The Renaissance in Europe

The traditional story of Europe between the fall of Rome in 476 and the early Renaissance of the1100’s is one of limited civilization and little learning. This is a simplistic view, though it has a grainof truth. The word civilization has many meanings and interpretations,1 but in one sense Europe wascertainly civilizing: the rise of the nation state. Prior to the 1100’s, Europe largely consisted of smallshifting kingdoms whose lack of stability was not conducive to scholarship. Much learning andstudy indeed took place in monasteries, though monks were unlikely to publish and widely promotetheir understanding. With notable exceptions, learning has flourished when a significant proportionof the population are freed from worrying about the basic needs of life: food, shelter, etc. For much ofhistory, non-religious education was restricted to an elite with the necessary status, time and moneyto devote to it; even then, few of those with the opportunity would choose to fully engage. The Re-naissance saw a slow process of national consolidation which proved a boon for education: it is mucheasier for a secure king (even a tyrant) to invest in libraries and universities than it is for a threatenedwarlord. By the 1700’s, the major modern borders of Western Europe were largely recognizable; po-litical and social power had expanded so that a much larger proportion of the population, still tinyin comparison to today, were able to take advantage of and contribute to the growth of knowledge.

The dates of the European Renaissance (literally rebirth) vary depending on geographic location anddiscipline (Italy versus France, art versus philosophy, etc.): a very wide net would encompass the 12th

to 17th centuries. The maps below show the political composition of Europe in 979 and then 1215.Feudal societies (France, England, The Holy Roman Empire, etc.) are stabilizing while the ByzantineEmpire (the last vestige of the Eastern Roman Empire) is in decline and under threat from Islam to theeast. At the same time, Islam is retreating from Iberia as the Kingdoms of Aragon and Castile assertthemselves. The rise of Europe is often contrasted with the decline of the Islamic power, though thelink is far from causal. The European Renaissance began even as the Islamic world was experiencinga golden age; the gains for Islam in modern Turkey were offset by losses in Spain.

1It is often considered uniformly positive: for the peasantry, the arrival of ‘civilization’ often meant little more than theoverthrow of a small local tyrant by one larger and more brutal.

Communication of Knowledge into Europe

Between 900 and 1300 the population of Europe is estimated to have tripled to around 100 million.Peoples became more settled, internal trade increased and the political structures necessary to man-age such arose. External trade also increased: in particular, Venice2 was a large trading hub for theMediterranean and many new ideas came along with the physical goods. During the same period,the Crusades began with the intent of ‘reclaiming’ the Holy Land from Islam. These wars, togetherwith the push-back against Islam in Spain, resulted in the acquisition of knowledge as well as terri-tory. It helped that Islamic scholars had so venerated the Greeks: Europeans could chauvinisticallyassert that they were merely reclaiming the lost knowledge of their own ancient past, a past whichhad been ‘stolen’ by their cultural and religious enemies. Not only was there no ‘theft’—Europe’scultural and scholastic decline predated Islam—but Islam gained its knowledge from both west andeast, consolidating and greatly improving on its inheritance. Without the Islamic scholars, little ofthe knowledge of the Greeks, Egyptians, Babylonians and Indians might have survived and the Eu-ropeans of the middle ages would have had much less to work with.

The high point of Islamic conquest in Europe came with the fall of Constantinople to Mehmet theConqueror in 1453 and the subsequent rise of the Ottoman Empire. This also marks the start of thedecline of Islamic scientific dominance. Many intellectuals fled Constantinople (which, under theByzantines, had preserved the last European vestiges of Alexandria’s knowledge) for Rome helpingto further fuel learning. With large powerful enemies to the east and continuing crusades, Europeansbegan travelling greater distances by sea,3 beginning the period of colonization and global empire.

As a very simple summary, two conditions provided the foundation for the increase in learning inRenaissance Europe:

• Greater political stability and riches providing the space and resources for scholarship.

• An influx of knowledge from outside Europe.

