7 The Renaissance in Europe
7.1 Fibonacci and Notational Development
Between the fall of Rome in 476 and the early renaissance
38
(c. 1100) came Europe’s dark ages. The
one-dimensional view is that this was a time of limited learning and technological progress; as ever,
the reality was more complex. Learning persisted in monasteries, though new researches took second
place to a conservative focus on preserving the wisdom of the ancients.
By 1100, the small shifting kingdoms that had characterised Europe were starting to come together
as more stable feudal states.
39
The maps below show the major feudal societies of medieval Eu-
rope (France, England, Holy Roman Empire, Poland, Austria) consolidating. While it would have
meant little to the average peasant, a large stable nation is able to produce and support a larger elite
population with the time and money to pursue education, and fund libraries and universities. By
the 1700s, the borders of western Europe are largely recognizable; political and social organization
had expanded so that a much larger proportion of the population—though small in comparison to
today—were able to take advantage of and contribute to the growth of knowledge. The European
renaissance is often contrasted with a decline in Islamic power, but again the story is more complex:
Islam retreats from Spain, while the Ottoman Empire becomes dominant in south-east Europe.
Between 900 and 1300, the population of Europe roughly tripled to around 100 million. Trade in-
creased, along with which came knowledge.
40
Learning (and land) came via wars with Islam (the
Crusades, Spain, etc.). It helped that Islamic scholars so venerated the Greeks; Europeans could tell
themselves that they were merely ‘reclaiming’ ancient knowledge which had been ‘stolen’ by their
cultural and religious enemies. The fall of Constantinople to Mehmet in 1453 marks both the high
point of Islamic conquest and the start of the decline of Islamic scientific dominance. Many intellec-
tuals fled Constantinople—where the Byzantines still preserved much of Alexandria’s learning—for
Rome, helping to further fuel developments. With powerful enemies to the east, Europeans began
travelling greater distances by sea,
41
beginning the colonial era of global European empire.
38
Literally rebirth. Dates vary by location and discipline (Italy vs. France, art vs. philosophy, etc.) but a wide net would
encompass the 12
th
to 17
th
centuries.
39
An arrangement where powerful landowners could demand service (rent, food, warriors) from their tenants.
40
A particularly important trading hub was Venice, from where Marco Polo (1254–1324), perhaps the most famous trader
of the period, travelled the silk road to China.
41
Christopher Columbus (born Genoa 1451) famously ‘discovered’ America in 1492 while looking for sea routes to Asia.
68
Scientific and philosophical progress was spurred by the translation of works from Arabic and an-
cient Greek into Latin, with the first universities being formed to teach this canon: Bologna 1088,
Paris 1150, and Oxford 1167. The typical student was a young man of wealth who had been privately
tutored in grammar, logic & rhetoric (the trivium). At university he would study the Greek-influenced
quadrivium (geometry, astronomy, arithmetic & music). While Islamic improvements were incorpo-
rated, scholars gave pre-eminence to the Greeks: Euclid for geometry, Aristotle for logic/physics,
Hippocrates/Galen for medicine, Ptolemy for astronomy. Early universities were often funded by
the Church and ‘research’ was more likely to involve the justification of biblical passages using Aris-
totle than the conduct of experiments.
Leonardo da Pisa (Fibonacci c. 1175–1250) Fibonacci
42
likely first encountered the Hindu–Arabic
numerals while trading with his father in North Africa. He was impressed by the ease of calculation
they afforded and is the first European known to use them (contemporary Europeans used Roman
numerals and Egyptian fractions). Fibonacci’s 1202 text Liber Abaci was written to instruct traders
in their use. The first page below explains how to compute with decimal fractions, with the two
columns at the bottom of the page showing how to repeatedly multiply 100 (and then 10) by the
fraction
9
10
. Thus:
100, 90, 81,
9
10
72 (= 72.9),
1
10
6
10
65 (= 65.61),
9
10
4
10
0
10
59 (= 59.049), etc.
Note how the fractional part is written backwards on the left, using a bar to separate numerator and
denominator; the Indians wrote fractions without a bar and it is thought Islamic scholars inserted it
for clarity in the 1100s. The second picture is of Fibonacci’s famous sequence: read top-to-bottom 1, 2,
3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Amongst other inheritances from the Hindu–Arabic tradition,
Fibonacci is the first known European to work with negative numbers, provided these represented
deficiencies or debts in accounting.
42
The name was given to Leonardo by French scholars of the 1800s: filius Bonacci means son of Bonacci.
69
Algebraic Notation and Development
To a modern reader, the most obvious mathematical development of the renaissance is notational.
Fibonacci’s fractional notation was cutting-edge for the 1200s but essentially everything else except
numbers and fractions was written in sentences. The next 500 years slowly brought notational im-
provements which eventually allowed algebra to eclipse geometry as the primary language of rea-
soning. Here is a brief summary.
