District Court for the District of Kansas at Topeka. In the fall term, Judge Cassius G. Foster quashed the indictments and ruled that settling on the Unassigned Lands was not a criminal offense. Boomers celebrated, but the government refused to accept the decision.

Between expeditions, Payne often spoke to crowds of admirers and recruited more Boomers. The day after such an address in Wellington, Kansas, on November 27, 1884, Payne collapsed and died of heart failure. His funeral filled the Methodist Episcopal Church in Wellington and thousands filed past his grave.

Payne's family moved his remains to Oklahoma in 1995. On April 22, 1996, a monument was dedicated at his final resting place in Stillwater, Oklahoma. Payne County, of which Stillwater is the county seat, is named in his honor.

William Couch succeeded Payne as leader of the Boomer movement.

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

title: David L. Payne


text:
Seigneur (disambiguation)

Seigneur (English: Lord), was the name formerly given in France to someone who had been granted a seigneurie (fief) by the crown, with all its associated rights over person and property.

Seigneur may also refer to:
 * The seigneurial system of New France
 * The hereditary feudal ruler of the island of Sark (see list of seigneurs of Sark)
 * Eddy Seigneur (born 1969), French road racing cyclist
 * Rivière du Seigneur, a tributary of St. Lawrence River in Les Éboulements, Charlevoix Regional County Municipality, Capitale-Nationale, Quebec, Canada

title: Seigneur (disambiguation)


text: Category:Indonesian people

title: Category:Indonesian people


text:
Interstate 195 (Florida)

Interstate 195 (I-195) is a 4.42 mi auxiliary Interstate Highway connecting I-95, its parent route, in the west with Miami Beach in the east. It crosses Biscayne Bay by traveling over the Julia Tuttle Causeway. The causeway is named after Miami founder Julia Tuttle.

It is part of the longer State Road 112 (SR 112), which continues to the west as the Airport Expressway and to the east as Arthur Godfrey Road (41st Street).

As part of a pilot program, the Florida Department of Transportation painted the shoulders as bike lanes east of US Highway 1 (US 1). Pedestrians are still prohibited.

Route description
I-195 begins at the eastern end of the I-95 and SR 112 interchange, also known as the 36th Street Interchange, heading east with interchanges with Miami Avenue and US 1 before heading onto the Julia Tuttle Causeway, where the Interstate crosses Biscayne Bay. At the eastern end of the causeway in Miami Beach, it has an interchange with SR 907 before terminating at the intersection of SR 907A and Arthur Godfrey Road, about west of SR A1A.

History
On December 23, 1961, three signed roads along the route of SR 112 were opened: the 36th Street Tollway, now the Airport Expressway, I-195, and the I-195 Spur along with a stretch of I-95 in Miami. The I-195 Spur was the surface portion of the west–east state road along Arthur Godfrey Road in Miami Beach, connecting I-195 and SR A1A east of the causeway. The I-195 Spur signs disappeared from the road shortly after the designation was decommissioned by the newly formed U.S. Department of Transportation in the late 1960s.

In popular culture
Famously, in early 1975, the rhythm of their car on this road was the inspiration for the Bee Gees's song "Jive Talkin'".

title: Interstate 195 (Florida)


text:
Talk:Spheroid

surface area of spheroids
Can somebody please check the validity of the following formulae for the surface area of spheroids??

Prolate spheroid ( a > b ):

A = { 2πab2/ (a2 – b2 )1/ 2 }ln{(a + (a2 – b2 )1/ 2 ) / ( a - (a2 – b2 )1/ 2 )}

Oblate spheroid (a < b) :

A = { 4πab2/ (b2 – a2 )1/ 2 }arctan {( (b2 – a2 )1/ 2 ) / a}

Shameek

Is there a reference for the surface area formulas? I would like to verify them.

Nathalie —Preceding unsigned comment added by 145.94.17.181 (talk) 14:38, 10 February 2010 (UTC)

I agree, the current formulas look wrong. I only checked the oblate spheroid case. If you plug in semi-major/minor axes of Earth (a=6378km,c=6357km) to this formula, you get approximately 255E6 km^2 which is about half of the published surface area of 510E6 km^2 (which is relatively close to the spherical approximation). If you start with the more authentic looking formulas on https://en.wikipedia.org/wiki/Ellipsoid, and do some algebra you find that a sign has been flipped from positive to negative.

I think that you should use the following for the oblate spheroid case:
 * $$\begin{align}

S_\text{oblate} &= 2\pi a^2\left(1 + \frac{c^2}{ea^2} \cdot \operatorname{arctanh}(e)\right) &&\text{where }\quad e^2 = 1 - \frac{c^2}{a^2}\quad (c < a), \\ \end{align}$$ Note that if you change the inner plus sign (+) to minus (-), you get half the value.

Frank 108.211.109.173 (talk) 07:49, 2 September 2020 (UTC)

 * I can’t say where your calculations might have gone wrong, but I don’t find any problem with the results using formulas in the text. Your formulation is equivalent. Strebe (talk) 17:11, 2 September 2020 (UTC)

In oblate version when e is zero the surface should be a sphere $$S = 4\pi a^2$$

In prolate version when a is zero the surface should be a sphere $$S = 4\pi c^2$$

In both cases there seems to be a factor of 2 when in a sphere it should be 4. Astrojed (talk) 02:50, 2 November 2020 (UTC)

 * The limit as e approaches 0 in the formulas given in the article for both oblate and prolate cases is 4πr², not 2πr², and therefore gives the expected result for the sphere.

Volume is indeed given by a single formula - check def of a and b
Another editor replacd the formula for the volume by this paragraph:
 * When the spheroid in question is oblate, the volume is $$\frac{4}{3}\pi a^2b$$, where a represents the major axis of the ellipse which, when rotated about its minor axis, b, produces the oblate spheroid. When the spheroid in question is prolate, the spheroid is produced by the rotation of an ellipse about it's major axis, hence the volume formula becomes $$\frac{4}{3}\pi b^2a$$ , where b represents the minor axis of the ellipse which, when rotated about its major axis, produces the prolate spheroid. Hence, the volume of the oblate spheroid which results from the rotation of an ellipse about it's minor axis is always greater than the volume of the prolate spheroid which results from that same ellipse rotated about it's major axis; this is the case whenever a represents the major axis and does not equal b, which represents the minor axis.