Education in the Renaissance

Scientific and philosophical progress was spurred by the translation of classic works, mostly fromfrom Arabic and ancient Greek, into Latin. The first universities were formed to teach this canon:Bologna 1088, Paris 1150 and Oxford 1167. Universities quickly formed curricula deriving fromthose of ancient Greece. The typical student was a young man of wealth who had previously beentutored in grammar, logic & rhetoric (the trivium). At university he would study the quadrivium (ge-ometry, astronomy, arithmetic & music). While Islamic improvements were incorporated, scholarstypically gave pre-eminence to the Greek scholars: Euclid for geometry, Aristotle for logic/physics,Hippocrates/Galen for medicine, Ptolemy for astronomy, etc. The veneration of Aristotle in par-ticular almost precluded the questioning of ancient knowledge, and these works were often taughtby rote. Moreover, early universities were often funded by the Church: ‘research’ was more likely toinvolve the justification of biblical passages using Aristotle than the conduct of physical experiments.

2The Venetian Marco Polo (1254–1324) is perhaps the most famous trader of this time, travelling the silk road to Mon-golia and China.

3Christopher Columbus (born Genoa 1451) famously ‘discovered’ America in 1492 while looking for sea routes to Asia.

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Leonardo de Pisa (Fibonacci c.1175–1250)

Fibonacci4 was perhaps the influential mathematician of the early Renaissance. In his time, the Re-public of Pisa was an independent mini-state and one of the most important trading centers in Eu-rope. Pisa had a small navy and its merchants, including Fibonacci’s father, traded across Europe andround the Mediterranean. Fibonacci joined his father on his travels and, reportedly, first encounteredthe Hindu-Arabic numerals while in North Africa.Fibonacci was impressed by the ease of calculation afforded by the Hindu-Arabic system and wrotea text Liber Abaci (1202) to instruct traders in its use. The pictures below are from the text. In thefirst, Fibonacci explains how to compute with decimal fractions; the two columns at the bottom ofthe page show how to repeatedly multiply 100 (and then 10) by the fraction 9

10 . Thus:

100, 90, 81,9

1072 (= 72.9),

110

610

65 (= 65.61),9

10410

010

59 (= 59.049), etc.

Note how the decimal part is written backwards on the left and using fractions with a bar to separatenumerator and denominator, both following Hindu-Arabic practice. The Indians wrote fractionswithout a bar and it is thought that Islamic scholars inserted the bar for clarity in the 1100’s. Fibonacciis the first known European to use this notation. The second picture is a close-up of Fibonacci’sfamous sequence: read top-to-bottom 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.Amongst many other inheritances from the Hindu-Arabic tradition, Fibonacci was also willing towork with negative solutions to problems, provided these represented deficiencies or debts in ac-counting. He is therefore the first known European to work with negatives.

4This sobriquet was given to Leonardo by French scholars of the 1800’s: it literally means ‘son of Bonacci.’

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Algebraic Notation in the Renaissance

While many new mathematical techniques were created during the Renaissance, the most obviousdevelopment to a modern reader is notational. Fibonacci’s fractional notation was cutting-edge forthe 1200’s but everything other than numbers and fractions was written in sentences. The slow math-ematical revolution of 12–1700 is the story of how notational improvement eventually allowed alge-bra to eclipse geometry as the primary language of reasoning. Here is a brief summary of notationaldevelopments over this period.

• The Italian Abacists were a group of mathematicians in the 14th century. They continued Fi-bonacci’s advocacy for the Hindu-Arabic system and for the use of accompanying algorithms.They were pitted against the traditional use of Roman numerals for business accounts. Theirapproach was highly practical, with little-to-no abstraction, and largely for the purposes ofconducting trade. For example,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 eightmonths?

The general idea of their texts was that a trader could find a worked solution to a problemsimilar to the one they needed, and simply switch the numbers. The Abacists went further thansimply restating old practical problems. They introduced the use of shorthands and symbolsfor unknowns and mathematical operations. In particular, the word cosa (‘thing’) was used foran unknown, while censo, cubo and radice meant, respectively, square, cube and (square-)root.Various shorthands for these expressions could be combined, for example ‘ce cu’ (read censo dicubo) referred to the sixth power of an unknown (x3)2 = x6.