Italian Abacists, 14
th
C. This group continued Fibonacci’s advocacy for the Hindu–Arabic system
against the traditional use of Roman numerals, and also for the use of accompanying algo-
rithms. Their approach was highly practical and largely for conducting trade. Here is a typical
problem described by the group:
The lira earns three denarii a month in interest. How much will sixty lire earn in eight
months?
The Abacists introduced shorthands and symbols for unknowns and certain mathematical op-
erations. Cosa (‘thing’) was used for an unknown. Censo, cubo and radice meant, respectively,
square, cube and (square-)root. These expressions could be combined, for example ‘ce cu’ (read
censo di cubo) referred to the sixth power of an unknown (x
3
)
2
= x
6
.
Luca Pacioli, Italy late 1400s. Introduced p, m (piu, meno) for plus and minus. For example 8m2 de-
noted eight minus two.
Nicolas Chuquet, France 1484. His text Triparty en la science des nombres borrowed Pacioli’s p and m,
and introduced an R-notation for roots. For instance R
4
7 meant
4
√
7, while
5
q
4 −
√
2 would be written R
5
4mR2
The underline indicates grouping (modern parentheses).
Christoff Rudolff, Vienna 1520s. Introduced symbols similar to x and ζ for an unknown and its
square. He had other symbols for odd powers and produced tables showing how to multiply
these. The words he used show Italian and French influence: algebra in the German-speaking
world was known as the art of the coss (German for thing). Rudolff also introduced ± symbols
as algebraic operations, though these had been used for around 30 years as a prefix denoting an
excess or deficiency in a quantity (profit/loss in accounting). A period denoted equals and he is
also credited with the first use of the square-root sign
√
, which is nothing more than a stylized
r. This would be written in front of a number to denote a root, e.g.
√
23 rather than
√
23.
Robert Recorde, England 1557. Introduced the equals sign in The Whetstone of Witt, asserting, ‘No
two things are more equal than a pair of parallel lines.’
Francois Vi`ete, France 1540–1603. Before Vi
`
ete, mathematicians typically described how to solve
particular equations algorithmically via examples (e.g. x
3
+ 3x = 14 rather than x
3
+ bx = c),
expecting readers to change numbers to fit a required situation. Vi
`
ete pioneered the modern
use of abstract constants, using letters to represent both unknowns and constants.
70
Simon Stevin, Holland 1548–1620. De Thiende (The Tenths) demonstrated how to calculate using
decimals rather than fractions. Stevin arguably completed the journey whereby the concept
of number subsumed that of magnitude, asserting that every ratio is a number. This increased
the application of algebra by permitting the description of any geometric magnitude.
William Oughtred, England 1575–1660. Introduced × for multiplication, though he often simply
used juxtaposition. Oughtred combined Vi
`
ete’s general approach (abstract constants) with
symbolic algebra. For instance, to solve a quadratic equation A
q
+ BA + C = 0, where A
q
means ‘A-squared’ (A-quadratum in Latin) and B, C are constants, he’d write the quadratic for-
mula as
A =
√
:
1
4
B
q
−C : −
1
2
B
In Oughtred’s notation, colons were parentheses.
Thomas Harriot, England 1560–1621. Made several steps towards modern notation including juxta-
position for multiplication and a modern encompassing root-sign. For example,
4
q
cccc + 27aa
3
√
2 + b meant
4
q
c
4
+ 27a
2
3
√
2 + b
Ren´e Descartes, France/Holland 1596–1650. Used exponents for powers (a
2
, a
3
) and solidified the
convention of using letters at the end of the alphabet (x, y, z) for unknowns and those at the
beginning (a, b, c) for constants.
While modern mathematics uses many specialized symbols (∅, ⇒, e, π, etc.), basic notation is essen-
tially unchanged from the mid 1600s. This is not to say that all mathematicians uniformly used the
most modern notation: for instance, papers of Leonhard Euler (1700s) used Harriot’s juxtaposition
notation for exponents, and some published works from the late 1800s still wrote equations in words.
It is also worth mentioning in this context Gutenberg’s 1439 invention of the printing press, which
naturally aided the dissemination of all learning. The relative ease of production meant a great
increase in the availability of written material, but also in the rejection/abandonment of some older
texts when money could not be found to make an updated printed edition.
Exercises 7.1. 1. Repeatedly divide 10 by 5 a total of five times (to
10
5
5
), expressing the results using
Fibonacci’s notation.
2. What would Nicolas Chuquet have meant by the expression 4pR
3
7 mR5?
3. How would William Oughtred have expressed the solution to the quadratic equation A
q
=
BA + C? What about Thomas Harriot?
4. I am owed 3240 florins. The debtor pays me 1 florin the first day, 2 the second day, 3 the third
say, etc. How many days does it take to pay off the debt?
5. (A problem of Antonio de Mazzinghi) Find two numbers such that multiplying one by the
other makes 8 and the sum of their squares is 27.
