I'm pretty sure it's Latin.

It is Latin.

Bismi Allahi alrrahmani alrraheemi

Alhakumu alttakathuru

Hatta zurtumu almaqabira

Kalla sawfa taAAlamoona

Thumma kalla sawfa taAAlamoona

Kalla law taAAlamoona AAilma alyaqeeni

Latarawunna aljaheema

Thumma latarawunnaha AAayna alyaqeeni

Thumma latus-alunna yawma-ithin AAani alnnaAAeemi

The word "translation" comes from Latin roots trans- "across" and latus "carried". It has two major uses in English

• the restating of a message in different terms, usually from one language to another (the meaning is "carried across" the language boundary)
• a uniform movement from one position to another without change of orientation (the object is "carried across" the intervening space)

"Flat land" may be the original meaning of the Italian word Lazio.

Specifically, the Italian word is the name of the central Italian region in which Rome (Roma) the national capital is located. The name comes from the Latin word Latium, as the area inhabited by the Latini, speakers of the ancient Latin language. It is possible that Latium comes from the Latin word latus for the "wide" flat lands of the Latini.

The pronunciation is "LAHT-tsyoh."

y = 3x2 - 18x + 16

y = 3(x2 - 6x + 9) + 16 - 27 complete the square

y + 11 = 3(x - 3)2

(4/4)(y + 11)/3 = (x - 3)2

4(1/12)(y - -11) = (x - 3)2

(x - h)2 = 4a(y - k); vertex (h, k), focus (h, k +a), parabola, axis of symmetry parallel to y-axis, opens up.

V(3, -11), F(3, -11 + 1/12) = (3, -131/12)

A parabola is the collection of all points P in the plane that are the same distance from a fixed point F (focus) as they are from a fixed line D (directrix).

If you are looking for the distance of focus from the vertex, it is 1/12, the value of a.

If you are looking for the distance of focus from directrix, it is 1/6, the value of 2a.

If you are looking for the length of the latus rectum, it is 1/3.