• In the late 1400’s, Luca Pacioli introduced the symbols p, m (piu, meno) for plus and minus. Forexample 8m2 denoted eight minus two.

• In 1484 in France, Nicolas Chuquet wrote Triparty en la science des nombres, a book of practicalmathematics. He borrowed the Italian p, m notation and introduced an R-notation for roots.For instance R47 meant 4

√7 while

5√

4− 3√

2 would be written R54m3R2

where the underline indicates grouping (essentially parentheses).

• In 1520’s Vienna, Christoff Rudolff introduced a symbols similar to x and ζ for an unknownand its square. He had other symbols for odd powers and produced tables showing how tomultiply his abstract symbols. The words he used show the influence of the Italian and Frenchmathematicians: algebra in the German-speaking world was known as the art of the coss (Ger-man for thing). More famously, Rudolff introduced ± symbols as algebraic operations. Thesehad been used for around 30 years as a prefix denoting an excess or deficiency in a quantity(profit/loss in accounts). A period denoted equals and he is also credited with the first use ofthe 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.√

2 rather than√

2.

• The equals sign comes from Robert Recorde in 1557’s The Whetstone of Witt, the first seriouswork of British mathematics. Recorde borrowed Germanic notation otherwise. To justify hischoice, he asserted, ‘no two things are more equal than a pair of parallel lines.’

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• Francois Viete (1540–1603) introduced the use of letters to represent both unknowns and con-stants. He was thus able to approach algebra very differently to his predecessors. WhereasCardano (and others) had described how to solve particular equations algorithmically, theytypically presented only examples (e.g. x3 + 3x = 14 rather than x3 + bx = c), expecting read-ers to simply change numbers to fit any required example. Viete pioneered the use of abstractconstants, though he continued to use words to describe constructions such as squaring.

• Simon Stevin (1548–1620, Holland) wrote De Thiende (the Tenths), demonstrating how to cal-culate using decimals: prior to this mathematicians used only fractions although, as we’veseen, Fibonacci was essentially using decimal fractions. Stevin arguably completed the journeywhereby the concept of number subsumed that of magnitude. Recall the ancient Greek conceptof incomensurability: the ratio of the side to the diagonal of a square could be compared (Eu-doxus) to integer ratios, or even ‘computed’ using Theatetus’ quotient-algorithm approach, butthe Greeks never considered it to be a number. Stevin is perhaps the first mathematician to as-sert that every ratio is a number. This increased the application of algebra by permitting thedescription of any geometric magnitude.

• William Oughtred (1575–1660, England) gave us × for multiplication, though he often sim-ply used juxtaposition. Oughtred combined Viete’s general approach (abstract constants) withsymbolic algebra. For instance, to solve a quadratic equation Aq + BA + C = 0, where Aqmeans ‘A-squared’ (A-quadratum in Latin) he’d write the quadratic formula as

A =√ :

14

Bq − C : −12

B

In Oughtred’s notation, colons were parentheses.

• Thomas Harriot (1560–1621, England/Virginia) made several steps towards modern notationincluding juxtaposition for multiplication and a modern encompassing root-sign. For example,

4√

cccc + 27aa 3√

2 + b meant4√

c4 + 27a2 3√

2 + b

With the exception of exponents, we essentially have modern notation.

• Rene Descartes (1596–1650, France) introduced exponents for powers (a2, a3, etc.) and the con-vention of using letters at the end of the alphabet (x, y, z) for unknowns and those at the begin-ning (a, b, c) for constants.

While modern mathematics contains many, many specialized symbols (∅, =⇒ , etc.), basic notationis essentially unchanged from Descartes’ time. This is not to say that all mathematicians uniformlyused the most modern notation. For instance, there are papers of Leonhard Euler from the 1700’susing Thomas Harriot’s juxtaposition notation for exponents. There are indeed published worksfrom the late 1800’s where equations are written out in words.It is also worth mentioning in this context the invention of the printing press in 1439. Along withevery other form of knowledge, this naturally aided the dissemination of mathematics. This momentis indeed a faultline in history: the sudden relative ease of production meant a great increase inwritten material; if new editions were not produced, older works were not so easily preserved.

