Astrology & Astronomy in the Middle
By Michael Johnathan McDonald.
Why are they Important?
is credited with the most accurate astronomical observations
of his time, and the data was used by his assistant J. Kepler to derive
the laws of planetary motion.
Key Phrase: so
many redundant observations.
Tychonic system became the major competitor
with Copernicanism, and was adopted by the Catholic Church
for many years as
its official astronomical conception of the
Tycho Brahe , born Tyge Ottesen
Brahe (December 14, 1546 – October 24, 1601), was a Danish nobleman astronomer as well as an astrologer and alchemist. He
was granted an estate on the
island of Hven and the funding to
build the Uraniborg, an early research institute, where he built large astronomical instruments and took many careful
measurements. As an astronomer, Tycho
worked to combine what he
saw as the geometrical benefits of the Copernican system with
the philosophical benefits of the Ptolemaic system into his own
model of the universe, the Tychonic
system. From 1600 until his
death in 1601, he was assisted by Johannes Kepler, who would
later use Tycho's
astronomical information to develop his own theories of
He purchased an ephemeris and
books such as Sacrobosco's Tractatus de
Sphaera. Tycho's naked eye measurements of
planetary parallax were accurate to the arcminute. These jealously guarded measurements
became the possessions of Kepler following his
death. Tycho was the last major astronomer to
work without the aid of a telescope, soon to be
turned toward the sky by Galileo.
11, 1572, Tycho observed a supernova for a ' new
'star.' This discovery was decisive for his
choice of astronomy as a profession.
Kepler tried, but was unable, to
persuade Tycho to adopt the heliocentric
model of the solar system. Tycho believed in
a modified geocentric model known as the
Tychonic system, for the same reasons that he
argued that the supernova of 1572 was not near
the Earth. He argued that if the Earth were in
motion, then nearby stars should appear to shift
their positions with respect to background
stars. In fact, this effect of parallax does
exist; it could not be observed with the naked
eye, or even with the telescopes of the next two
hundred years, because even the nearest stars
are much more distant than most astronomers of
the time believed possible. The Tychonic
is very similar to the Copernican one, except
that it has a static earth instead of a static sun.
Brahe 's religiosity and firm belief in
astrology including the desires to reform
by these methods inspires to calculate
more accurate positions for the planets. He
and constant physical observation.
He divides the universe again, with the
induction of what are
comets and paths of the
tail. Therefore the unchanging and changing are
trimmed down a little as
comments link the
corruptible to the incorruptible - the changing
and the unchanging.
had trouble with his water clock, apparently the mercury flux,
so this little problem helped him create new methods to
calculate the 'natural positions' of the stars.
Problem sets for Tycho
Problem? A lousy clock at Uraniborg.
Significance: This problem' solution created the Following technique
to correct this problem.
Couldn't do Right Ascension.
Most reliant on the sun.
Measured from the sun.
Needed intermediate marker because one
cannot look at the sun and keep one's eyes in tact.
Used Venus. In 1582 he had a good view of
Then he measured the angular distance
before Venus to the sun.
Now he has a reference point.
After sun goes down, he uses Venus to
make another reference point. This created a good
Then once he gets a star for a reference
point, he later used a Ran star, in the zodiac
constellation of Aries.
This technique created a good calculation
of arc. In fact, down to seconds of arc. Ptolemy would
be happy with four-minuets of arc.
Now a problem with the tilt of the earth
How do I figure in the variables for the tilt of axis of earth?
Tycho had to revert to trigonometry.
Finding the moving sun.
= 90 - Ф - Є
= 90 - Ф + Є
+ αss = 180° - 2Ф
≠ Ф (north pole).
To get around the seven
digit calculations, a tedious process,
he decided to cheat. Here is the
cosA cosB = ½
[ cos (A-B) - cos (A+B) ]
This is the way he got
around the long seven digit
Refraction of light raises
the star from the earth's vantage-point
because of the atmospheric disturbance. However, to
make correct observations, the stars at the zenith
remained with a radius of strait-light, i.e. no
refraction. Therefore, Tycho's refraction tables
remain pretty good.
his parallax measurements were not so good.
Parallax: Parallax depresses the
Comets and Tycho
the first to put a circle trajectory with a comet that goes around
the sun and the tail opposite of the sun. Comet of 1577,
corresponded to his parallax studies. Tycho's
system of comets helped his theory of orbits of planets around the
around the sun but the earth is still the center of the universe. Jesuits
adopted this until the 1700s.
was the supreme existence in the universe, to place it not at the
center of the universe created a heretical issue.
improved the accuracy of arc by a factor of 10 over Copernicus.
