1. The German astronomer Johannes Kepler (1571-1630) supported the Copernican concept that
the Sun is at the center, but gave to the planets elliptical orbits, with the Sun in one of the foci of
each ellipse, to describe their complicated motions more correctly. The direct observations by
Galileo Galilei (1564-1642), who showed that Venus has phases like the moon (using the
telescope he invented and built) clinched the matter for the heliocentric system. (The Church
disagreed. It put Copernicus' book on the index of forbidden works in 1616, and left it there till
1835. Also, Galileo had to recant and was forbidden to teach and to leave his home.) Galileo was
one of the great minds of all time. As the French philospher Yves Bonnefoy has said: "With
Galileo, the Moon ceased to be an object of adoration and became an object for scientific study."
Another Italian astronomer, Giovanni Domenico Cassini (1625-1721) (whose name is associated
with the large gap between the inner and outer rings of Saturn) determined the size of Earth's
orbit. His value was only 7 percent short of the modern one (150 million km). He also
established the size of the solar system. (Aristarchus had been off by a factor of 20 in estimating
the distance to the Sun.)
Then came Isaac Newton (1642-1727) who brought the laws of physics to the solar
system. Isaac Newton explained why the planets move the way they do, by applying his laws of
motion, and the force of gravitation between any two bodies, letting the force decrease with the
square of the distance between the two bodies. (Besides formulating the laws of motion, Newton
invented the concept of gravitational force and a new kind of math to calculate planetary
motions. The math is now called calculus. It was invented independently, and published earlier,
by the German mathematician Gottfried Wilhelm Leibniz (1646-1716) whose notation is still
used in textbooks today. Newton also built the first reflecting telescope to survey the sky with.)
Knowing that the Sun is in the center of the system, and the rotating planets move around it in
their proper orbits following Newton's laws became the basis for further exploration of "celestial
mechanics". Many details of the motions remained to be worked out. Major contributions came
from the French mathematicians and astronomers Pierre Simon de Pierre Simon Laplace (1749-
1827), Joseph Louis Lagrange (1736-1813), and Urbain Leverrier (1811-1877). (Even now
celestial mechanics is an active field of study, because of the amount of computation it takes to
work out planetary positions for millions of years.)
Laplace, Pierre Simon -(1749-1827): Marquis de Laplace was a French mathematician and
astronomer. He worked on celestial mechanics and perturbations in the motions and rotations of
planet and determined that the obliquity of Earth's axis is not constant, but varies cyclically. In a
paper read before the Academy of Sciences, on the 10th of February 1773, Laplace announced
his celebrated conclusion of the invariability of planetary mean motions, carrying the proof as far
as the cubes of the eccentricities and inclinations. This was the first and most important step in
the establishment of the stability of the solar system. It was followed by a series of profound
investigations, in which Lagrange and Laplace alternately surpassed and supplemented each
other in assigning limits of variation to the several elements of the planetary orbits. The
analytical tournament closed with the communication to the Academy by Laplace, in 1787, of an
entire group of remarkable discoveries. It would be difficult, in the whole range of scientific
literature, to point to a memoir of equal brilliancy with that published (divided into three parts) in
the volumes of the Academy for 1784, 1785 and 1786. The long-sought cause of the "great
inequality" of Jupiter and Saturn was found in the near approach to commensurability of their
mean motions; it was demonstrated in two elegant theorems, independently of any except the

2. most general considerations as to mass, that the mutual action of the planets could never largely
affect the eccentricities and inclinations of their orbits; and the singular peculiarities detected by
him in the Jovian system were expressed in the so-called "laws of Laplace." He completed the
theory of these bodies in a treatise published among the Paris Memoirs for 1788 and 1789; and
the striking superiority of the tables computed by Delambre from the data there supplied marked
the profit derived from the investigation by practical astronomy. The year 1787 was rendered
further memorable by Laplace's announcement on the 19th of November (Memoirs, 1786), of the
dependence of lunar acceleration upon the secular changes in the eccentricity of the earth's orbit.
The last apparent anomaly, and the last threat of instability, thus disappeared from the solar
system.