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Cardano and the Cubic As a concrete example, in 1545 Girolamo Cardano published Ars Magna(Great Art or the Rules of Algebra), in which he describes algebraically how to solve quadratic andcubic equations.5 As was typical for the time, Cardano describes negative solutions to equationsas fictitious, though he even writes the square root of −15 in one ‘solution,’ if only to mention itsabsurdity. His primary approach, which is still known as Cardano’s method, was for the equationx3 + bx = c where b, c > 0. Here it is in modern notation:

Let u, v satisfy u3 − v3 = c and uv = b3 . Then x = u− v is easily seen to solve the cubic.

x3 + bx = (u− v)3 + b(u− v) = u3 − 3u2v + 3uv2 − v3 + b(u− v)

= (u3 − v3) + (u− v)(b− 3uv) = c

However u and v also satisfy

(u3 + v3)2 = (u3 − v3)2 + 4(uv)3 = c2 + 4(

b3

)3

thus

u3 + v3 =

√c2 + 4

(b3

)3

u3 − v3 = c=⇒

u =

3

√√( c2

)2+(

b3

)3+ c

2

v =3

√√( c2

)2+(

b3

)3− c

2

=⇒ x =3

√√√√√( c2

)2+

(b3

)3

+c2− 3

√√√√√( c2

)2+

(b3

)3

− c2

Even though Rudolff’s x’s, ± and √ symbols were in use, Cardano described almost everything inwords with regular uses of fractional notation and Pacioli’s p and m. There was no equivalent of theabstract constants b and c in our formula. Imagine how hard this was to follow!

As an easy example, consider x3 + 3x = 14: we have{u3 − v3 = 14uv = 3

3 = 1=⇒ (u3 + v3)2 = 142 + 4 · 13 = 200 =⇒ u3 + v3 = 10

√2

=⇒ u3 =12(10√

2 + 14) = 5√

2 + 7, v3 = 5√

2− 7

=⇒ x = u− v =3√

5√

2 + 7− 3√

5√

2− 7 = (√

2 + 1)− (√

2− 1) = 2

As the last step shows, Cardano’s formula might leave you with a very ugly expression for a simpleanswer! Try solving other easy examples yourself.

5See in particular page 9 of the pdf where the solution to x2 + 6x = 91 (‘quadratum & 6 res aequalie 91’) is describedvia completing the square.

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Being unable or unwilling to work directly with negative numbers, Cardano modified his methodto solve other cubics such as x3 + c = bx, and moreover described how to remove a quadratic termfrom a cubic using a substitution:

x3 + ax2 + bx + c =(

y− a3

)3+ a

(y− a

3

)2+ b

(y− a

3

)+ c = y3 − a2

3y + · · · (∗)

Lodovico Ferrari (Cardano’s student) extended the method to solve quartic equations in terms of thesolution of a resultant cubic.

While negative solutions to equations were generally held to be absurd until the late 1600’s, thisdidn’t stop mathematicians from playing with them. For instance, Rafael Bombelli (1526–1572, Rome)introduced notation for complex numbers and described their algebra and how they could be usedto find solutions to any quadratic or cubic equation. For example, he would write 4 + 3i as 4 p di m 3‘quattro piu di meno tre,’ literally ‘four plus of minus three.’ Similarly, 4− 3i was ‘quattro meno di menotre.’ Bombelli himself did not believe that complex numbers were real. That he introduced themat all, and bothered to describe their interactions, is an early example of pure abstraction: posit theexistence of something, then see what it can do!