This paragraph assumes that the parameters a and b are the major and minor semi-diameters of the ellipse which is the vertical section. However, in the formula given earlier in the article, a and b are the equatorial radius and the polar radius. With this nomenclature, indeed the formula is always (4π/3)a2b for any kind of spheroid. Incidentally this nomenclature is more sensible because it also gives *one* equation (implicit or parametric) for the surface, *one* equation for the area, and so on. All the best, --Jorge Stolfi (talk) 01:07, 28 December 2009 (UTC)

Tumor volume example
I moved the following example to grading (tumors) since it was far too specialized:
 * In experimental biology, tumor growth is approximated to take the shape of a spheroid. Often, cancer studies involve the implantation of tumors subcutaneously in mice.  Such studies require a simple mechanism by which to evaluate tumor burden.  One such method is for two blinded researchers to measure tumor dimensions length and width with calipers.  The depth is not measured.  Tumor volume in cubic millimeters can be approximated with the following formula:$$ Volume = 0.52(Width^2)Length$$

Moreover, in this layman's opinion that method is extremely crude (as it does not measure depth and assumes that the shape is a spheroid). However I retained the approximate formulas, in terms of radii and in termd of diametes (as in the example above). --Jorge Stolfi (talk) 02:25, 28 December 2009 (UTC)

Revised formulae
The above discussion bears witness to much confusion. If the discussion related to an oblate spheroid only it would make sense to use a,b on the spheroid as conforming to the 'standard' notation for the ellipse. For the prolate case however major and minor are reversed. I have changed notation to agree with current texts which consider both types of spheroid: in particular I agree with the Mathworld discussion. Typos permitting this article now agrees with mathworld. I have removed 'angular eccentricty' since the concept is not used at all in the current literature of mathematics, geodesy, map-projections (according to academic colleagues). The expressions are now written in conventional form. Peter Mercator (talk) 22:55, 7 January 2012 (UTC)

Question Here...
Yeah, I don't know the freaking elite math grammar, so is tanh-1 e "tahn(-1)*e" or is it some other concept entirely? Because I'm getting improbably small surface areas by assuming the former. — Preceding unsigned comment added by 68.190.144.120 (talk) 23:35, 28 April 2014 (UTC)

 * Use $$\textstyle \tanh^{-1}e = \operatorname{artanh}\, e = \tfrac12\ln\left(\frac{1+e}{1-e}\right)$$, from the article on inverse hyperbolic functions. cffk (talk) 10:01, 29 April 2014 (UTC)

Merger proposal
I propose Oblate spheroid and Prolate spheroid be merged into this page, and turned into redirects. Much of the content is redundant except for the real-world examples of Oblate/Prolate spheroids.

67.252.103.23 (talk) 01:48, 8 June 2014 (UTC)

 * I think this is a good idea. cffk (talk) 03:29, 8 June 2014 (UTC)

 * I agree. Fgnievinski (talk) 00:05, 13 August 2014 (UTC)

 * Great idea. + m t  22:40, 1 September 2014 (UTC)

 * I agree. Bharat Singhvi 12:48, 5 September 2014 (UTC)

 * Agreed. Jaywilson (talk) 21:15, 17 November 2014 (UTC)

 * Yes BUT. Why stop there? The article Ellipsoid also covers the same material. I suggest that the three spheroid articles be merged into Ellipsoid using the figures therein and dropping some pretty pictures and possibly some of the uses/applications. We also need to include a discussion of nomenclature in the article, ie usage of spheroid/ ellipsoid of revolution/ ellipsoid as a function of time and subject area. Who will volunteer? — Peter Mercator (talk) 14:18, 18 November 2014 (UTC)

 * Disagree; in fact, portions of ellipsoid detailing ellipsoid of revolution would better be moved into spheroid. Fgnievinski (talk) 18:45, 18 November 2014 (UTC)

 * Peter, the key phrase is "Who will volunteer?" I think many editors realize that these pages need work but few are prepared to do the necessary grunt work.  So if you're willing to put in the work, I'm sure that most will be happy to defer to your choices.  A few thoughts: (1) Maybe include ellipsoidal coordinates, see Geodesics on an ellipsoid.  (2) A standard convention for triaxial ellipsoids is a ≥ b ≥ c; so oblate and prolate ellipsoids are given by a = b > c and a > b = c, respectively.  On the other hand if you're only considering ellipsoids of revolution, you would naturally choose a = b > c and a = b < c. cffk (talk) 15:07, 6 January 2015 (UTC)

✅ —David Eppstein (talk) 20:34, 21 April 2015 (UTC)

Sphere vs perfect sphere
I came here from a request at WikiProject math. Which is the better phrase, sphere or perfect sphere'? It depends on the context. In mathematics, a sphere is an ideal geometric object; it is already perfect and the perfect adjective is redundant. In colloquial parlace, outside of mathematical exposition, a sphere may a real world object that is fairly round and may be solid or hollow. In this case perfect sphere or most perfect sphere means a round object whose surface is remarkably close to that of a mathematical sphere. The usage of the perfect adjective is fairly common; searching for "perfect sphere" on Wikipedia nets 93 hits, including a number of our science articles and even in Sphere itself.

This article is firmly in the mathematical domain, so I don't see any need for perfect. But I think it is reasonable to include the colloquial term the first time sphere is used, to connect with lay readers. Perhaps something like

 * If the generating ellipse is a circle, the result is a sphere (commonly called a perfect sphere).