With these brilliant performances the first period of Laplace's scientific career may be said to
have closed. If he ceased to make striking discoveries in celestial mechanics, it was rather their
subject matter than his powers that failed. The general working of the great machine was now
laid bare, and it needed a further advance of knowledge to bring a fresh set of problems within
reach of investigation. The time had come when the results obtained in the development and
application of the law of gravitation by three generations of illustrious mathematicians might be
presented from a single point of view. To this task the second period of Laplace's activity was
devoted. As a monument of mathematical genius applied to the celestial revolutions, the
Mécanique céleste ranks second only to the Principia of Newton.
Huygens, Christiaan (1629-1695)Dutch physicist who was the leading proponent of the wave
theory of light. In Traité de la Luminère (1690), he developed the concept of the wavefront, but
could not explain color. The wave theory, however, was supported by the observation that two
intersecting beams of light did not bounce off each other as would be expected if they were
composed of particles. In contradiction to Newton, Huygens correctly believed that light must
travel more slowly when it is refracted towards the normal, although this was not proven until
experiments by Foucault in the nineteenth century.Huygens also made important contributions to
mechanics, stating that in a collision between bodies, neither loses nor gains "motion" (his term
for momentum ). He stated that the center of gravity moves uniformly in a straight line, and gave
the expression for centrifugal force as, he studied pendula. He discovered Titan and was the first
to correctly identify the observed elongation of Saturn as the presence of Saturn's rings. Huygens
was also the mentor of Leibniz in math and mechanics.
Leonhard Euler, (born April 15, 1707, Basel, Switz.—died Sept. 18, 1783, St. Petersburg,
Russia), Swiss mathematician and physicist, one of the founders of pure mathematics. He not
only made decisive and formative contributions to the subjects of geometry, calculus, mechanics,
and number theory but also developed methods for solving problems in observational astronomy
and demonstrated useful applications of mathematics in technology and public affairs.Euler’s
mathematical ability earned him the esteem of Johann Bernoulli, one of the first mathematicians
in Europe at that time, and of his sons Daniel and Nicolas. In 1727 he moved to St. Petersburg,
where he became an associate of the St. Petersburg Academy of Sciences and in 1733 succeeded
Daniel Bernoulli to the chair of mathematics.By means of his numerous books and memoirs that
he submitted to the academy, Euler carried integral calculus to a higher degree of perfection,

3. developed the theory of trigonometric and logarithmic functions, reduced analytical operations to
a greater simplicity, and threw new light on nearly all parts of pure mathematics. Overtaxing
himself, Euler in 1735 lost the sight of one eye. Then, invited by Frederick the Great in 1741, he
became a member of the Berlin Academy, where for 25 years he produced a steady stream of
publications, many of which he contributed to the St. Petersburg Academy, which granted him a
pension. In 1748, in his Introductio in analysin infinitorum, he developed the concept of function
in mathematical analysis, through which variables are related to each other and in which he
advanced the use of infinitesimals and infinite quantities. He did for modern analytic geometry
and trigonometry what the Elements of Euclid had done for ancient geometry, and the resulting
tendency to render mathematics and physics in arithmetical terms has continued ever since. He is
known for familiar results in elementary geometry; for example, the Euler line through the
orthocentre (the intersection of the altitudes in a triangle), the circumcentre (the centre of the
circumscribed circle of a triangle), and the barycentre (the “centre of gravity,” or centroid) of a
triangle. He was responsible for treating trigonometric functions—i.e., the relationship of an
angle to two sides of a triangle—as numerical ratios rather than as lengths of geometric lines and
for relating them, through the so-called Euler identity (eiθ = cos θ + i sin θ), with complex
numbers (e.g., 3 + 2 √(-1) ). He discovered the imaginary logarithms of negative numbers and
showed that each complex number has an infinite number of logarithms.Euler’s textbooks in
calculus, Institutiones calculi differentialis in 1755 and Institutiones calculi integralis in 1768–
70, have served as prototypes to the present because they contain formulas of differentiation and
numerous methods of indefinite integration, many of which he invented himself, for determining
the work done by a force and for solving geometric problems; and he made advances in the
theory of linear differential equations, which are useful in solving problems in physics. Thus, he
enriched mathematics with substantial new concepts and techniques. He introduced many current
notations, such as Σ for the sum; ∫n for the sum of divisors of n; the symbol e for the base of
natural logarithms; a, b, and c for the sides of a triangle and A, B, and C for the opposite angles;
the letter “f ” and parentheses for a function; the use of the symbol π for the ratio of
circumference to diameter in a circle; and i for √(-1) .After Frederick the Great became less
cordial toward him, Euler in 1766 accepted the invitation of Catherine II to return to Russia.