With a modern understanding of complex numbers, the three roots of the cubic are u − v, ζu −ζ2v, ζ2u − ζv, where ζ = e2πi/3 is a primitive cube root of unity. Together with the Tschirnhaustransformation (∗), Cardano’s formula is all you need to solve all cubic equations. The solutionshowever are often very ugly and it is usually more useful in applications to simply approximateusing a numerical method. For example, to solve x3 = 3x + 1 we might write

x3 − 3x = 1 =⇒{

u3 − v3 = 1uv = −1

=⇒ (u3 + v3)2 = 12 + 4(−1)3 = −3 =⇒ u3 + v3 =√

3i

=⇒ x = u− v =3

√√3i + 1

2− 3

√√3i− 1

2=

3

√1 +√

3i2

+3

√1−√

3i2

This looks ugly, but it is in fact a real number, being the sum of complex conjugates. Indeed if youknow Euler’s formula, you can quickly see that this is x = 2 cos 20° ≈ 1.8794 :)

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Astronomy and Trigonometry in the Renaissance

The first genuinely European progress in trigonometry came courtesy of Johannes Muller (1436–1476), better known to history as Regiomontanus.6 His text De Triangulis Omnimodius (Of all kinds oftriangles 1463) was essentially a more rigorous and axiomatic translation of Ptolemy’s Almagest andits Islamic improvements. Even though the title of the book refers to triangles, the approach is stillbased on circles (chords and half-chords). Regiomontanus was a renowned astronomer, famouslytracking a comet from late 1471 to spring 1472 and (badly!) estimating its distance. The appearanceof the comet produced controversy as its existence and movement provided concrete evidence thatobjects could move between the, supposedly fixed, heavenly spheres of ancient Greek theory.

Domenico Novara (1454–1504), one of Regiomontanus’ students, inherited much of his unpublishedwork (De Triangulis Omnimodius wasn’t published for 70 years, though it’s content was widely dis-seminated). Novara was also a student of Luca Pacioli in Florence and became an astronomer at theUniversity of Bologna. Novara is now perhaps most famous as an adviser of Nicolaus Copernicus(1473–1543) who studied in Bologna from 1496, with the ostensible intent of joining the priesthood. . .

Copernicus, originally from Krakow, Poland, had stud-ied the Ptolemaic earth-centric model and decided thatit could not be reconciled with observations. It tookCopernicus a lifetime to formulate his great work Derevolutionibus orbium celestium (On the revolutions of theheavenly spheres) which was finally published a year af-ter his death. The text describes how to compute withina heliocentric (sun-centered) model of the universe. This,Copernicus believed, was the obvious solution to theproblem of retrograde motion which had plagued the an-cient Greeks and led to their creation of complex epicy-cle and equant models.

The animation demonstrates Copernicus’ solution:with the sun at the center, the large retrograde motionof Mars (red orbit) and smaller motion of Jupiter (greenorbit) are easily explained. The outer circle representsthe ‘fixed stars’ against which the motion of the planetsare observed.

Copernicus’ work is now described as a revolution, and it is popularly believed that Copernicuswas the first to promote heliocentrism: neither claim stands up; he was neither the first,7 nor did heengage in much promotion! Indeed, De revolutionibus was dedicated to the Pope, welcomed by theChurch and used by Vatican astronomers to aid in calculation. The difficulty and narrow readershipof his work made it unthreatening to contemporary dogma. Copernicus did not present heliocen-trism as ‘reality’ or advocate for long-believed truths to be upended. Perhaps he was sensibly awarewhat the result would have been had he chosen to do so. Within a century, however, the Copernicantheory had found its bulldog in Galileo and the crisis in the Church became unavoidable.

6His grand-sounding name is a latinization of his birthplace, Konigsberg (Bavaria, Germany), literally King’s Mountain.7Several ancient Greek scholars embraced heliocentrism, Aristarchus of Samos (c.310–230 BC) being credited with the

first presentation. Aristarchus’ views were rejected by the Greeks, and it is likely Copernicus never encountered his work.

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Trigonometry is finally about triangles! Georg Rheticus(1514–1574 Austria/Hungary) was the first mathematicianto define trigonometric functions purely in terms of tri-angles. He referred to sine as the perpendiculum and co-sine as the basis of a right-triangle with a fixed hypotenuse:these were still fixed lengths rather than the modern ratios oflengths. Rheticus was a student of Copernicus and helpedpublish and Copernicus’ work after his death.

In 1595, Bartholomew Pitiscus finally introduced the mod-ern term with his book Trigonometriæ, in which he purpose-fully sets out to solve problems related to triangles. The pic-ture is the title page from the second edition (1600—MDCin Roman numerals).