--Mark viking (talk) 22:54, 9 June 2016 (UTC)

 * I was thinking about adding the following paragraph to the article:
 * In common English usage, "sphere" may refer to objects that are only approximately shaped like mathematical spheres. For this reason, spheroids may be called "spherical" even though they are not in the mathematical sense.  To distinguish mathematical spheres, they may be referred to as perfect spheres.
 * But then I realized that it really fit better at sphere. This article shouldn't need to address the linguistics of "sphere".  I think it reads fine with or without "perfect" in front of "sphere".  Ozob (talk) 01:32, 10 June 2016 (UTC)

 * I can see the points of both arguments. However, as Mark viking implicitly points out, when he uses the phrase, ". . . remarkably close to that of a mathematical sphere," this is a mathematics article.  Hence, I agree with David Eppstein:  the phrase "perfect sphere" is redundant in this article.  It may be reasonable to refer to a sphere in non-mathematics articles as, "a sphere (commonly called a perfect sphere)." — Anita5192 (talk) 03:48, 10 June 2016 (UTC)

 * Additionally, per WP:OSE, just because there are many instances of the phrase "perfect sphere" in Wikipedia does not make it correct usage. I believe that every time this phrase occurs, it should be clarified parenthetically as Mark does above. — Anita5192 (talk) 03:55, 10 June 2016 (UTC)

 * We need not even resort to the mathematical argument. It's a matter of simple logic: If we say “very nearly spherical”, the phrase has no meaning unless the sphere referred to is perfect. Otherwise, the meaning would be, “very nearly a sphere which is not quite perfect”, which is simply a solecism. Using “perfect sphere” here is wrong, rather than a question of taste or even brevity. Strebe (talk) 08:19, 10 June 2016 (UTC)

To Anita and all above, you're reverted again, and here's why. I already discussed this at length and fully in the user's talk page, and also in that special "mathematics" page...now I'll transfer and paste all that I wrote onto here below:

hi. I don't like genuine redundancy either, but frankly you're just plain wrong with your comparisons of "wet water" and other things to the phrase "perfectly spherical". Astronomers and degreed people themselves have used that phrase. People who would never say the phrase "wet water". Not all spheres are necessarily perfect, is the point. What's the problem here?? And as I said, other WP articles have used that phrasing, as well as outside Reliable Sources. From another Wikipedia article that I had nothing to do with, these exact words:

"The Earth is not perfectly spherical but an oblate spheroid, so the length of a minute of latitude increases by 1% from the equator to the poles. Using the WGS84 ellipsoid, the commonly accepted Earth model for many purposes today, one minute of latitude at the WGS84 equator is 6,046 feet and at the poles is 6,107.5 feet. The average is about 6,076 feet (about 1,852 metres or 1.15 statute miles)."

And from an article on physics.stackexchange.com, these words:

"By this measure, the Sun is a near-perfect sphere with an oblateness estimated at about 9 millionths, which means that its polar diameter differs from its equatorial diameter by only 10 kilometres (6.2 mi)."

Are these scientists being "redundantly redundant" as you put it? Or do you see them saying "wet water"? (And that's just some examples; there are a lot more.) You accused me of "edit-warring" for simply not putting up with rude unwarranted reverts, for excuses that simply don't hold up, and keeping to 3RR. (One of my comments on the page was just an edit comment with no real edit...so I kept right at 3RR, and won't cross that.)  YOU are edit-warring by imposing and removing a valid mod (provably valid mod), and clarity, that is NOT really "redundant"...as I kind of just proved with just a sample of places that rightly use the phrase that you have an issue against. The edit and qualifier was for clarity and is correct and used phrasing, and does not qualify for abrupt removal on the grounds of "redundant". That might be true if all "spheres" were considered always "perfect". Apparently not all of them are. Redzemp (talk) 21:22, 9 June 2016 (UTC)
 * In fact, all spheres are spherical. Otherwise they would be some other shape, not a sphere. That is the standard mathematical usage, and the spheroid article is a mathematics article. Your mention of astronomical bodies that are not actually spheres is irrelevant, because this is an article about mathematics, not astronomy. But as I said, you should take this to Talk:Spheroid. —David Eppstein (talk) 21:44, 9 June 2016 (UTC)
 * The phrase in question is not "spherical sphere" (THAT would be "redundant") but rather the phrase that you keep removing is "perfectly spherical" or "perfect sphere", and sorry, that simply is not redundant...as sources etc prove...as not all spheres are necessarily "perfect" is the point.  Redzemp (talk) 22:20, 9 June 2016 (UTC)
 * In mathematics, this is false. It is possible to distinguish between different meanings of the word sphere (geometric versus topological, or in different metric spaces) but there is no such thing as an imperfect sphere. —David Eppstein (talk) 22:28, 9 June 2016 (UTC)
 * In "mathematics" or any other context, would you call the sun a "sphere" then?  Even if it's not "perfect"?   Would you call the earth a "sphere" even though it's not a perfect sphere?  The sun and earth are both spheres, though "near perfect" or "not perfect".   Why are they called "sphere" even if not 100% "perfect" in absolute circularity in every part?    Yes we call the earth a "spherOID" but isn't the earth also called a sphere too?    Maybe not in strict mathematics, I guess is your point.  My point is that even you'd have to admit that the phrase "spherical sphere" is WAY MORE "redundant" than "perfectly spherical" or "perfect sphere".    Remember, Wikipedia is NOT JUST for technical experts and semantical types, but also for average readers who may need elaboration and clarity.   Again, OTHER Wikipedia articles dealing with distances etc, regarding the earth, say phrases like "perfectly spherical" etc...    And so do some outside sources...written by degreed scientists.   My point is that the phrase "perfectly spherical" is PROVABLY NOT the same as "wet water", as you were saying, with that comparison.   The phrase "spherical sphere" would be more comparable to "wet water".   As both those phrases are truly redundant and needless.   Redzemp (talk) 22:36, 9 June 2016 (UTC)
 * The sun is definitely not a sphere. Its rotation makes its shape more of an oblate spheroid, even if one doesn't count solar flares, sunspots, and other irregularities. But also, the sun is a physical object, not a mathematical one, so asking whether it is a mathematical sphere is a context error, not even wrong. —David Eppstein (talk) 22:41, 9 June 2016 (UTC)
 * Here's the problem. I disagree with you in thinking that this article is strictly narrowly just a "mathematics article" in every single line or paragraph.   And the context of the immediate paragraph and sentences in here were dealing with astronomy matters, the planet earth.   (Even that means what, since isn't the whole universe mathematical in various ways anyway?)   You say that this article is not about "astronomy".    (As if astronomy has no math in it, which we all know that that's not true).   But even so, how can you say that the Spheroid article is ONLY a "mathematics article" in every single line, sentence, or paragraph... when a paragraph about the PLANET EARTH (an astronomical thing) is in the article, etc?    And also "gravity".  Which also includes physics.    All are connected.     Such as:  "Because of the combined effects of gravity and rotation, the Earth's shape is not quite a perfect sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles."    (And sorry, the sun and earth are "spheres" broadly, and are called that by scientists, when not being so overly-technical, and use the phrase "near perfect sphere" or "not a perfect sphere" etc.   Both the sun and earth (and the moon too) are circular objects, even if not perfectly so on every side necessarily.   Why does the word "spheroid" have the very word "sphere" in it?   But anyway, you see the paragraph's wording dealing with astronomy matters.   So even if this article is maybe mainly a "mathematics article", it's not 100% strictly just that.   At least not totally in this paragraph in question.  Redzemp (talk) 22:53, 9 June 2016 (UTC)