Still small discrepancies needed addressing in which Kepler will
True founder of the New
True founder of the New
A new celestial geometry. Three laws of
Most important fact: Used the true
sun and not the mean sun.
Very important cultural significance on
schools: He entered school on a pilot program to give
under representative students a chance at an education.
This was a
limited experiment, but an early example of public programs to
represent good students who studied but were otherwise too poor to
pay for school.
Personal-cultural significance: He is a
product of the Reformation.
The three laws:
- Kepler's elliptical orbit
law: The planets orbit
the sun in elliptical orbits with the (true) sun
at one focus.
- Kepler's equal-area law:
The line connecting a planet to the sun sweeps
out equal areas in equal amounts of time.
- Kepler's law of periods:
The time required for a planet to orbit the sun,
called its period, is proportional to the long
axis of the ellipse raised to the 3/2 power.
constant of proportionality is the same for all
the planets. (more in-depth below).
Johannes Kepler (December 27, 1571 – November 15, 1630), a key
figure in the scientific revolution and the
true founder of the New
Astronomy, was a German
mathematician, astrologer and imperial astronoger, and astronomer. He is best known for his
laws of planetary motion, based on his Astronomia nova,
Mundi and the textbook Epitome of Copernican Astronomy.
Through his career Kepler was a mathematics teacher at a Graz
(Southern Austria and heavily Catholic) seminary school (later the University of Graz), an assistant to
Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics
teacher in Linz, and court astrologer to General Wallenstein. He
also did fundamental work in
the field of optics and helped to
legitimate the telescopic discoveries of his contemporary Galileo Galilei.
after moving through grammar school, Latin school, and lower and
higher seminary in the Lutheran education system, Kepler began
attending the University
of Tübingen as a theology student,
where he proved himself to be a superb mathematician and earned a
reputation as a skillful astrologer. It is here he met and then
learned under a teacher he says profoundly influenced him.
Michael Maestlin (1550-1631) was a German astronomer and
mathematician. He was a professor at the
University of Tübingen for
47 years. The two corresponded for some. Although he primarily
taught the Ptolemaic view solar system, he was also one of the first
accept and teach the Copernican view in which Kepler fell in love
left Graz after a new leader decided to kick-out all protestants,
and gave him one week to leave. He had roots and a family. He left
for Prague. About the same
time Tycho had a similar fate, in that
his King patron died and his King's son had a different attitude
about Tycho's investigations and its significance. So Tycho
up his equipment and moved to Prague. This is were the two met.
Kepler needed to win Tycho's trust and finally was given the job of
finishing the Tychonic
Universe before Tycho passes away from a
As Imperial Mathematician,
Kepler inherited Tycho's responsibility for the Emperor's horoscopes
as well as the commission to produce the Rudolphine Tables
(1627). The Rudolphine Tables, his final project and dedicated
to his former patron King Rudolpho II, proved the
greatest measurements of planet’s true positions up till this time.
He manages to predict them within one second of arc. Ptolemy would
be happy with a few minuets of arc.
His most significant
achievements came from the realization that the planets moved in
elliptical, not circular, orbits. This realization was a direct
consequence of his failed attempt to fit the planetary orbits within polyhedra. Kepler's willingness to abandon his most cherished theory
in the face of precise observational evidence also indicates that he
had a very modern attitude to scientific research. Kepler also made
great steps in trying to describe the motion of the planets by
appealing to a force which resembled magnetism, which he believed
emanated from the sun. Although he did not discover gravity, he
seems to have attempted to invoke the first empirical example of a
universal law to explain the behaviour of both earthly and heavenly
Architectonics come to him after contemplating on the musical
theory and his old teachings of the astrology phenomenon called
the trigon cycle.
True founder of the New
He opposed an infinite Universe.
Thought the universe contained,
or more aptly, composed of ice.
Philosophy: Cosmos not formed by chance.
God had a reason and a order. God had a plan, not random. No
Sought underlying reason.
Concerned with finding the
constructional laws, also referred too as 'Architectonics'.
Kepler was the first, and the
only one up to the time of Descartes, to ask for a physical
Kepler always raised questions
that others never thought of or were to scared to ask. This
significance brought progress and a ' New Astronomy'.