Both Rheticus and Pitiscus had problems which look veryfamiliar to modern readers, such as solving for unknownsides of triangles. Here is an example from Pitiscus.

A field has five straight edges of lengths 7, 9,10, 4 and 17 in order. The distance from thefirst to third vertex is 13 and from the thirdto fifth is 11. What is the area of the field?

The problem can be solved very quickly using Heron’sformula, but Pitiscus opts for trigonometry: we give amodernized version.

7

9

13

10

11

4

17

ab

cα

β γ

Applying the law of cosines to the three large triangles,

cos α =72 + 132 − 92

2 · 7 · 13=

1372 · 7 · 13

=137182

cos β =172 + 132 − 112

2 · 17 · 13=

3372 · 17 · 13

=337442

cos γ =42 + 112 − 102

2 · 4 · 11=

372 · 4 · 11

=3788

The values of α, β, γ and therefore the altitudes of the three major triangles could be read off a table,or found exactly using Pythagoras’:

a = 7 sin α = 7√

1− cos2α =7

182

√1822 − 1372 =

326

√1595,

b = 13 sin β, c = 4 sin γ

The total area is then easily computed:

A =12(13a + 17b + 11c) =

14

(3√

1595 +√

81795 + 5√

255)

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Johannes Kepler (1571–1630) Kepler was a student of Tycho Brahe8 for the last two years of thelatter’s life. He inherited Brahe’s position, his decades of accurate data and his philosophy on theimportance of theory based on observation. Kepler also had a contrasting side, being influenced bymystical ideas of perfection. He was a devotee of the Pythagorean view that nature reflects harmony;any observation of a natural ratio was something of great import. For example, in observing that thedaily movement of Saturn at its furthest point from the sun was roughly 4/5 of that at its nearestpoint, his temptation was to assume that ‘roughly’ must be ‘exactly.’ His belief in hidden beautyindeed partly drove his scientific pursuits; far from a rare combination among modern scientists!

Thanks to Brahe, Kepler had accurate data on roughly 13orbits of Mars and 2 of Jupiter on which to base his calcu-lations. These ultimately resulted in his three laws.

1. Planets move in ellipses with the sun at one focus.

2. The radius of the orbit sweeps out equal areas inequal times. In the picture, the sectors all have thesame area: the planet therefore moves more slowlythe further it is from the sun.

3. The square of the orbital period is proportional to thecube of the semi-major axis of the ellipse: T2 ∝ a3.

Kepler’s laws are primarily empirical observations ratherthan the result of mathematical proof. His process of dis-covery however required great mathematical ability: hereis a summary:

2a

• Kepler began by assuming the essential correct-ness of Copernicus’ model; that all planets, in-cluding the Earth, travel round the sun in uniformcircular motion. He studied Mars first because ofhis wealth of data on its motion.

• He only explicitly knew directions from the Earthto each planet, not distances. Under his assump-tion of uniform circular motion, he could makean estimate of the relative distance to each planet.He knew the direction (gray line) from the Earthto Mars at equally spaced times: by drawing cir-cles of different radii, he could observe that thelocations of Mars were not equally spaced if itsorbit were one of the dashed circles.

• To fit the model, Kepler had to convert all observed data (longitude and latitude of each planeton each date relative to Prague) to absolute measures relative to the assumed center of motion,

8Tycho Brahe (1546–1601) was a Danish astronomer who worked for Austro-Hungarian Emperor Rudolph II in Praguefor 25 years, producing a wealth of extremely accurate astronomical measurements. While these helped burnish the Coper-nican theory, he is better known for his 1572 observation of a nova (a new star that later disappeared, now known to be thedeath of a star) and then a comet in 1577, which provided yet more evidence of the changeability of the heavens.

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the sun, thus requiring a huge number of trigonometric calculations. Note again that he had noabsolute notion of distance, so everything was relative and subject to errors of estimation.9

• Kepler now altered the model to reflect the Earth’s slightly non-circular orbit. He first tried anequant model(!), offsetting the center of the orbit slightly from the sun. Despite this, he keptfailing to fit his data for Mars to pure circular motion and so had to modify his assumptionsfurther: planets would be permitted to move in ovals.