And that's all there is...and on the mathematics special page, some people who admit they are mathematically biased admit that "perfect sphere" can be used in this context here. He wrote:

'''....While this discussion should be on Talk:Spheroid perhaps placing it here will generate a greater response and may point to a larger issue that is of interest to the project. First of all, a sphere is a sphere is a sphere. A non-perfect sphere is not a sphere, so the adjective is mathematically redundant. However, to the general populace, the term sphere may refer to anything that is almost spherical (technically a spheroid) and for those who incorporate this fudge factor in their terminology, a perfect sphere distinguishes a mathematical sphere from the misnamed spheroids. The issue, as I see it, is whether or not a page that is devoted to the mathematical presentation should strictly use mathematical terminology even though the audience may not appreciate the mathematical nuances. I see the current (endless) discussion going on at Talk:Area of a disk as being essentially the same issue in a slightly different context. As an editor with a mathematical background, I do see, in myself, a definite bias towards precise, correct mathematical terminology ... but I do see this as a bias and feel that in some articles I should loosen up and not try to be as exact as I normally would desire to be. Bill Cherowitzo (talk) 03:51, 10 June 2016 (UTC)....'''

That was on the special page. Therefore, regardless of Anita and David... Reverted again, and Regards.... Redzemp (talk) 00:27, 11 June 2016 (UTC)

 * Redzemp, at this point you’re ignoring the preponderance of opinion on this page. That’s not a recipe for editing success. To repeat, “not quite a sphere” is nonsensical if “sphere” is taken as imperfect. Hence, there is no ambiguity about its meaning as is. Hence, adding “perfectly” is redundant. Redundancy is wrong. Meanwhile you have yet to demonstrate that anyone, ever, has been confused by the text as it stand. You are solving no problem, at the cost of thousands of words and the waste of many people’s time. What is your purpose here? Strebe (talk) 02:23, 11 June 2016 (UTC)
 * I just reverted your revert seconds ago. With the words "stop edit-warring and stop meat-puppeting, see my points in talk".  Meaning, stop subbing for other people, in a tag-team fashion situation.     You just dodged all the points and facts.   There's no real "redundancy", insanely equating "perfect sphere" with "wet water" has been refuted.   The discussion just started, and at least Mark kind of agrees. No real "consensus" yet.  And your rude revert had no comment or explanation in it, against WP policy.   I keep to 3RR.      Thanks. Redzemp (talk) 02:26, 11 June 2016 (UTC)

Streb twice gives no comment explanation for his reverts, which is against WP policy. Ignoring the facts and points in my long comment and complaining about the length and words does not make an argument, it's just an evasion...and NO CONSENSUS WAS REACHED. Also Mark kind of agrees. Reverted. Redzemp (talk) 02:31, 11 June 2016 (UTC)

 * Redzemp claims I have given no rationale for my edits. I have. They’re up above, where they belong. Redzemp has completely ignored what I wrote, twice now.
 * Redzemp claims I am “subbing for other people, in a tag-team fashion”. I have no idea what “he” is talking about. I am the one who originally reverted the pointless injection of “perfect”, and I don’t know any of the other people involved. That and “meat-puppeting” are paranoid accusations.
 * Redzemp wants to imagine “his” edit takes precedence over what was in the article. It doesn’t. Every argument he wishes to make about the sanctity of his own edit applies to the original text as well.
 * Redzemp wants to imagine he makes a credible accusation of “edit warring” when he is the one who has reverted six times over the past 48 hours, overriding the judgment of three other editors, while refusing to wait until the conversation plays out here, on the Talk page, where it belongs. There is no emergency.
 * Strebe (talk) 04:38, 11 June 2016 (UTC)

 * Yes, Strebe, you were the first to revert, and you notice that in your first revert, no explanation or rationale was given.  AGAINST WIKIPEDIA POLICY.  And you continued with that, in your subsequent reverts.   No rationale given...but only later and only in the talk page (later on), after other "expert editors" followed suit with wrong and refuted rationales of "wet water" redundancy claims, despite the fact that outside sources (not just other WP articles, but outside RS) say "perfect sphere" as well, etc.    Again, to the general populace, the term sphere may refer to anything that is almost spherical (technically a spheroid) and for those who incorporate this fudge factor in their terminology, a perfect sphere distinguishes a mathematical sphere from the misnamed spheroids. Redzemp (talk) 07:34, 11 June 2016 (UTC)

hello. You may have had a valid point about wrong placement, in the paragraph about earth (possibly), but that's why the more respectful edit would be to simply re-locate to the more apropos paragraph, just before...but you didn't do that, because you just don't like or want the phrase "perfect sphere" in any sense, even when said to be "commonly called" etc...anywhere anyhow. That part is NOT repetitive, so that part (alone) I restored. So removing all of it was not called for. Trimmed yes, and better located, yes, but the parenthetical part was the whole matter in the first place. But you removed that too. And that's ownership activity, and hogging, mainly for "I don't like" reasons, but always given front excuses of "redundant" and "not accurate" or 'not needed' or whatever. But look up NO OWN.... Seriously. TOO many contributors on Wikipedia commit that, and always deny they're doing it of course. This is a wiki. This was MARK'S own wording... You have no business deleting stuff you don't like.... Non-valid removal restored. see article Talk... If the statement was not in the best paragraph, that's a valid point, but deleting it completely instead of relocating it better, with the excuse of "repetition" is not valid cuz YOU JUST DON'T LIKE "PERFECT SPHERE" anywhere in the article. That's really what it boils down to, despite Mark's comments and points on Talk. No consensus against putting Mark's compromise suggestion. Anyway, I did better placement...instead of wholesale removal. Redzemp (talk) 19:08, 12 June 2016 (UTC)