(Hint: let the numbers be x ±
√
y)
71
7.2 Polynomials: Cardano, Factorization & the Fundamental Theorem
As an example of contemporary algebraic notation, we consider Girolamo Cardano’s 1545 Ars Magna
(Great Art or the Rules of Algebra), in which he describes how to solve quadratic and cubic equations.
The example on the right (start of Caput V, page 9 of the linked
pdf), is the beginning of Cardano’s description of how to solve
x
2
+ 6x = 91, quadratum & 6 res aequalie 91 (square and 6 things
equals 91), by completing the square. He employs several single-
letter abbreviations but still writes in sentences and provides a
pictorial justification. In modern algebra, the argument is nothing
more than completing the square:
x
2
+ 6x = 91 =⇒ x
2
+ 2 ·3x + 3
2
= 91 + 3
2
=⇒ (x + 3)
2
= 100
=⇒ x + 3 = 10
=⇒ x = 7
where the picture justifies 7
2
+ 2 ·21 + 3
2
= 10
2
.
The quadratic algorithms were well-established by this time; on
the following page of his text, we find a picture from the work
of al-Khw
¯
arizm
¯
ı (Exercise 6.2.2), after which comes more obvious
mathematical notation. Even though Rudolff’s ± and
√
were in
use, Cardano wrote almost everything in words augmented with
fractional notation and Pacioli’s p and m.
As was typical for the time, Cardano describes negative solutions
as fictitious; he even writes the square root of −15 at one point,
though only to mention its absurdity. He also follows the Islamic
approach of solving a concrete problem of each type rather than
proceeding abstractly, though observe the gratia exempli (“for the
sake of an example”) as acknowledgement that the general prob-
lem x
2
+ ax = b may be solved identically.
It is for the solution of cubic (and quartic) equations that Cardano
is most famous. Below we describe, in modern notation, Car-
dano’s method for solving the cubic equation x
3
+ bx = c where
b, c > 0, though we stress (again) that Cardano only gave examples
not a general formula.
Let u, v satisfy u
3
−v
3
= c and uv =
b
3
. Then x = u −v is seen to solve the cubic.
x
3
+ bx = (u − v)
3
+ b(u − v) = u
3
−3u
2
v + 3uv
2
−v
3
+ b(u − v)
= (u
3
−v
3
) + (u − v)(b − 3uv) = c
However u and v also satisfy
(u
3
+ v
3
)
2
= (u
3
−v
3
)
2
+ 4(uv)
3
= c
2
+ 4
b
3
3
72
so we obtain a system of linear equations in the unknowns u
3
, v
3
which are easily solved:
u
3
+ v
3
=
r
c
2
+ 4
b
3
3
u
3
−v
3
= c
=⇒
u =
3
s
r
c
2
2
+
b
3
3
+
c
2
v =
3
s
r
c
2
2
+
b
3
3
−
c
2
=⇒ x = u − v =
3
v
u
u
t
s
c
2
2
+
b
3
3
+
c
2
−
3
v
u
u
t
s
c
2
2
+
b
3
3
−
c
2
Examples 1. We apply Cardano’s method to x
3
+ 6x = 7, which has the obvious solution x = 1.
(
u
3
−v
3
= 7
uv =
6
3
= 2
=⇒ (u
3
+ v
3
)
2
= 7
2
+ 4 ·2
3
= 81 =⇒ u
3
+ v
3
= 9
=⇒ u
3
=
1
2
(9 + 7) = 8, v
3
=
1
2
(9 − 7) = 1
=⇒ x = u − v =
3
√
8 −
3
√
1 = 2 − 1 = 1
2. This time we solve x
3
+ 3x = 14.
(
u
3
−v
3
= 14
uv =
3
3
= 1
=⇒ (u
3
+ v
3
)
2
= 14
2
+ 4 ·1
3
= 200 =⇒ u
3
+ v
3
= 10
√
2
=⇒
(
u
3
=
1
2
(10
√
2 + 14) = 5
√
2 + 7 = (
√
2 + 1)
3
v
3
= 5
√
2 −7 = (
√
2 −1)
3
=⇒ x = u − v = (
√
2 + 1) − (
√
2 −1) = 2
As this shows, Cardano’s formula might produce an ugly expression for a simple answer—of
course, the cube root of 5
√
2 ±7 is the tip of everyone’s tongue!
3. The equation x
3
+ 3x = 10 may be solved using Cardano’s method, though in this case the
answer is just ugly.
(
u
3
−v
3
= 10
uv =
3
3
= 1
=⇒ (u
3
+ v
3
)
2
= 10
2
+ 4 ·1
3
= 104 =⇒ u
3
+ v
3
= 2
√
26
=⇒ u
3
=
√
26 + 5, v
3
=
√
26 −5
=⇒ x = u − v =
3
q
√
26 + 5 −
3
q
√
26 −5 ≈ 1.6989
Being unable or unwilling to work directly with negative numbers, Cardano modified his method
to solve other cubics such as x
3
+ c = bx, and moreover described how to remove a quadratic term
from a cubic using what is now known as the Tschirnhaus substitution (x = y −
a
3
):
x
3
+ ax
2
+ bx + c =
y −
a
3
3
+ a
y −
a
3
2
+ b
y −
a
3
+ c = y
3
−
a
2
3
y + ··· (∗)
Cardano’s student, Lodovico Ferrari, extended the method to solve quartic equations in terms of the
solution of a resultant cubic.