• He decided to approximate the oval motion of Mars using an ellipse and set out to calculateits parameters, eventually stumbling on a perfect match when the sun was placed at a focus.Since the foci of an ellipse are geometrically significant, natural beauty was preserved! Thisestablished his first law for Mars. Now knowing what to look for, he repeated the exercise forthe other known planets (Mercury, Venus, Earth, Jupiter, Saturn) as well as he could given hisinferior data.

• Kepler’s second law followed an infinitessimal argument based on inspired guesswork. Heassumed that planetary velocity was inversely proportional to the distance from the sun (v = k

r )and approximated areas of sectors swept out by the radius vector. Over a small interval of time∆t, a planet would travel a distance v∆t and thus sweep out an approximate triangle of area12 rv∆t = 1

2 k∆t. In modern language, Kepler’s second law is simply the conservation of angularmomentum: k is proportional to this. Kepler had no other justification beyond the fact that itseemed to fit the data. In particular, he had no idea why a planet should move more slowlywhen further from the sun.

• The third law was stated without analysis: given the relative sizes of each of the planetaryorbits and their periods, he simply observed that T2

a3 is approximately the same value for each.

Kepler’s discoveries were published over many years in several texts, some more mystical than oth-ers. His masterwork on the matter, Epitome astronomiæ Copernicanæ, was published in 1621. WhileKepler’s laws are essentially empirical, within a century, Issac Newton could be described as havingrigorously proved them based on his own axioms: an inverse square law for gravitational accelera-tion, his own three laws of motion, and the theory of calculus.

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,10

now known as the counter-reformation. Of particular interest to science and mathematics was thecreation of the Index Librorum Prohibitorum, a list of prohibited books whose contents contradictedChurch doctrine. This was also a response to the technology of mass-printing, which made thedissemination of new ideas easier. The ultimate result was the slow ceding of scientific power tonorthern (Protestant) Europe where papal diktat had no effect. Heliocentrism came quickly underattack: Kepler’s book was banned immediately upon publication in 1621. His distance from Rome,however, meant that Kepler and his ideas were relatively safe.

9Brahe, following Ptolemy, thought that the Earth-Sun distance was around 1/10 of the true value. Kepler believed thisto be an underestimate by at least a factor of three. It was not until 1659 that Christiaan Huygens found the distance to anaccuracy of 3%.

10Martin Luther’s publication of his Ninety-five Theses in 1517 is generally considered the start of the Protestant Reforma-tion: over the next 150 years Europe saw several religious wars as various countries broke away from Catholicism.

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In contrast to the counter-reformation crackdown, Pope Gregory is also famous for shepherding anastounding scientific achievement: calendar reform. By 1500, observations and calculations were ac-curate enough to show that the solar year is roughly 11.25 minutes shorter than the 365.25 days ofthe Julian calendar.11 Over 12 to 13 centuries, the Church had been basing the date of Easter12 on thevernal equinox; the accumulated error in this date had grown to 10–11 days. The impetus to correctthe date of Easter meant that calendar reform became an important church project.

Over 100 years of effort on the part of the Church involving noted astronomers and mathematicians,13

resulted in the Gregorian calendar. Gregory instituted the new calendar in 1582, imposing it on allCatholic countries. It corrected the 10 day deficit by deleting October 5th–14th 1582 and reformingthe computation of leap years: every 4th year is a leap year, excepting non-millennial centuries. Thus1900 was a normal year and 2000 a leap year. The Gregorian calendar is astonishingly accurate, losingonly one day every 3000 years. Since it emanated from Rome, many Protestant parts of Europe tookdecades if not centuries to adopt the new calendar. The Eastern Orthodox Church still uses the Juliancalendar for the computation of Easter.