 * the refs were about the point of 'sometimes called'...unwarranted revert... For suppression and "I don't like" reasons....won't work, David....... We discussed this the other day, but obviously you won't even compromise coolly with no collaboration.   Just with the attitude of "the words 'perfect sphere' will not exist in this article, in any way form or fashion, period".   You have this big hang-up for some reason against the words "perfect sphere" even though they are sourced.   And instead of admitting that you personally don't like the phrase, you and Strebe come up with all these front excuses and cop-out reasons to remove a valid and sourced point...  And this was MARK's wording and suggestion (that even whittled down to instead of "commonly" called to "sometimes" called), but that's still not good enough for people who don't understand what the word "wiki" means, and that you don't own any article, no matter what your background is in "mathematics".    Or how many Admins get fooled and suckered by this nonsense.  I only go by references and true CONSENSUS...(even if the consensus is provably wrong, I still abide by it ultimately.)    You and Strebe and Anita versus me and Mark and others don't a consensus make.   Reverted, restored, and regards. Redzemp (talk) 20:41, 12 June 2016 (UTC)

Bad references
The references in this diff, which is continuing to edit-war to include (as part of his larger war to qualify the word "sphere" with the meaningless adjective "perfect"), both fail our criteria for reliable sources.
 * physics.stackexchange.com is an open forum, and as such is a self-published source and in addition fails WP:ELNO #10.
 * the phys.org story uses the word "perfect" in a different context ("perfectly round", not "perfect sphere") and only in a headline. The usual rules of reliable sourcing are that information must appear in the actual body of a source, not the headline, because headlines are too often sensationalized to be reliable. And in this case it fails to source the claim that a sphere (the abstract geometric object) is "sometimes called a perfect sphere". —David Eppstein (talk) 22:01, 12 June 2016 (UTC)
 * Yes, David Eppstein, I agree that the ref is not so good, which is I added the other one phys.org, which is RS. You have a problem with that one too, because it doesn't say "perfect sphere" but "perfectly round", even though the rest of the page is in the context of a spherical sun, even if not actually using the word.   (And if you're honest, you'd have to admit that.)    But what do you think of "New Scientist.com"?    Read these words here:   "Now, an international group of engineers and craftsmen has gone him one better and built a pair of nearly perfect spheres that are thought to be the roundest objects in the world."   (Click:  https://www.newscientist.com/article/dn14229-roundest-objects-in-the-world-created/)   You'll find some corny problem with that ref too, because in reality it's NOT about the source, as those are just convenient FRONT excuses that you (and your tag-team partner Strebe) are using, to hide the real reason of 'ME NO LIKE'.   Even confronted with proof and good sources (here) that simply use that phrase that you think is so "redundant"...   But the phrase, in whatever context, IS sometimes used by scientists and astronomers and physicists etc, "perfect sphere" etc.    That new scientist source is not the only one either.   Want another one?     Ever heard of universetoday.com?    These words:  "The ones on the left are pulling towards the right. With all points pulling towards the center of the mass you would get a perfect sphere."  Would you consider those "bad references"? (Click: http://www.universetoday.com/112805/why-is-everything-spherical/)   I'm sure you'll find some cop-out reason to diss those too, like maybe how it doesn't apply or "doesn't fit", or whatever.   Ignoring the point about "sometimes used"...in GENERAL.   Anyway, Strebe the presumed co-owner of this article did another revert.   I won't violate true 3RR.  Redzemp (talk) 22:19, 12 June 2016 (UTC)
 * Your newscientist source may source the claim that non-spheres may be called "nearly perfect" but does not source the claim that spheres may be called "perfect". And AGAIN, it is about physical objects, while this article is primarily about abstract mathematical objects, so it is off-topic (wrong context) for claims about mathematical usage. And you have already hit at least 7RR today so your promise not to violate 3RR is too little too late. —David Eppstein (talk) 22:35, 12 June 2016 (UTC)
 * As I said, and true to form, which is why from now on I'll be ignoring you as you show no honesty or credibility on this particular matter, but just invested in a "no way will that phrase make it on this article" agenda.   I'm sure you'll find some cop-out reason to diss those too, like maybe how it doesn't apply or "doesn't fit", or whatever.   Ignoring the point about "sometimes used"...in GENERAL.  Now you're coming up with LAUGHABLE AND TORTURED nonsense of ""nearly perfect" versus "perfect sphere" etc.   Or maybe "this is primarily a mathematics article" even though some parts deal with physics and "earth" and "gravity".     Seriously, man, you showed days ago what you're all about on here.   Days from now, the edit goes back, with the refs.  And of course you and Strebe, who ignore "no own" and "I don't like" all the time, will get in and disrespect as usual, with your edit-warring and nit-picking and wiki-battering and suppression.   I'm done for now though.  The point though is that the basic phrase of "nearly perfect" or "perfect sphere" WHICH YOU CLAIMED WAS "REDUNDANT" is "sometimes" used by authorities and people in these fields.  Regards......  Redzemp (talk) 22:39, 12 June 2016 (UTC)
 * Please stay WP:CIVIL. Your personal attacks have no place here. —David Eppstein (talk) 22:45, 12 June 2016 (UTC)
 * True enough, but your actions and attitudes are not really all that civil here.  How you (and Strebe) have been carrying on.   About this minor thing, that is sourced and proven.   It's not a cool attitude.   What is the BIG DEAL of having (and in Mark's modified whittled down form even) and in parentheses, this simple sourced thing?  I DON'T GET IT.   You come up with all these reasons and problems and excuses, but I've addressed those.    The phrase is sourced in GOOD references. I already conceded that you might have had a point about the open forum source, though the phys.org was a lot better, but these other two sources new scientist and universetoday are very good, and the matter is pretty clear there, and you're just blatantly dismissing them anyway, and LOOKING for ways to dismiss them.   That's not all that cool or civil.   Regards........   Redzemp (talk) 22:55, 12 June 2016 (UTC)

Compromise? I've showed it big time...whereas David and Strebe keep reverting to original, showing zero compromise.
I modified wording, changed it to make it "sometimes" instead of "commonly". I took MARK's suggested wording, of parenthetical, etc, from the Talk page. I looked for sources, and put better ones on the Talk page. I have gone out of my way to "compromise", big time. David flat-out lies when he says I haven't. On the notice board page. Unbelievable The only one "clearly not interested in compromise" is David Eppstein, because if you see what I've done I'VE BENT OVER BACKWARDS to "compromise". David's idea of "compromise" is don't change or modify it at all, and leave it the way it was. I provided good refs that show that "perfect sphere" is used by people in the field. David is unbelievable in saying that I am not interested in compromise, when he and Strebe are the ones who show ownership and "I don't like" attitudes and actions all over the place with ZERO compromise or give or take. Forgive my bluntness but he's been uncivil in his uncool actions and dissings of sourced mods and edits. Also this was MARK'S compromise and suggestion, that I tried putting in (with sources) that get rudely removed regardless. This is what I wrote and proved in the article talk page....