73
Negative Solutions and Complex Numbers
By the late 1500s, mathematicians were mentioning negative solutions to equations—these were usu-
ally described as fictitious, or false roots, but this didn’t stop them from being investigated. Rafael
Bombelli (1526–1572, Rome) even introduced a notation for complex numbers, described their alge-
bra, and showed how they could be used to find solutions to any quadratic or cubic equation. For
instance, he wrote 4 + 3i and 4 −3i as follows:
4 p di m 3, read ‘quattro piu di meno tre’ (four plus of minus three), and,
4 m di m 3, ‘quattro meno di meno tre.’
Given Bombelli’s belief in the fictitiousness of complex numbers, the effort he expended in their
honor is extraordinary: this is a prime example of pure abstraction, math for the sake of math!
In modern language, the three roots of Cardano’s cubic x
3
+ bx = c are
u − v, ζu −ζ
2
v, ζ
2
u − ζv
where ζ = e
2πi/3
=
−1+
√
3i
2
is a primitive cube root of unity. Together with the Tschirnhaus substitu-
tion (∗), Cardano’s formula therefore solves all cubic equations.
Examples 1. Returning to one of our previous examples, if x
3
+ 3x = 14, then u =
√
2 + 1 and
v =
√
2 −1, from which the three solutions are
u − v = 2
ζu −ζ
2
v = (
√
2 + 1)
−1 +
√
3i
2
−(
√
2 −1)
−1 −
√
3i
2
= −1 +
√
6i
ζ
2
u − ζv = (
√
2 + 1)
−1 −
√
3i
2
−(
√
2 −1)
−1 +
√
3i
2
= −1 −
√
6i
2. To find a solution to x
3
+ 3x
2
= 3, we perform the Tschirnhaus substitution x = y −
3
3
= y −1
before applying Cardano’s method:
y
3
−3y
2
+3y −1 + 3(y
2
−2y + 1) = 3 =⇒ y
3
−3y = 1 (b = −3, c = 1)
=⇒
(
u
3
−v
3
= 1
uv = −
3
3
= −1
=⇒ (u
3
+ v
3
)
2
= 1
2
+ 4(−1)
3
= −3
=⇒ u
3
+ v
3
=
√
3i =⇒ u
3
=
1
2
(
√
3i + 1), v
3
=
1
2
(
√
3i −1)
=⇒ x = u − v −1 =
3
s
√
3i + 1
2
−
3
s
√
3i −1
2
−1
This is ugly! If you know Euler’s formula and you choose a compatible pair of cube roots (you
need uv = −1), this will evaluate to one of three real numbers: the positive solution is in fact
x = 2 cos 20° −1 ≈ 0.8794.
74
Factorization & the Fundamental Theorem of Algebra
By the late 1500s, Vi
`
ete’s abstraction allowed him to streamline Cardano’s methods. He also investi-
gated the relationship between the coefficients of a polynomial and its roots. For Vi
`
ete the roots had
to be positive, but later improvements by Thomas Harriot and Albert Girard (1629) applied this to
all polynomials with any roots. For instance, if ax
2
+ bx + c = 0 has roots r
1
, r
2
, then
(r
1
−r
2
)
a(r
1
+ r
2
) + b
= a( r
2
1
−r
2
2
) + b(r
1
−r
2
) =
ar
2
1
+ br
1
+ c
−
ar
2
2
+ br
2
+ c
= 0
Provided the roots are distinct,
43
we conclude that
b
a
= −(r
1
+ r
2
),
c
a
= −r
2
1
−
b
a
r
1
= −r
2
1
+ (r
1
+ r
2
)r
1
= r
1
r
2
These are the quadratic version of what are known as Vi
`
ete’s formulas; other versions exist for every
degree. Their use amounts to an early form of factorization.
Example To solve 3x
2
−2x −1 = 0, first spot that r
1
= 1 is a root. By Vi
`
ete’s formulas,
r
1
+ r
2
= −
b
a
=
2
3
=⇒ r
2
= −
1
3
or alternatively r
1
r
2
=
c
a
= −
1
3
=⇒ r
2
= −
1
3
Think about the relationship between this approach and factorization!
A nice side-effect is a method for obtaining the quadratic formula by solving a pair of simultaneous
equations analogous to Cardano’s cubic approach:
r
1
+ r
2
= −
b
a
r
1
−r
2
=
p
(r
1
+ r
2
)
2
−4r
1
r
2
=
r
b
a
2
−4
c
a
=⇒ r
1
, r
2
= −
b
2a
±
1
2
s
b
a
2
−4
c
a
Vi
`
ete’s formulas were central to the later development of Galois Theory (1830) and the Abel–Ruffini
theorem regarding the insolubility of quintic and higher-degree polynomials.