He was the first to conceive and
conscientiously follow, if not realize, a programme directed to
the scientific unification of the starry Universe and the
sublunary Universe, and substitute celestial dynamics for
kinematics of circles and the metaphysics of spheres in
traditional astronomy (Koyré
He based his dynamics on
by ascribing to the Sun a spiritual, perhaps even intelletive,
driving force, and to the Earth not only a spiritual,
but also a
sensory driving force (Koyré
placed the true sun instead of the mean sun as the measurement.
'Vicarious hypotenuse,' in trying a simple Ptolemy's scheme is
his first approach to mars.
laws of motion.
Kepler is called the calculator. He pursues his seven year war
Copernicus's center is the mean sun, not the true sun. The mean
sun is our clock, but everyone knows out clock is not accurate.
Aristotle became the first person to discuss and write upon the
universe and astronomy. His cosmology would gain much acceptance
up until the 1500’s, where
Copernicus and Galileo began to
figure out that Europe is not the center of the universe. As a
young man it is astrology that starts Kepler in the direction of
life, First question
was the calendar reform for him.
He is sometimes referred to as "the first theoretical
astrophysicist", although Carl Sagan also referred to him as the
last scientific astrologer.
Johannes Kepler Trivia:
He had a miserable beginning.
His mother was a witch and tried,
and he represented her in court and won.
His father was a vagabond,
He was premature at birth and
Poor and needed affirmative
action to get into school. The program
recognized his brilliance.
He did his horoscope to make sure
he came from a bad sign - he wanted to
understand why he was the way he was.
He said when he was little he
gnawed on bones and ate hardened bread -crust.
He received a good education,
mainly in Greek as the northern schools
He, of course, trained as an
astrologer as was the curriculum which helped
him calendar problems.
Kepler began attending the
University of Tübingen as a theology student,
where he proved himself to be a superb
mathematician and earned a reputation as a
He learned Copernicus in school,
at this time a few places were beginning to use
his work. He fell in love with the ideas of
He said this is religion. The Sun
is a better proxy of God, then the earth.
We need a triune God. The Sun,
the Earth and the emptiness is the holy ghost.
Kepler wanted to become a pastor.
Why are there six planets in Kepler's universe when Ptolemy
dropped the moon because it couldn't fit into his new
geometrical system of distance.
Sothern Austria was heavily Catholic, but for a brief period
of time in certain regions Protestants moved in while the king
allowed this due to peace. He was a
teacher, lecturer of
astrology and math. He took Stadius' position. Then the king died and his son reverting
back to the old norms asked for everyone to get out. He
seven days to move, although, Kepler had roots with family and kids
in Graz. Kepler noted the astrology charts of bad omens for
Protestants in Graz. At
Graz, Kepler publishes prophesying
almanacs and predicts Turkish events. The archduke kicked him
out of Graz because he wanted to impose his religion on the
He left to Prague. Tycho left
for Prague after being kicked out/ or leaving his island observatory
do to political changes too. His patron, the king died and the
one decided the island was more useful for other reasons. They met
up in Prague. They argued together because instantly Kepler wanted to see all
the data that
Tycho had compiled, but Tycho was apprehensive and a
little secretive. Rudolpho II (1552-1612, Roman emperor, son of the
emperor Maximilian II ) employed Tycho
and possibly the
astronomer/astrologer didn't want his data open to others to get
work. However, Kepler finally signed on to Tycho's work and after
nobleman died he retained all his data and Rudolpho hired
him. Rudolpho was fond of astrology.
died and he went back to Tubingen and talked for a commission of his
architectonic structure in a cup of silver, but it didn't happen.
This why we only have a
written record of this.
not wanting to be rude at state function and get up from the
table and go to the bathroom, split his gut and died.
an Imperial Mathematician, Kepler took responsibility for
the Emperor's horoscopes as well as the commission to produce
the Rudolphine Tables.
Jesuits, ironically, noticed his brilliance and asked him to return,
in which he did.
longer taught at a school, but was used as a computer to calculate
mathematical sets. This experience gave him confidence and expertise
along with speed.
Kepler and Religious references to his work.
Kepler wanted to become a pastor before he was side tracked by the
love of Copernicus' universe. But his
theories remain cemented in
religion. Kepler: two universes the 'mobile', now only five
bodies and the unmovable (stars). There are six bodies in motion
Sun. The Sun is motionless
and the driving force.