"New Scientist.com"?

Read these words here:  "Now, an international group of engineers and craftsmen has gone him one better and built a pair of nearly perfect spheres that are thought to be the roundest objects in the world."   (Click:  https://www.newscientist.com/article/dn14229-roundest-objects-in-the-world-created/)

universetoday.com?

These words: "The ones on the left are pulling towards the right. With all points pulling towards the center of the mass you would get a perfect sphere."

Would you consider those "bad references"? (Click: http://www.universetoday.com/112805/why-is-everything-spherical/)  I'm sure you'll find some cop-out reason to diss those too, like maybe how it doesn't apply or "doesn't fit", or whatever. Ignoring the point about "sometimes used"...in GENERAL.

Anyway, You see the sources? And I took the time and effort to make points and show proofs etc, making effort to discuss and make the case, and instead of appreciating any of that, David disses and belittles it and accuses me of "textwalling" putting a negative dishonest spin on everything. But look at what happened. That shows that the statement is justified and "sometimes used" by reliable sources, that David simply does not want in, regardless of sources. He'll find some excuse to diss or dismiss those valid sources anyway.. Showing NO compromise at all. I've tried. He's pot-kettle-black on this, big time.

I've done all I can to show compromise, from taking Mark's suggested wording. And then even changing that from "commonly used" to "sometimes used". Tell me. And then finding sources. One was not that good, so I got another one. And now I found two VERY good ones. Tell me. Where in any of that did I show "no compromise"?

But David keeps removing and reverting all the time, to bring it back to the original with no addition whatsoever. Tell me. Where is David showing any compromise with that? And he has the nerve to talk about "civility"? And "no compromise". That's why I made the blunt statement that I will be ignoring him, because he showed clearly that he has no real credibility on this (along with Strebe), but is blatantly dishonest on this, from start to finish. I have solid proof that I've showed BIG TIME "compromise". He said I showed none. (???)  I have solid proof that David showed literally ZERO compromise. Yet he's making out like he's the cool collaborator and compromiser, when he'd been nothing of the kind. The proof is in the edits and in the talk page comments. Regards. Redzemp (talk) 12:22, 13 June 2016 (UTC)

The solid?
What is the name of a solid whose surface is a spheroid? This should be mentioned in the opening two sentences. Koro Neil (talk) 03:17, 22 February 2021 (UTC)

title: Talk:Spheroid


text:
Eumir Deodato

Eumir Deodato de Almeida (born 22 June 1943) is a Brazilian pianist, composer, arranger and record producer, primarily in jazz but who has been known for his eclectic melding of genres, such as pop, rock, disco, rhythm and blues, classical, Latin and bossa nova.

Deodato has arranged and produced more than 500 records for acts such as Frank Sinatra, Roberta Flack, Björk and Christophe, as well as produced Kool & the Gang's hits "Celebration", "Ladies' Night" and "Too Hot".

Deodato was nominated for three Grammy Awards and won the Grammy Award for Best Pop Instrumental Performance in 1974 for "Also Sprach Zarathustra (2001)." The song peaked at number 2 on the weekly Billboard Hot 100 in March 1973. It reached number 3 in Canada and number 7 on the British charts.

Biography
Deodato, born in Rio De Janeiro on June 22, 1943, began his musical life on accordion when he was 12 years old, and then piano two years later. He studied orchestration, conducting and arranging. He played bossa nova in bands with Paulo Silvino and Orlandivo, then formed his own band, which featured João Palma, Sergio Barroso, Ugo Marotta, Henri Ackselrud and guitarist Celso Brando, who was later replaced by Roberto Menescal in 1962. Eumir’s paternal grandfather was a marble craftsman from Custonaci, Sicily.

Recording career
Deodato often plays the Fender Rhodes electric piano. He became successful as a keyboard player in the 1970s. Since then, he has produced and arranged music on more than 500 albums for artists such as Kool & the Gang, Con Funk Shun, Björk, Christophe, Ithamara Koorax and k.d. lang. Guitarist John Tropea and flautist Hubert Laws appeared on his early albums.

Prelude, his first album in the U.S., was released in 1973. His track Also Sprach Zarathustra (2001) gained big hit in Billboard pop chart. This album was crossover music style that attracted a large audience and was produced by Creed Taylor for his label CTI. The album sold 5 million copies worldwide and earned Deodato the 1974 Grammy Award for Best Pop Instrumental Performance for the track Also Sprach Zarathustra (2001) as well as a Grammy nomination for Best New Artist.

His second album, Deodato 2, reached number 19 on the Billboard album chart, and the single "Rhapsody in Blue" reached No. 41 on the Billboard Hot 100 in 1973. His interpretation of Pavane pour une infante défunte ("Pavane for a Dead Princess") by Maurice Ravel was used in the 1970s by an Australian television station as background music.

In 1978, he had an orchestral hit with "Whistle Bump" from the LP titled Love Island. The track promoted the widespread use of whistles in nightclubs at the time. However, his popularity in the discos was solidified when he released the 1979 single Night Cruiser from the album of the same name, which earned him a third Grammy nomination for Best R&B Instrumental Performance. Deodato continued recording through the 1980s. In 1985, he had two hits, "S.O.S., Fire in the Sky" and "Are You For Real", on Billboard magazine's top 20 Dance chart.

In 1998, Deodato invited his old friend and fellow Brazilian jazz impresario Arnaldo DeSouteiro to co-produce with him the first official CD reissues of all his albums from the 1960s, in a total of nine releases. Among them, Inútil Paisagem (reissued by JSR in co-production with Universal Music), Idéias, and all the albums from the Os Catedráticos series that were digitally remixed and remastered by audio engineer Rodrigo de Castro Lopes under the direct supervision of both Deodato and DeSouteiro.

He recorded Live in Rio in 2007. In 2009, Prelude was reissued for the first time in Asia on SACD and SHM-CD formats under the supervision of Arnaldo DeSouteiro. In 2011, he released the album The Crossing, which he produced with Lino Nicolosi and Pino Nicolosi at Nicolosi Productions, with guest vocalist Al Jarreau. In 2010 he played the rhodes piano in the album Oasis produced and performed by Marita Pauli. In 2018 he arranged and conducted the strings orchestra in the album The First released by Riccardo Dalli Cardillo.