Let y = -131/12. Then,

y = 3x2 - 18x + 16

y = -131/12

3x2 - 18x + 16 = -131/12

3x2 - 18x = -323/12

x2 - 6x = -323/36

x2 - 6x + 9 = 9 - 323/36

(x - 3)2 = 1/36

x - 3 = ±√1/36

x = 3 ± 1/6

x = 19/6 or x = 17/6

19/6 - 17/6 = 2/6 = 1/3

Nicolas Copernicus (1473 - 1543) Nicolas Copernicus was born in Torun, Poland in 1473 and died in 1543. He studied both law and medicine in Italy, but spent most of his life as Cannon of Frombork Cathedral in Poland. He published his great work "On the Revolutions of the Heavenly Spheres" in 1543, the year he died. In this book he detailed a heliocentric system (sun centered) and described advantages it had over the geocentric system of Ptolemy. Over the space of about 75 years, this new view of the solar system gradually gained acceptance throughout Europe. Copernicus' system:Main features: 1. Planets move in circular orbits around the sun 2. Earth is one of the planets 3. Night and day on the earth are due to the rotation of the earth on its axis (one rotation every 24 hours). 4. The apparent motion of the sun along the ecliptic is due to the revolution of the earth around the sun (one complete revolution in one year). 5. Retrograde motion of the planets is an illusion produced by the earth passing the planets as it journeys around the sun (much like the way the car you pass on the freeway seems to be moving backward relative to your car). 6. Mercury and Venus are always found close to the sun because their orbits lie between the earth and the sun 7. The scale of the solar system can be established in astronomical units(AU - the distance from the earth to the sun). Using the inner planets this is accomplished as in the following diagram: This is how it works: When Venus is at its maximum distance from the sun, the triangle defined by the sun, the earth and Venus is as shown above. The angle a is known because that is the angular distance between Venus and the sun as viewed from the earth. Geometrical theorems can show that the angle at Venus is always 90 degrees when a is a maximum. The length of the hypotenuse of this right triangle is one AU by definition. Knowing all the angles and one side of a triangle allows us to determine the length of all the other sides, in this case the distance from Earth to Venus and Venus to the sun. Scholars were initially attracted to Copernicus' system because it (1) it explained retrograde motion in an elegant way (2) it explained why Mercury and Venus were always close to the sun, and (3) it provided a way for establishing the scale of the solar system. There were, however, two drawbacks: (1) it was hard to explain physically (for example, why don't we notice that the earth is moving?) And (2) although elegant, it didn't predict the future position of the planets any better than Ptolemy's system. Johannes Kepler (1571 - 1630) With respect to social status and personality, Johannes Kepler was as far from Tycho Brahe (a noble Danish astronomer) as is possible to imagine. Whereas Tycho was a wealthy aristocrat with vast resources and had a voracious appetite for life's pleasures, Kepler was born into abject poverty and practiced a strict and pious form ofProtestantism. Yet Kepler and Tycho ultimately collaborated to sweep away the ancient concept of perfectly circular motion in the heavens and to replace it with planets moving in elliptical orbits. Kepler developed a fascination with the sky and its movements as a student of mathematics in Tübingen, Germany and became a convert to Copernicus' newheliocentric system. He was determined to show how the Copernican system could lead to more accurate predictions than Ptolemy's. Kepler began working with Tycho in 1600 to take advantage of the fact that Tycho had the most accurate planetary position data available anywhere. Using this data, he began trying to fit the orbit of Mars into a curve that could be used to predict positions of that planet in the future as well as to specify its position in the past. Tycho died in 1601, but Kepler stayed with Tycho's organization and wasultimately successful in demonstrating that planets must move in elliptical orbits. With that innovation, Copernicus' heliocentric model was much better at prediction than Ptolemy's and the number of scholars who believed in a sun centered universe began to rise. Kepler was able to formulate three laws of motion that describes how planetsmove about the sun. Kepler's First Law

The first law says: "The orbit of every planet is an ellipse with the sun at one of the foci". The mathematics of the ellipse is as follows. The equation is: where (r?) are heliocentric polar coordinates for the planet, p is the semi latus rectum, and e is the eccentricity, which is less than one. For? =0 the planet is at the perihelion at minimum distance: for? =90º: r=p, and for ?=180º the planet is at the aphelion at maximum distance: The Semi-major axis is the arithmetic mean between rmin and rmax: The Semi-minor axis is the geometric mean between rmin and rmax: and it is also the geometric mean between the semi major axis and the semi latus rectum: Kepler's second law

The second law: "A line joining a planet and the sun sweeps out equal areas during equal intervals of time". This is also known as the law of equal areas. Suppose a planet takes one day to travel from points A to B. The lines from the Sun to A and B, together with the planet orbit, will define a (roughly triangular) area. This same amount of area will be formed every day regardless of where in its orbit the planet is. So the planet moves faster when it is closer to the sun. This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out, but Kepler did not know that reason. The two laws permitted Kepler to calculate the position of the planet, based on the time since perihelion, t, and the orbital period, P. The calculation is done in four steps. 1. Compute the mean anomaly M from the formula 2. Compute the Eccentric anomaly E by numerically solving Kepler's equation: 3. Compute the true anomaly ? by the equation: 4. Compute the heliocentric distance r from the first law: The proof of this procedure is shown below. Kepler's third law

The third law : "The squares of the orbital periods of planets are directly proportional to the cubes of the Semi-major axis of the orbits". P = orbital period of planeta = semi major axis of orbit So the expression P2a-3 has the same value for all planets in the solar system as it has for Earth where P= 1 Sidereal year and a=1 astronomical unit, so in these units P2a-3 has the value 1 for all planets. With P in seconds and in meters: . Thus, not only does the length of the orbit increase with distance, the orbital speed decreases, so that the increase of the Orbital period is more than proportional. The general equation, which Kepler did not know, is be derived by equating Newton's law of gravity with Uniform Circular Motion (a = (4p2r) / t2), which is valid for (near) circular orbits. G = gravitational constant M = mass of sunm = mass of planet Note that P is time per orbit and P/2p is time per radian. See the actual figures: attributes of major planets. This law is also known as the harmonic law.