Soon after his arrival at St. Petersburg, a cataract formed in his remaining good eye, and he spent
the last years of his life in total blindness. Despite this tragedy, his productivity continued
undiminished, sustained by an uncommon memory and a remarkable facility in mental
computations. His interests were broad, and his Lettres à une princesse d’Allemagne in 1768–72
were an admirably clear exposition of the basic principles of mechanics, optics, acoustics, and
physical astronomy. Not a classroom teacher, Euler nevertheless had a more pervasive
pedagogical influence than any modern mathematician. He had few disciples, but he helped to
establish mathematical education in Russia.
JOSEPH-LOUIS LAGRANGE
Both France and Italy claim Joseph-Louis Lagrange (January 25, 1736 – April 10, 1813), the
greatest
analytical mathematician of his time, as their own. In 1788, hepublished his masterpiece
Méchanique analitique (Analytic Mechanics), which is significant for its use of differential
equations.In it mechanics is developed algebraically and a wide variety of problems are solved

4. by the application of general equation. s. He laid the foundation for finding a different way of
expressing Isaac Newton’s Equations of Motion with what is now known as Lagrangian
Mechanics. Given the subject we might find it odd that the book did not have a single diagram or
construction. But Lagrange explained: “One will not find figures in this work. The methods that I
expound require neither constructions, nor geometrical or mechanical arguments, but only
algebraic operations, subject to a regular and uniform course.”
Lagrange’s mathematical reputation was established when, in 1755, he wrote a letter to Leonhard
Euler
in which he outlined his idea of a general method of dealing with “isoperimetric” problems, one
of the
motivating problems for the development of the calculus of variations, a name supplied by Euler
in
1776. Euler, who had also researched the problem, recognized that the younger man’s solution
was
superior to his own, and withheld publication of his result so Lagrange could get full credit for
the
invention. The Euler-Lagrange equation is a fundamental differential equation that gives the
necessary
conditions for solving classical problems dealing with paths, curves, and surfaces in the calculus
of
variations.
Caroline Hershel In 1781, William discovered the planet Uranus and astronomy became his
livelihood, with his sister by his side. It was only while William was away that Caroline was able
to make her own observations, discoveries which guaranteed her place in history.
On August 1, 1786, Caroline discovered her first comet and became history's first woman with
this distinction. Her comet came to be known as the "first lady's comet" and brought with it the
fame that secured her own place in history books.
Major Contributions
The primary discoveries Caroline Herschel made were the foundings of three nebulae in 1783.
The first one, NGC 2360, was an open cluster and an original discovery made February 26, 1783.

5. M 48, also known as NGC 2548, was an open cluster and an individual discovery made March 8,
1783. The third nebulae, NGC 6866, was an original discovery made July 23, 1783 and was
found to be an open cluster as well. Other space sky objects revealed by Caroline include the
well-known "elliptical M110(NGC 205), the evident spiral galaxy NGC 253, and the second
satellite of the Andromeda galaxy M31." The spiral galaxy NGC 253 is acknowledged as a
"starburst
The second satellite of the Andromeda galaxy. Photo courtesy of seds.org
galaxy, where stars form and explode" at a rate that is immense. It is also one of the brightest
galaxies. M110(NGC 205) is a rather small spheroidal instead of a regular elliptical. The
Andromeda galaxy is large and comparable by its looks to our galaxy. It is so large that its size is
"more than four times the width of the full moon."[⁶]
In the period of 1786-1797, Caroline discovered eight comets. Six of the eight comets are named
after Herschel. The comets ranged from a 3 to a 7.5 magnitude. Caroline recorded all of her
observations and discoveries in her own catalog. The evening of August 1, 1786, was when she
first layed eyes through her telescope on her original comet. This comet is known as "Comet
C/P1 (Herschel)." The sky that night was not exactly in the best conditions, therefore delaying
confirmation till the following evening. "Comet C/P1 (Herschel) was a brighter comet than most.