The remaining key results regarding solutions of polynomials also appeared around this time:
Fundamental Theorem of Algebra Including multiplicity, a degree n polynomial has n complex
roots. Girard offered the first version in 1629, though a complete proof didn’t appear until
the work of Argand, Cauchy and Gauss in the early 1800s.
Factor Theorem In 1637, Descartes proved that p(r) = 0 ⇐⇒ p(x) is divisible by x −r.
For instance, to find the roots of p(x) = x
3
+ 2x
2
−13x + 10, Descartes would observe:
• p(1) = 0 =⇒ x − 1 is a factor; use long-division to obtain p(x) = (x − 1)(x
2
+ 3x −10).
• q(x) = x
2
+ 3x −10 has q(2) = 0; divide to get q(x) = (x − 2)(x + 5).
• The roots of p(x) are therefore 1, 2 and −5 (Descartes called this last a false root).
43
In fact the formulas hold even when r
1
= r
2
.
75
Exercises 7.2. 1. (a) Find Vi
`
ete’s formulas for the polynomial p(x) = x
3
+ ax
2
+ bx + c with roots
r
1
, r
2
, r
3
; that is, find the coefficients a, b, c in terms of the roots.
(Hint: Multiply out p(x) = (x −r
1
)(x − r
2
)(x − r
3
). . . )
(b) Solve x
3
− 6x
2
+ 9x − 4 = 0 using Girard’s method: first, determine one solution by in-
spection, then use Vi
`
ete’s formulas for the cubic to investigate the relationship between
the remaining roots.
2. (a) Apply Cardano’s method to the equation x
3
+ 6x = 20.
(Hint: to finish, compute (1 +
√
3)
3
)
(b) If b, c > 0, Cardano’s method finds a single positive solution to x
3
+ bx = c. Explain why
such an equation always has exactly one real solution which is moreover positive.
3. Prove that if t is a root of x
3
= cx + d, then
r
1
=
t
2
+
r
c −
3t
2
4
and r
2
=
t
2
−
r
c −
3t
2
4
are both roots of x
3
+ d = cx. Use this to solve x
3
+ 3 = 8x.
4. Consider the cubic equation aaa −3raa + ppa = 2xxx (as written by Harriot). Show that the
substitution a = e + r reduces this to an equation without a square term.
As an example, reduce the equation aaa − 18aa + 87a = 110 to a cubic in e without a square
term. Find all three solutions in e and therefore find the solutions to the original equation in a.
5. Find all roots of the cubic p(x) = 2x
3
−3x
2
−3x + 2 using Descartes’ factor theorem.
6. Use Cardano’s method (with Tschirnhaus substitution and complex numbers!) to find the so-
lutions to the equation
x
3
+ 4 = 3x
2
Verify that you get the same solutions using Descartes’ factor theorem.
(Hint: all solutions are integers!)
7. (If you are very comfortable with complex numbers) Use Euler’s formula to verify that the
equation x
3
+ 3x
2
= 3 has the positive solution x = 2 cos 20° −1. It also has two negative real
solutions: find them!
76
7.3 Astronomy and Trigonometry in the Renaissance
Distinctly European progress in trigonometry came courtesy of Johannes M
¨
uller (a.k.a. Regiomon-
tanus,
44
1436–1476). De Triangulis Omnimodius (Of all kinds of triangles, 1463) provided an axiomatic
update of Ptolemy’s Almagest and its Islamic improvements. Though the title refers to triangles,
his approach remains circle-based (chords and half-chords). Regiomontanus was a renowned as-
tronomer; his tracking of a comet from late 1471 to spring 1472 provided controversial evidence that
objects could move between the, supposedly fixed, heavenly spheres of ancient Greek theory.
Domenico Novara (1454–1504), Regiomontanus’ student, inherited much of his unpublished work.
He became a student of Pacioli in Florence and an astronomer at the University of Bologna, though
he is now perhaps best known as adviser to a young Pole, Nicolaus Copernicus (1473–1543), who
studied in Bologna from 1496 with the ostensible intent of joining the priesthood. . .
Copernicus concluded that Ptolemy’s geocentric model could not be
reconciled with astronomical observation. De revolutionibus orbium ce-
lestium (On the revolutions of the heavenly spheres), published a year after
his death, describes how to compute within a heliocentric (sun-centered)
model. This, Copernicus believed, was the obvious solution to the
problem of retrograde motion that had plagued the ancient Greeks.
45
The animation demonstrates Copernicus’ solution: with the sun at the
center, the retrograde motion of Mars and Jupiter are easily explained.
The outer circle represents the ‘fixed stars’ against which the motion
of the planets are observed.