His trinity theory appears
inspired by Nicholas of Cusa - architectonic...( Koyré, 154). The
reasoning mearly points out that Copernicus'
universe is not heliocentric, but heliostatic.
Therefore the Sun plays a larger role and the
earth plays a less significant role in the
Universe contrary to Ptolemy as it does in
Copernicus' work. (Koyré,
God the father was the Sun. The
empty space the holy ghost. Earth was imperfect
and so it goes was hell. The sun was paradise.
Johannes Kepler's Polyhedra
Trigon of astrology
"I see the
secret of the Universe" - Kepler.
Trigon of astrology leads him in a direction to explains why
there are only six planets. Kepler creates
charts of trigon conjunctions of Jupiter and Saturn and he
something that becomes somewhat of a satori for him. When the table
is complete he noticed a circle in the center absent of anything but
space. He therefore
knew that this was the difference in diameter of
Jupiter and Saturn. Therefore, he quickly decided to understand if
all the planets could be geometrified so that they
would all relate
to each other and in a form and also correspond to a geometric
structure. However, this proved inaccurate or a point that beauty
and nature are not
always complimentary. It is not perfect, but
close, and he eventually let it go which was a first example of how
modern astronomers should work. If it doesn't work,
then move on.
However, it had significant meaning to his life and his discoveries.
leads to his great discovery on 'Architectonics'
five essential shapes are the only
conceivable solids whose sides and edges
are exactly the same. They are also the
only perfectly symmetrical
Icosahedron (20 sides) :Dodecahedron (12
sides) :Octahedron (8 sides) :Hexahedron,cube
(6 sides) :Tetrahedron (4 sides)
Kepler lectured on trigons and he no doubt
became an excellent intellect upon its mysteries. This, of course,
fitted into astrology as he discussed fiery trigons and repeatable
element conjunctions. When he came upon his satori, and
decided to place a triangle into the orbit of schemes, he came
across his brilliant discovery. What he needed to do he did
and dropped our moon as a planet to a mere satellite (Copernicus was
the first to drop the moon as a planet) and implanted
the six remaining spheres into these above geometric perfect shapes. Then
calculated their radiuses.
- Take 1000. Calculations below.
- The relationship between the radii and the
|| By Solids. Kepler
agreement is decidedly not perfect, but it is too significant to be
accidental. The discrepancy between the data and calculation is
considerable in the case of
Jupiter and Mercury, though we ought
not to be surprised thereat in view of the enormous distance which
separates us from the former, and the difficulty in providing
satisfactory theory of the motion of Mercury" ( Koyré
geometric relationship served Kepler as a mental model of the solar
system" (1.1 Kepler). The crack-pot theory proved genius.
Copernicus, Kepler concluded had
made the mistakes. This was a
confident move, knowing that Copernicus' intellect remained admired.
" Kepler said. " This crackpot idea is better than Copernicus."
must be wrong, not me." This initial astrology knowledge led to the
three laws of motion. Copernicus happened to use the mean sun
instead of the true sun,' " in
order to shorten the
calculations' and ' in order not to alarm his readers by departing
too much from Ptolemy' "(Koyré
of the planets believed in ancient times to have the same velocity.
Kepler disagreed. " The periods of orbits are 'not' directly
proportional to the distance, but differ considerably from therefrom..." (Koyré
150-1). See the soul and the further away from
the sun the weaker the 'motive.' For animal Motive and material motive and adoption then late change of mind, see Joseph Scaliger
influence with Kepler. The further away from the sun the slower the
souls ( Force) moved the 'movables.' The planets had moderate
activity and the stars no activity, but the Sun had the greatest
because it was the force that moved the planets around itself, just
as God is the prime mover of life.
trinity. God is the trinity. Therefore,
image is revealed in the sphere by God the Father at the center, God
the Son on the surface, God the holy Ghost mid-way ( emptiness)
between the center point and the circumference (Koyré
143), ( See satori of trigon, above) .
Why did the ancients think the Earth was the center of the Universe?
understand that the Earth spun on its axis. Therefore, if the
earth went all the way around the sun in one day - it would
fling-it-out into the universe in a violent motion. Thus, the
earth was the center of the Universe.
" 'If the Earth revolved?, said Ptolemy of Alexandria, that is
to say made one revolution in a
day.... this motion, which is twenty-four hours passes over the
whole circumference of the Earth, would-be extremely violent and
of unsurpassable speed.. Now things move in a violent motion
seem to be totally unable to gather together, but rather to
disperse...'" (Koyré 112).