Caroline's final comet discovery came on August 14, 1797 and it faded quickly. She could not
have made these discoveries possible without learning from her brother. The eight comets
discovered by Herschel remained a record up until the 1980s. Carolyn Shoemaker, who's name is
fairly close to Caroline, broke that record for women astronomers.Caroline Herschel contributed
majorly to recording the research between William and herself in catalogues that were later
published. Without the assist of Caroline's strong ability to calculate equations from the
observations, William's research would not have accurate results. Caroline put in enormous effort
and time into her study of astronomy and in return got outcomes that gave her success. Her major
contributions were recognized and awarded with tremendous honors. She was even granted a
salary by the King and also became appointed into several distinguished societies and academies.
Caroline Herschel earned the title as "The First Lady of Astronomy" because of her eight comet
discoveries and the research work she shared with her brother William. Not only did she
contribute in the field of science but in the field of music by sharing the gift of her talented
soprano voice that landed her performances at opera houses across England.
LEGENDRE

6. Most of his work was brought to perfection by others: his work on roots of polynomials inspired
Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in
statistics and number theory completed that of Legendre. He developed the least squares method,
which has broad application in linear regression, signal processing, statistics, and curve fitting;
this was published in 1806 as an appendix to his book on the paths of comets. Today, the term
"least squares method" is used as a direct translation from the French "méthode des moindres
carrés".
In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, which was also proven by
Lejeune Dirichlet in 1828.
In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss;
in connection to this, the Legendre symbol is named after him. He also did pioneering work on
the distribution of primes, and on the application of analysis to number theory. His 1798
conjecture of the Prime number theorem was rigorously proved by Hadamard and de la Vallée-
Poussin in 1896.
Legendre did an impressive amount of work on elliptic functions, including the classification of
elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions
and solve the problem completely.
He is known for the Legendre transformation, which is used to go from the Lagrangian to the
Hamiltonian formulation of classical mechanics. In thermodynamics it is also used to obtain the
enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy. He is also the
namegiver of the Legendre polynomials, solutions to Legendre's differential equation, which
occur frequently in physics and engineering applications, e.g. electrostatics.
Legendre is best known as the author of Éléments de géométrie, which was published in 1794
and was the leading elementary text on the topic for around 100 years. This text greatly
rearranged and simplified many of the propositions from Euclid's Elements to create a more
effective textbook.

7. Music and behavior. OK, add kids and teenagers to the equation. Music leads tofeelings, which lead
to behavior. And sometimes bad behavior! Or alternatively, kids experience music in the context
of entertainment - kids are smart enough to enjoy it without transferring the emotional effects to family and
relationships, and it does not necessarily affect how young people act in real life. This has been an
ongoing public debate for decades.
Music Can Rev Up Our Emotional Brain
Dr. Rich talks about how music can affect the limbic system of the brain, which is the seat of our
emotions. When fully engaged emotions can be very powerful, and can overwhelm the other brain
functions such as rational thinking of the neocortex.
There’s no doubt that our moods and attitudes can be affected by the music we listen to. Dr. Rich gives
the example of how music affected and helped to define a generation and the anti-war movement of the
60′s. I know in my own life that music played a big part in developing my values, and in shaping the ways
I contribute to the world creatively as an adult. There’s a lot of controversy about the effects of aggressive
and jarring music on kids - such as metal andgangsta rap - and very little research that offers clear
conclusions.
A Balancing Act
There is a suggestion here that adults should be involved in their children’s music and media
consumption, and of course, it’s a balancing act. Teenagers must be able to express their individuality
and join in peer activities like they’ve always done. And at the same time adults need to engage with their
children and help guide them where needed, in order to make sure their lives and important activities stay
in balance.
Awareness and Guidance
The bottom line is that we need to be aware of, and responsible for the media influences on our families.
The music industry and government regulators, such as the Federal Communications Commission,
cannot be responsible for policing what is proper or improper for all of us to be exposed to. That’s up to us
to be aware of, and guide the music and media consumption habits of our families. Of course this will vary
from one family to the next according to our tastes, values and comfort level with pop culture.