M
J
Copernicus’ work is now described as a revolution, though it was not perceived so at the time. De
revolutionibus was dedicated to the Pope, welcomed by the Church, and used by Vatican astronomers
to aid in calculation. The difficulty and narrow readership of his work made it unthreatening to
contemporary dogma. Copernicus did not present heliocentrism as reality nor did he advocate for
overturning long-held beliefs. Within a century, however, the Copernican theory had found its bull-
dog in Galileo, and conflict between science and the Church became unavoidable.
Trigonometry is finally about triangles!
Georg Rheticus (1514–1574) defined trigonometric functions purely
in terms of triangles, referring to the perpendiculum (sine) and ba-
sis (cosine) of a right-triangle with fixed hypotenuse. Rheticus was
a student of Copernicus and helped posthumously to publish his
work.
In 1595, Bartholomew Pitiscus finally introduced the modern term
with his book Trigonometriæ, in which he purposefully sets out to
solve problems related to triangles. The picture is the title page from
the second edition (1600—MDC in Roman numerals). Both Rheti-
cus and Pitiscus had problems which look very familiar to modern
readers, such as solving for unknown sides of triangles.
44
His grand-sounding name is a latinization of his birthplace K
¨
onigsberg (King’s Mountain), Bavaria, Germany.
45
Copernicus wasn’t the first to suggest such a model. Several ancient Greek scholars embraced heliocentrism, with
Aristarchus of Samos (c. 310–230 BC) credited with its first presentation. However, Aristarchus’ views were rejected by the
mainstream Greeks, and it is likely Copernicus never encountered his work.
77
Example (Pitiscus) A field has five straight edges of
lengths 7, 9, 10, 4 and 17 in order. The distance from the
first to third vertex is 13 and from the third to fifth is 11.
What is the area of the field?
The problem can be solved easily using Heron’s formula,
but Pitiscus opts for trigonometry. We give a modernized
version that depends on applying the law of cosines to the
three large triangles.
7
9
13
10
11
4
17
a
b
c
α
β
γ
cos α =
7
2
+ 13
2
−9
2
2 ·7 · 13
=
137
182
cos β =
17
2
+ 13
2
−11
2
2 ·17 · 13
=
337
442
cos γ =
4
2
+ 11
2
−10
2
2 ·4 · 11
=
37
88
The values of α, β, γ and therefore the altitudes a, b, c of the three major triangles could be read off a
table, or found exactly using Pythagoras’:
a = 7 sin α = 7
p
1 −cos
2
α =
7
182
p
182
2
−137
2
=
3
26
√
1595,
b = 13 sin β =
1
34
√
81795, c = 4 sin γ =
5
22
√
255
The total area is easily computed:
A =
1
2
(13a + 17b + 11c) =
1
4
3
√
1595 +
√
81795 + 5
√
255
≈ 121.4
Kepler’s Laws
Johannes Kepler (1571–1630) was a student of Tycho Brahe
46
for the last two years of Brahe’s life. He
inherited Brahe’s position, decades-worth of astronomical data, and his philosophy on the impor-
tance of observation-based theory. Kepler also embraced the mystical Pythagorean view that nature
reflects harmony, a belief that partly drove his scientific pursuits. To Kepler, any observation of a nat-
ural, simple ratio was something of great import. For example, in observing that the daily movement
of Saturn at its furthest point from the sun was roughly 4/5 of that at its nearest point, his temptation
was to assume that ‘roughly’ must be ‘exactly.’
Thanks to Brahe, Kepler had data on roughly thirteen orbits of
Mars and two of Jupiter. From these data, he posited three laws.
1. Planets move in ellipses with the sun at one focus.
2. The orbital radius sweeps out equal areas in equal times. In
the picture, the sectors all have the same area and the planet
moves more slowly the further it is from the sun.
3. The square of the orbital period is proportional to the cube of
the semi-major axis of the ellipse: T
2
∝ a
3
.
2a
46
Tycho Brahe (1546–1601) was a Danish astronomer who worked for Austro-Hungarian Emperor Rudolph II in Prague
for 25 years, producing a wealth of accurate astronomical measurements. While these helped burnish the Copernican
theory, he is also known for his 1572 observation of a nova (a new star that later disappeared, now understood to be the
death of a distant star), and then a comet in 1577; both provided yet more evidence for the changeability of the heavens.
78
Kepler’s laws are empirical observations rather than the result of mathematical proof. However, the
process of their discovery demonstrates Kepler’s tremendous mathematical abilities.
Starting point Kepler began by assuming the essential correctness of the Copernican model in that
all planets exhibit uniform circular motion round the sun.
Orbital Estimation Kepler’s data told him the direction to each
planet, but not the distance, though his Copernican assump-
tion allowed him to estimate the relative distance.
47
For exam-
ple, using the direction from Earth to Mars at equally spaced
times and by drawing circles of different radii for possible or-
bits of Mars, he was able estimate an orbit where Mars’ mo-
tion would also be uniform. This required an enormous num-
ber of trigonometric calculations: each measurement of plan-
etary longitude/latitude relative to Prague had to be converted
to measurements relative to the sun. Everything was subject
to errors of estimation.