Galileo Galilei (1564–1642) Based in northern Italy, Galileo was close to the center of Church powerand, unlike Copernicus and Kepler, he openly challenged its orthodoxy. While undoubtedly a greatmathematician (we shall see some in a moment), he is more importantly considered the father of theScientific Revolution for his reliance on experiment and observation. He famously observed Jupiter’smoons with a telescope (of his own invention) and noted that the presence of other objects orbitingan alien body was counter to Ptolemaic theory: skeptics, when shown this image, preferred to assertthat the image must be somewhere inside the telescope!

In 1632 Galileo published Dialogue Concerning the Two Chief World Systems, a Socratic discussion be-tween two philosophers: Salviati argued for Copernicus, while Sagredo was an independent ques-tioner. In addition, a layman Simplicio argued for the Ptolemaic system. The character of Simpliciowas modeled on various conservative philosophers who refused to look at experiments: more dan-gerously, he bore a resemblance to the Pope. While Galileo ostensibly strove to present a balanceddiscussion, Salviati almost always came out on top and Simplicio was made to appear foolish. Thetext 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.

Despite the best efforts of the Church, Galileo’s previous works continued to be distributed by hissupporters and he continued working. His most important scientific text, Discourses Concerning TwoNew Sciences (materials science and kinematics) was smuggled out to be published in Holland in 1638.In this book he resurrects his characters from Two Chief World Systems, but is more even-handed inhis treatment of the arguments. In particular, Galileo famously refutes Aristotle’s claim that heavierobjects fall more rapidly than lighter ones. Here are two mathematical results from this text.

11Named for Julius Caesar, a year has 365 days with an extra (leap-) day every 4th year. Months are the same as in ourmodern calendar, with varying lengths, all approximating the lunar cycle ≈ 29 1

2 days.12The Bible fixes Easter (the resurrection of Jesus) via reference to the moon. For roughly 1200 years, the Church had

decreed Easter to be the Sunday after the first full moon after the vernal equinox of March 20th/21st. This moment could,by 1500, be determined and predicted with great accuracy, yet it was happening roughly 10 days early.

13Pope Sixtus IV tried to recruit Regiomontanus to the cause in 1475, though the mathematician died first. Copernicuswas among those invited to consider proposals in the early 1500’s, though he distanced himself, perhaps because he knewthat his developing heliocentric ideas would not be accepted in the mainstream.

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Theorem. If acceleration is uniform, then the average speed is the average of the initial and final speeds.

Proof. Galileo argues pictorially; we augment with a little algebra.

• CD is the time axis, increasing downward.

• Velocity is measured horizontally as the distance from CD to the uniformlysloped line AE.

• As the object passes point A it has speed vA.

• Its speed increases to vB uniformly over the time interval t = |AB|.Clearly

area(4ABE) =12

t(vB − vA) = area(�ABFG)

But then

t · average speed = distance travelled under motion

= t · vA + area(4ABE) =12(vB + vA)

= t · average of initial and final speeds

Corollary. A falling object dropped from rest will traverse distance in proportion to time-squared. Otherwisesaid, d1 : d2 = t2

1 : t22.

This is Galileo’s way of stating the well-known kinematics formula d = 12 gt2.

Proof. Suppose that d1, d2, v1, v2 represent the distances travelled and the speeds of a body at times t1and t2. Since acceleration is uniform, we have

v1 : v2 = t1 : t2

By the Theorem,

d1 =0 + v1

2t1 =

12

v1t1, and d2 =12

v2t2

whence

d1 : d2 = v1t1 : v2t2 = t21 : t2

2

Galileo then proves that projectiles follow parabolic paths by asserting that their motion can be de-composed into a horizontal (uniform speed) motion and a vertical (uniform acceleration) motion.

Several other important mathematical topics were broached in his text; we shall return to these whenwe discuss calculus. While his mathematical ideas were cutting-edge for his time, it is his insistenceon testing theory against real-world data that makes him a true revolutionary. It might seem amaz-ing to us to consider how few of Aristotle’s easy-to-refute claims had ever been tested, though thehostility Galileo provoked by so-doing perhaps explains why. This is the core of the scientific rev-olution: scientists acknowledging the primacy of experiment and observation over ancient ‘wisdom.’

Galileo was finally cleared of heresy by the Church in 1992.

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