The 7 Year War on
Mars was actually 9 years
'Vicarious hypotenuse,' in trying
a simple Ptolemy's scheme is his first approach
to mars. 900
pages of calculations comes to
naught, but the process leads to understandings
that will factor into his discoveries about
motion. Kepler talked to Tycho on his deathbed
and was giving Tycho's works with promises that
he finish the Tycho universe.
Three laws of planetary motion.
Architectonics, the platonic solids and
the placements of the planets and their orbits.
Music of the spheres (two day's
before the publication of Harmonice Mundi, Rudolpho paid for
the publishing but not Kepler's salars ): Kepler, decided to
take a break from his long war with mars and picked up a book on
Music theory. He needed a change of pace from his 900 logins from
his pages on Mars. Chronologically this is his second law. By using
notes visualized on a music sheet he maps out the eccentricities of
the planets. Therefore, Mars has the greatest musical variation
while Venus has the least. The object was to get the pitch ( tone)
of the planets. Venus is almost a circle in its orbit
therefore the notes on the music page are all the same notes.
would indicate a pitch for Venus.
Area/time = constant= total area/period
is chronologically his third law, but it is called his second law.
- Kepler's elliptical orbit law: The
planets orbit the sun in elliptical orbits with
the sun at one focus.
- Kepler's equal-area law: The line
connecting a planet to the sun sweeps out equal
areas in equal amounts of time.
- Kepler's law of periods: The time
required for a planet to orbit the sun, called
its period, is proportional to the long axis of
the ellipse raised to the 3/2 power. The
constant of proportionality is the same for all
Titius-Bode Law is rough rule that predicts the spacing of the
planets in the Solar System. The relationship was first pointed out
by Johann Titius in 1766 and was formulated as a mathematical
expression by J.E. Bode in 1778. It lead Bode to predict the
existence of another planet between Mars and Jupiter in what we now
recognize as the asteroid belt [2.8] (Bode).
||Actual distance (A.U.)
Uranus, but he knew it was out there by this law. Somehow this law
skips Neptune and the next value is right smack on Pluto, but many
believe it is an actual planet.
amount of "flattening" of the ellipse is termed the eccentricity".
(The Online ).
Law one is actually his second law in regards to chronology of
I. Elliptical orbit law.
The orbits of the planets are ellipses,
with the Sun at one focus of the ellipse.
Law is illustrated in the image
shown above. The Sun is not at
the center of the ellipse, but
is instead at one focus
(generally there is nothing at
the other focus of the ellipse).
The planet then follows the
ellipse in its orbit, which
means that the Earth-Sun
distance is constantly changing as the planet goes around its
orbit. For purpose of
illustration we have shown the
orbit as rather eccentric;
remember that the actual orbits
are much less eccentric than
this" (The Online ).
circle, bi-sect and equant point). He knew it wasn't true, but still
leads him in a correct direction.
back out of his failure for Mars a rule. Equal times sweep out equal
distances: arcs = areas . He gets rid of the eqaunt point.
to the Area law ( the absence of an equant). He asks what is the
shape of this stuff ( the area)? He uses computations from Tycho,
and now defines the shapes of the sectors in a circumference. XC=CS,
as X increases with time. He wrote 900 pages on this.
The eccentric circle area doesn't work either, but it
gives him a clue. What about equal quadrants of time.
Therefore = arcs= areas. The Area Law is now absent of
all equants. He asks "what is the shape of this stuff?"
( that is the quadrants?). After he defines the shape.
In a letter of a friend in playfulness
says. " Circles are the definitions of Astronomy," his
friend says, " you destroyed Astronomy."
Kepler writes back: "if matter were more
tractable, then of course, stars and planets would
travel in circles." He is saying that if the planets and
stars traveled in a circle Astronomy would have been
solved many millennia ago, in which he is correct.
Astronomia nova (1609) has the
presentation of the law.
Equal areas in
Chronology this is his
II. The Area Law.
The line joining the
planet to the Sun
sweeps out equal areas in equal time
as the planet travels around the ellipse.
Arcs = Area
"Kepler's second law is illustrated in the preceding figure. The
line joining the Sun and planet sweeps out equal areas in equal
times, so the planet moves faster when it
is nearer the Sun. Thus, a
planet executes elliptical motion with constantly changing angular
speed as it moves about its orbit. The point of nearest approach of
planet to the Sun is termed
perihelion; the point of greatest separation is termed aphelion.