Modifying the model Kepler altered his model to reflect Earth’s slightly non-circular orbit. He first
tried an equant model (in the style of the ancient Greeks), offsetting the center of the orbit
slightly from the sun. Despite this, he failed to fit his data for Mars to pure circular motion.
The First Law Abandoning circles, Kepler now permitted planets to move in ovals. He decided to
approximate Mars’ orbit with an ellipse and set out to calculate its parameters, stumbling on
an almost perfect match when the sun was placed at a focus. The geometric significance of the
focus was exactly the natural beauty sought by Kepler. Having now established the first law
for Mars, he repeated the exercise for the other known planets (Mercury, Venus, Earth, Jupiter,
Saturn) as well as he could given his inferior data.
The Second Law Kepler’s second law followed an infinitesimal argument based on inspired guess-
work. To fit his elliptical orbits, planetary velocity was non-constant, appearing inversely pro-
portional to the distance from the sun (v =
k
r
). Kepler used this to approximate the area of a
sector swept out by the radius vector: over a small time-interval ∆t, a planet travels a distance
v∆t and thus sweeps out an approximate triangle of area
1
2
rv∆t =
1
2
k∆t. In modern language,
this is the conservation of angular momentum. Kepler had no justification beyond the fact that
it seemed to fit the data. In particular, he did not know why a planet should move more slowly
when further from the sun.
The Third Law This was stated with very limited analysis. Given the relative sizes of each orbit and
its period, some inspired guesswork allowed him to observe that
T
2
a
3
is approximately the same
value for each.
Kepler’s discoveries were revealed over many years in several texts, and his magnum opus Epitome
astronomiæ Copernicanæ was published in 1621. Within a century Issac Newton had provided a math-
ematical justification of Kepler’s laws based on the theory of calculus and his own axioms: an inverse
square law for gravitational acceleration and his own three laws of motion.
47
Following Ptolemy, Brahe thought the Earth-Sun distance was around 1/10 of its true value. Kepler thought this an
underestimate by at least a factor of three. In 1659, Christiaan Huygens found the distance to an accuracy of 3%.
79
A Religious Interlude: Protestantism, the Counter-Reformation and Calendar Reform
In 1563, Pope Gregory began the Catholic Church’s push-back against the spread of Protestantism,
48
the counter-reformation. Of particular interest to science and mathematics was the newly created In-
dex Librorum Prohibitorum, a list of books contradicting Church doctrine. This was also a response
to the new technology of mass-printing, which made disseminating controversial new ideas easier.
Heliocentrism came quickly under attack: Kepler’s book was banned immediately upon publica-
tion in 1621. However, his location far from Rome meant that Kepler and his ideas were relatively
safe. The ultimate result of Gregory’s crackdown was the slow ceding of scientific power to northern
(Protestant) Europe where papal diktat had no effect.
In contrast to the anti-scientific book-banning fervor of the counter-reformation, Pope Gregory is
also famous for shepherding an astounding scientific achievement: calendar reform. By 1500, as-
tronomers knew the solar year to be roughly 11
1
4
minutes shorter than the 365
1
4
days of the Julian
calendar.
49
For 1200 years, Easter had been decreed to be the Sunday after the first full moon after
the vernal equinox (March 20
th
/21
st
), but by 1500 the equinox was happening 10–11 days earlier. The
impetus to correct the date of Easter meant that calendar reform and astronomical modelling were
now an important Church project.
A century of effort
50
resulted in the Gregorian calendar (designed by Aloysius Lilius and Christopher
Clavius). Gregory imposed the new calendar in all Catholic countries in 1582. The 10 day deficit was
corrected by eliminating October 5
th
–14
th
1582. To prevent the error re-occurring, the computation
of leap-years was also changed: centuries are now leap-years only if divisible by 400, thus 1600 was
a leap year, but 1900 was not. The Gregorian calendar is astonishingly accurate, losing only one day
every 3000 years. Since it emanated from Rome, many Protestant parts of Europe took decades if not
centuries to adopt the new calendar. The Eastern Orthodox Church still computes Easter using the
Julian calendar, which is now 13 days behind the Gregorian.
Galileo Galilei (1564–1642)
Based in northern Italy, Galileo was close to the center of Church power; unlike Copernicus and
Kepler, he openly challenged its orthodoxy. While undoubtedly a great mathematician, he is more
importantly considered the father of the scientific revolution for his reliance on experiment and ob-
servation. He famously observed Jupiter’s moons with a telescope of his own invention, noting that
objects orbiting an alien body was counter to Ptolemaic theory. Skeptics, when shown this image,
preferred to assert that it must be somewhere inside the telescope!
In 1632 Galileo published Dialogue Concerning the Two Chief World Systems, a Socratic discussion be-
tween three characters: Salviati argued for Copernicus, Simplicio was for Ptolemy, and Sagredo was
an independent questioner. The character of Simplicio was provocatively modeled on conserva-
tive philosophers who refused to consider experiments and bore a notable resemblance to the Pope.