Arranging and producing
Since the 1960s, Deodato has been in demand as a producer and arranger. He has worked on more than 500 albums, and 15 have reached platinum status as defined by the RIAA.

In the early 1960s, he worked as a freelance arranger for Odeon Records. He wrote arrangements for Wilson Simonal, Marcos Valle, Leny Andrade, Pery Ribeiro, Tito Madi and for his own debut album, Inútil Paisagem (1964), which was dedicated to the work of Antônio Carlos Jobim and recorded in Rio when Deodato was 22. Jobim praised him in the album's liner notes. Deodato played piano with guitarists Oscar Castro-Neves and Roberto Menescal, drummer Juquinha and an all-star cast including Copinha, J.T. Meirelles, Cipó, Paulo Moura. He has been credited for helping to start the career of Milton Nascimento; Deodato was part of a committee tasked with choosing songs for a festival and chose three by Nascimento.

He moved to New York City in 1967 to work with guitarist Luiz Bonfá, with whom he recorded seven albums and countless jingles. Bonfá also introduced him to vocalist Astrud Gilberto and record producer Creed Taylor, when they were reunited on the making of Astrud's "Beach Samba" album for Verve Records. Later on, Creed Taylor hired him to write arrangements for musicians at CTI Records who included Wes Montgomery, Antônio Carlos Jobim, Milton Nascimento, Walter Wanderley, Stanley Turrentine, Paul Desmond and the CTI All Stars. Deodato also recorded with Frank Sinatra, (1969), João Donato (1970), Youg, Holt Unlimited (1973), Aretha Franklin (1973), and recorded three albums with Roberta Flack from 1971 to 1973.

He had other collaborations, including producing four million-selling albums for Kool and the Gang from 1979 to 1982, the first solo album by Kevin Rowland of Dexys Midnight Runners in 1988, three albums four singer Ithamara Koorax, as well as records by Chuck Mangione, Larry Graham, Tania Maria, Eliane Elias, Pretty Poison, One Way, Con Funk Shun, Brenda K. Starr, Kleeer, and kd Lang. Deodato also developed a close collaboration with Iceladinc singer Bjork from 1996 to 1998, arranging her albums Post, Telegram and Homogenic. For Deodato's Love Island, he and Maurice White co-wrote the instrumental "Tahiti Hut", which (with lyrics added to it) was later recorded by the band Switch with guest vocals by Jermaine Jackson.

Deodato wrote orchestral arrangements for the films The Gentle Rain (with original soundtrack by Luiz Bonfá in 1965), and ""The Adventurers" (with soundtrack by Antonio Carlos Jobim and additional score by Deodato in 1969), as well as the original soundtracks for ""Target Risk", "The Reporter", The Black Pearl (1977), The Onion Field (1979), ""Beat Street (1980) and Bossa Nova (2000), among many others. He also produced and arranged for the movies "Body Rock" and "The Ghostbusters II".

Personal life
Eumir Deodato was married to Ruth Deodato from 1963 to 1999. Their daughter Kennya Deodato (b. 1968) is married to actor Stephen Baldwin. Their granddaughter Hailey Bieber is married to Canadian singer Justin Bieber.

As sideman
With Luiz Bonfá
 * The Bonfa Magic (Milestone, 1993)
 * Jacaranda (Ranwood, 1973; reissued JSR, 1998)
 * Black Orpheus Impressions (Dot, 1968; reissued Sony, 2000)
 * Luiz Bonfá Plays Great Songs (Dot, 1967)
 * Steve & Eydie, Bonfa & Brazil (Columbia, 1968)

With Astrud Gilberto
 * Gilberto with Turrentine (CTI, 1971)
 * Now (Perception, 1972)
 * Windy (Verve, 1968)

With Antônio Carlos Jobim
 * The Daventurers - Original Soundtrack (Paramount, 1969)
 * Stone Flower (CTI, 1970)
 * Tide (A&M/CTI, 1970)
 * Antonio Carlos Jobim's Finest Hour (Verve, 2000)

With Kool & the Gang
 * Ladies' Night (De-Lite, 1979)
 * Celebration (De-Lite, 1980)
 * Something Special (De-Lite, 1981)
 * As One (De-Lite, 1982)

With Ithamara Koorax
 * Ithamara Koorax Sings the Luiz Bonfa Songbook (Paddle Wheel, 1996)
 * Exclusively For My Friends (BCE, 1999)
 * Serenade in Blue (Milestone, 2000)

With Roberto Menescal
 * A Nova Bossa Nova (Elenco, 1965)
 * The Boy from Ipanema Beach (Kapp, 1965)
 * Bossa Nova (Odeon, 1965)
 * A Bossa Nova De Roberto Menescal  (Elenco, 1964)

With Milton Nascimento
 * Courage (A&M/CTI, 1969)
 * Clube Da Esquina (EMI, 1972)

With Frank Sinatra
 * Sinatra & Company (Reprise, 1971) (side one arrangements)

With Brenda K. Starr
 * Brenda K. Starr (MCA, 1987)
 * By Heart (Epic, 1991)

With Stanley Turrentine
 * Salt Song (CTI, 1971)
 * The Baddest Turrentine (CTI, 1973)
 * The Sugar Man (CTI, 1975)

With Marcos Valle
 * Braziliance! (Warner Bros., 1966)
 * Samba '68 (Verve, 1968)