Hence, by Kepler's second law, the planet moves fastest when it is
perihelion and slowest when
it is near aphelion" (The Online ).
III. The Harmonic law -
The time required for a
planet to orbit
the sun, called its period, is proportional
to the long axis of the ellipse raised to
the 3/2 power. The constant of
proportionality is the same for all the
planets. (more in-depth below).
Kepler's 3rd Law:
P2 = a3
= total area/period= (t2/r3)
see that column one is off and Mercury clearly must be
downsized and Saturn must be increased. So Kepler, took t2/r3
and found out all the r=1.0.This became his third law.
law is a mathematical formula.
It means that if you know the
period of a planet's orbit (P =
how long it takes the planet to
go around the Sun), then you can
determine that planet's distance
from the sun (a = the semimajor
axis of the planet's orbit),
because P2 = a3.
t also tells us that planets
which are far away from the Sun
have longer periods than those
close to the Sun. They move more
slowly around the Sun. Look at
the formula: if you make the
distance, a, large, then P must
also be large
" Kepler's Third
Law implies that the period for
a planet to orbit the Sun
increases rapidly with the
radius of its orbit. Thus, we
find that Mercury, the innermost
planet, takes only 88 days to
orbit the Sun but the outermost
planet (Pluto) requires 248
years to do the same" (The
Universe is 4 million Solar diameters. He makes it three times larger than Tycho's.
saturn /d saturn = d saturn
after Rudolpho dies, dedicated to his parton's name, his planetary
tables, called the Rudolpho tables.
final work is the Epitome of Copernican Astronomy. He tried
to derive something that Newton about 70 years later get on too. The
magnification attraction of the planets and the sun. This was
mathematical physics applied to astronomy, which would end up as a
new field with Newton.
this was the essence of God's thoughts on creating the universe. He
moved after a Protestant crackdown and hooked up with Wallenstein
who gave him shelter.
disdained astrologers who pandered to the tastes of the common man
without knowledge of the abstract and general rules, but he saw
compiling prognostications as a justified means of supplementing his
meager income. At least 800 horoscopes and natal charts drawn up by
Kepler are still extant, several of himself and his family,
accompanied by some unflattering remarks.
Bode's Law Department
of Astronomy. ( Ithaca: NY Cornell University, 2006),
Khun S. Thomas. The Copernican
Revolution 'Planetary Astronomy in the Development of Western Thought'. by
the President of Fellows of Harvard. 1957. r. 1985. 30,32.
Heilbron, J. L. Astronomy and Astrology in
Medieval and Early Modern Europe. Represented in University of California
Berkeley Reader. History 181A. January 2006.
( Astrolabe/23 J. L. Heilbron).
1.1 Kepler. <http://cti.itc.virginia.edu/~meg3c/classes/tcc313_inuse/Book/chap1/kepler.html>2006.
Koyré, Alexander , The Astronomical
Revolution: Copernicus-Kepler- Borelli 2nd ed. ( Ithaca: Dover Publications,
Inc, New York),1992.
Wilson, Jonathan .
Symmetry:two sides to every story
The Online Journey Through Astronomy,
Johannes Kepler: The Laws of Planetary Motion ,
University Corporation for Atmospheric Research, Windows
to the Universe, , 1995-1999, 2000, <http://www.windows.ucar.edu/tour/link=/the_universe/uts/orbits_data.html
Trigon and history with it.
Orbital Data for the Planets
2007 Michael Johnathan McDonald. Bookoflife.org . All rights
updates Darwin Plagerized theory of evolution 2014 A.D.
Calculus developed side-by-side by independently.
Linear calculus is stacking functions upon a Descartes grid system, cutting them later with conical linear frames.
but Leibniz did not write 'Principia' to which we call him ( Sir, Isaac Newton) the father of Physics. u just hate him because newton was white, bisexual and a genius Christian.
ur jews are the plagiarizers, bitch. It is not the white. in fact, can you look it up. a white scot figured out the electron count to make the first " Periodic table, ' and some French wasa Arab white dude, stole his idea ( he was a poor scot farmer with white dna, genius) and he worked for a while for this French aristocrat who had labs and money to make proofs of the electron counts.
so again, jews or cross breeds steal and plagiarize whites.