Kepler's Laws are illustrated in the adjacent animation. The red arrow indicates the instantaneous velocity vector at each point on the orbit (as always, we greatly exaggerate the eccentricty of the ellipse for purposes of illustration). Since the velocity is a vector, the direction of the velocity vector is indicated by the direction of the arrow and the magnitude of the velocity is indicated by the length of the arrow. Notice that (because of Kepler's 2nd Law) the velocity vector is constantly changing both its magnitude and its direction as it moves around the elliptical orbit (if the orbit were circular, the magnitude of the velocity would remain constant but the direction would change continuously). Since either a change in the magnitude or the direction of the velocity vector constitutes an acceleration, there is a continuous acceleration as the planet moves about its orbit (whether circular or elliptical), and therefore by Newton's 2nd Law there is a force that acts at every point on the orbit. Furthermore, the force is not constant in magnitude, since the change in velocity (acceleration) is larger when the planet is near the Sun on the elliptical orbit.

Since this is a survey course, we shall not cover all the mathematics, but we now outline how Kepler's Laws are implied by those of Newton, and use Newton's Laws to supply corrections to Kepler's Laws. # Since the planets move on ellipses (Kepler's 1st Law), they are continually accelerating, as we have noted above. As we have also noted above, this implies a force acting continuously on the planets. # Because the planet-Sun line sweeps out equal areas in equal times (Kepler's 2nd Law), it is possible to show that the force must be directed toward the Sun from the planet. # From Kepler's 1st Law the orbit is an ellipse with the Sun at one focus; from Newton's laws it can be shown that this means that the magnitude of the force must vary as one over the square of the distance between the planet and the Sun. # Kepler's 3rd Law and Newton's 3rd Law imply that the force must be proportional to the product of the masses for the planet and the Sun. Thus, Kepler's laws and Newton's laws taken together imply that the force that holds the planets in their orbits by continuously changing the planet's velocity so that it follows an elliptical path is (1) directed toward the Sun from the planet, (2) is proportional to the product of masses for the Sun and planet, and (3) is inversely proportional to the square of the planet-Sun separation. This is precisely the form of the gravitational force, with the universal gravitational constant G as the constant of proportionality. Thus, Newton's laws of motion, with a gravitational force used in the 2nd Law, imply Kepler's Laws, and the planets obey the same laws of motion as objects on the surface of the Earth!

The ellipse is not the only possible orbit in a gravitational field. According to Newton's analysis, the possible orbits in a gravitational field can take the shape of the figures that are known as conic sections (so called because they may be obtained by slicing sections from a cone, as illustrated in the following figure). For the ellipse (and its special case, the circle), the plane intersects opposite "edges" of the cone. For the parabola the plane is parallel to one edge of the cone; for the hyperbola the plane is not parallel to an edge but it does not intersect opposite "edges" of the cone. (Remember that these cones extend forever downward; we have shown them with bottoms because we are only displaying a portion of the cone.)

We see examples of all these possible orbitals in gravitational fields. In each case, the determining factor influencing the nature of the orbit is the relative speed of the object in its orbit. * The orbits of some of the planets (e.g., Venus) are ellipses of such small eccentricity that they are essentially circles, and we can put artificial satellites into orbit around the Earth with circular orbits if we choose. * The orbits of the planets generally are ellipses. * Some comets have parabolic orbits; this means that they pass the Sun once and then leave the Solar System, never to return. Other comets have elliptical orbits and thus orbit the Sun with specific periods. * The gravitational interaction between two passing stars generally results in hyperbolic trajectories for the two stars. Thus, Kepler's elliptical orbitals are but one example of the possible orbits in a gravitational field. Only ellipses (and their special case, the circle) lead to bound orbits; the others are associated with one-time gravitational encounters. For a given central force, increasing the velocity causes the orbit to change from a circle to an ellipse to a parabola to a hyperbola, with the changes occurring at certain critical velocities. For example, if the speed of the Earth (which is in a nearly circular gravitational orbit) were increased by about a factor of 1.4, the orbit would change into a parabola and the Earth would leave the Solar System.