Salviati almost always came out on top and Simplicio was made to appear foolish. The text resulted
in Galileo’s conviction for heresy; all his publications, past and future, were banned, and he spent
the remainder of his life under house arrest.
48
Martin Luther’s Ninety-five Theses (1517) is generally considered the start of the Protestant Reformation. Europe saw
several gruesome religious wars over the next 150 years as various countries broke from Catholicism and Rome.
49
Named for Julius Caesar, the Julian year has 365 days with a leap-day added every four years.
50
Pope Sixtus IV tried to recruit Regiomontanus to the cause in 1475, though the mathematician died first. Copernicus
was among those invited to consider proposals in the early 1500s, though he distanced himself, perhaps because he knew
that his developing heliocentric ideas would not be accepted.
80
Despite Church efforts, Galileo’s works continued to be distributed by his supporters and he contin-
ued working. His most important scientific text, Discourses Concerning Two New Sciences (materials
science and kinematics) was smuggled out of Italy to be published in Holland in 1638. In this book
he resurrects his characters from Two Chief World Systems and famously refutes Aristotle’s claim that
heavier objects fall more rapidly than lighter ones.
51
Here are two results from this text.
Theorem. If acceleration is uniform, then the average speed is the average of the initial and terminal
speeds.
Proof. Galileo argues pictorially.
In essence, CD is the time-axis, increasing downward. Velocity is measured hor-
izontally from the time-axis to the uniformly sloped line AE, of which I is the
midpoint.
The distance travelled is the area between AE and the time-axis, which plainly
equals the area of the rectangle between GF and the time-axis. The velocity cor-
responding to GF is the average of those corresponding to A and E.
Here is the calculation using modern algebra. Suppose the object has velocity
v
A
as it passes A and v
B
as it passes v
B
, and that t =
|
AB
|
. Then the distance
travelled is
v
av
t = v
A
t + area(△ABE) = v
A
t +
1
2
(v
B
−v
A
)t =
1
2
(v
A
+ v
B
)t
whence v
av
is the average of the initial and final speeds.
Corollary. A falling object dropped from rest will traverse distance in proportion to time-squared,
d
1
: d
2
= t
2
1
: t
2
2
This is Galileo’s version of the well-known kinematics formula d =
1
2
gt
2
.
Proof. Let d
1
, d
2
, v
1
, v
2
represent the distances travelled and the speeds of the dropped body at times
t
1
and t
2
. Since acceleration is uniform,
v
1
: v
2
= t
1
: t
2
By the Theorem,
d
1
=
0 + v
1
2
t
1
=
1
2
v
1
t
1
, and d
2
=
1
2
v
2
t
2
whence
d
1
: d
2
= v
1
t
1
: v
2
t
2
= t
2
1
: t
2
2
51
Supposedly by dropping weights off the leaning tower of Pisa, though take this story with a pinch of salt. . .
81
Galileo follows this by decomposing the motion of a projectile into horizontal (uniform speed) and
vertical (uniform acceleration) components, thereby proving that projectiles follow parabolic paths.
Galileo covered several other important mathematical topics, some of which we’ll mention when we
discuss calculus. While his mathematical ideas were cutting-edge for the time, it is his insistence
on testing theory against data that makes him a true revolutionary. By 1600 very few of Aristotle’s
easy-to-refute claims had been rejected due to experimental testing; the hostility Galileo provoked
by doing so perhaps explains why. This is the core of the scientific revolution: primacy is given to
experiment and observation over ancient ‘wisdom,’ whatever the source.
Galileo was finally cleared of heresy by the Catholic Church in 1992.
Exercises 7.3. 1. Compute today’s date in the Julian calendar and explain your calculation.
2. (A problem of Copernicus) Given the three sides of an isosceles
triangle, to find the angles.
Suppose AB and AC are the equal legs of the triangle. Circum-
scribe a circle around the triangle and draw another with center A
and radius AD =
1
2
AB.
(a) Why is Copernicus introducing the second circle?
(b) Explain why the ratio of each of the equal sides to the base of
△ABC equals that of the radius AD to the chord DE.
(c) If
|
AB
|
=
|
AC
|
= 10 and
|
BC
|
= 6, use modern trigonometry
to find the three angles of the original triangle.
A
B
C
D E
3. Verify that Heron’s formula gives the same solution to Pitiscus’ problem (pg. 78).
4. Given that Earth’s orbital period is 1 year and that the mean distance of Mars from the sun is
1.524 times that of Earth, use Kepler’s third law to determine the orbital period of Mars.
5. According to Kepler’s second law, at what point in a planet’s orbit will it be moving fastest?
6. Galileo states the following.
A projectile fired at an angle α = 45° above the horizontal at a given initial speed
reaches a distance of 20,000. Then, with the same initial speed it will reach a distance
of 17,318 when α = 60°, or α = 30°.
Check this statement: if you want a challenge, try to do without the standard Physics formulæ
and instead use ratios!
82