With others
 * Anthony and the Camp, Suspense (Warner Bros., 1988)
 * Charles Aznavour, Toujours (EMI, 2011)
 * Camp Lo, Uptown Saturday Night (Profile, 1997)
 * Tony Cicco, ...E Mo' Parlamm'e Musica (RCA 1987)
 * Billy Cobham, Drum 'n' Voice Vol. 4 (Nicolosi, 2016)
 * Dazz Band, Rock the Room (RCA Victor, 1988)
 * Milton Delugg, Accordion My Way-Ole! (RCA Victor, 1967)
 * Paul Desmond, Summertime (A&M, 1968)
 * Eliane Elias, So Far So Close (Blue Note, 1989)
 * Roberta Flack, Killing Me Softly (Atlantic, 1973)
 * Aretha Franklin, Let Me in Your Life (Atlantic, 1974)
 * Michael Franks, Burchfield Nines (Warner, 1978)
 * Fun Lovin' Criminals, Come Find Yourself (EMI, 1996)
 * Fun Lovin' Criminals, The Grave and the Constant EP (Chrysalis, 1996)
 * Larry Graham, Fired Up (Warner Bros., 1998)
 * Kleeer, Seeekret (Atlantic, 1985)
 * Kleiton & Kledir, Dois (Som Livre, 1996)
 * Tito Madi, Balanco Zona Sul E Outros Sucessos (Odeon, 1966)
 * Chuck Mangione, Disguise (Columbia, 1984)
 * Maysa, Maysa (Elenco, 1964)
 * Lisa Ono, Pretty World Suite! (Supuesto! 2000)
 * Jorge Pescara, Grooves in the Temple (2005)
 * Pretty Poison, Catch Me I'm Falling (Virgin, 1988)
 * Quarteto em Cy, Quarteto em Cy (Forma, 1964)
 * Ranieri, Meditazione (CGD, 1976)
 * Dom Um Romão, Lake of Perseverance (JSR/Irma, 2001)
 * JSR All-Stars, Friends From Brazil 2001 (JSR/Irma, 2001)
 * Carlos Pingarilho, Histórias E Sonhos (JSR,2002)
 * Brazil All Stars, Rio Strut (JSR, 2002)
 * Kevin Rowland, The Wanderer (Mercury, 1988)
 * Wanda Sá, Wanda Vagamente (RGE, 1964)
 * Steve and Eydie, Steve & Eydie, Bonfa & Brazil (CBS/Sony, 1987)
 * Riccardo Dalli Cardillo, The First (DCP, 2018)

title: Eumir Deodato


text:
Charles E. Townsend

Charles Elroy Townsend (August 15, 1856 – August 3, 1924) was an American lawyer who served as both a U.S. Representative and U.S. Senator from the state of Michigan. He served in the United States Congress from 1903 to 1923.

Early life and career
Townsend was born near Concord, Michigan and attended the common schools in Concord and Jackson and the University of Michigan at Ann Arbor. He taught school at Concord 1881-1886 and was Jackson County Register of Deeds 1886–1897. He studied law and was admitted to the bar in 1895 and commenced practice in Jackson.

Tenure in Congress
Townsend was elected as a Republican to the United States House of Representatives for the Fifty-eighth and for the three succeeding Congresses, serving from March 4, 1903, to March 3, 1911.

He was elected to the United States Senate in 1910 and was reelected in 1916, serving from March 4, 1911, to March 3, 1923. In 1914 and while holding the office of Senator, he was on the Central Committee of the First National Conference on Race Betterment, a conference on eugenics held at the Battle Creek Sanatorium. He was an unsuccessful candidate for reelection in 1922. He was chairman of the U.S. Senate Committee on Coast and Insular Survey in the Sixty-second Congress, the U.S. Senate Committee on Expenditures in the War Department in the Sixty-fifth Congress, and the U.S. Senate Committee on Post Office and Post Roads in the Sixty-sixth and Sixty-seventh Congresses.

Career after Congress
Townsend was appointed in 1923 as a member of the International Joint Commission created to regulate the use of the boundary waters between the United States and Canada, in which capacity he served until his death in Jackson. He is interred in Maple Grove Cemetery, in Concord.

title: Charles E. Townsend


text:
Ringling Brothers Circus

Ringling Bros. World's Greatest Shows is a circus founded in Baraboo, Wisconsin, United States in 1884 by five of the seven Ringling brothers: Albert, August, Otto, Alfred T., Charles, John, and Henry. The Ringling brothers were sons of a German immigrant, August Frederick Rüngeling, who changed his name to Ringling once he settled in America. Four brothers were born in McGregor, Iowa: Alf T., Charles, John and Henry. The Ringling family lived in McGregor, Iowa, for twelve years, from 1860 until 1872. The family then lived in Prairie du Chien, Wisconsin, and moved to Baraboo, Wisconsin, in 1875. In 1907 Ringling Bros. acquired the Barnum & Bailey Circus, merging them in 1919 to become Ringling Bros. and Barnum & Bailey Circus, promoted as The Greatest Show on Earth. Ringling Bros. and Barnum & Bailey closed on May 21, 2017, following weakening attendance and high operating costs.

On Wednesday, May 18, 2022, the company announced that it will officially return, with its first show on Sept. 28, 2023, and a tour of more than 50 cities, but without any animals.

History

In 1882, before the Ringling brothers created their first circus, the five brothers performed skits and juggling routines in town halls around the state of Wisconsin. Their first show was on November 27, 1882, in Mazomanie, Wisconsin. They called this the "Ringling Bros. Variety Performance" when they took the show to the next town. With two wandering performers the next year, the brothers toured the Northwest. After the Northwest tour, they used the money earned for suits.

They expanded their acts into a one ring show in 1884. The show added a trick horse and a bear at the end of the season. The circus started traveling by trains in 1888 allowing the show to consistently expand.

Ringling Circus purchased the Yankee Robinson Circus and opened a joint show on May 19, 1884. This brought them to the attention of James Anthony Bailey of Barnum and Bailey's Circus as a viable competitor. The brothers met with Bailey thus agreeing to a division of areas. This was followed by them purchasing a half share of the Adam Forepaugh Sells Brothers Circus from Bailey. Bailey, under the area division, prohibited the Ringlings from playing at the Madison Garden, a location it was the brothers' ambition to perform at. In 1887 Ringling Circus changed its official title to the "Ringling Bros. United Monster Shows, Great Double Circus, Royal European Menagerie, Museum, Caravan, and Congress of Trained Animals."

In 1906, Bailey died, which led to the Ringlings taking over Forepaugh–Sells, which continued to operate separately. In October 1907, the stockholders of Barnum and Bailey's Circus approved the sale of the circus to the Ringlings. Due to declining audiences and employees being drafted for World War I, Ringling Circus and Barnum and Bailey's Circus were merged in 1919 as Ringling Bros. and Barnum & Bailey Circus.

title: Ringling Brothers Circus


text:
List of rulers of the Akan state of Denkyira

This is a list of rulers of the Akan state of Denkyira, formerly known as Agona. Dates in italics indicate de facto continuation of office.