3 Renaissance mathematics

Question 1

RENAISSANCE MATHEMATICS

Answer 1

REBIRTH IN LITERATURE AND ART

Revival in mathematics slightly later

GREAT CUBIC CONTROVERSY
Time : 16th Century
Place : Northern Italy

solve cubic equations algebraically by formula (NOT BY DIAGRAM)

ALGEBRAIC SOLUTIONS

Question 2

Dramatis Personae

Answer 2

Scipione del Ferro (1465 - 1526)
 Antonio Maria Fior (c. 1506)
 Niccolo Tartaglia (1500 - 1557)
 Girolamo Cardano (1501 - 1576)
 Ludovico Ferrari (1522 - 1565)
 Rafael Bombelli (1526 - 1572)

Question 3

NUMBER SYSTEM FOR CUBICS

Answer 3

• integers used
• irrationals used
• couldnt cope with negative numbers
• no complex numbers but they became “accepted”
• RHETORICAL ALGREBRA- written in words

Question 4

Prologue: Sixteenth century

Answer 4

Prologue: Sixteenth century
University positions temporary
Public problem-solving contests

Question 5

Scipione del Ferro

Answer 5

In around 1515 Scipione del Ferro (1465 - 1526), a Mathematics professor at the University of Bologna, solved cubic equations of the form:

x^3 + ax=b
‘cube and things equal numbers’

His solution was never published due to the common practice of withholding new mathematical methods; Universities had no tenure and so problem solving contests were often held in sixteenth century Italy, with the winners likely to maintain their positions. Before Del Ferro died in 1526 he managed to tell his son-in-law della Nave and student Fiore.

Fiore replaced del Ferro at the University, however, he is described as a mediocre mathematician.
He became increasingly boastful about being able to solve cubics and challenged Tartaglia in 1535.

Question 6

Niccolò Tartaglia (1500 - 1557),

Answer 6

Niccolò Tartaglia (1500 - 1557), ‘the stammerer’, was particularly interested in ballistics and published the first italian translations of Euclid and Archimedes. He was also able to solve cubics such as:

x^2+bx^2=d
‘cube and squares equal numbers’

Tartaglia independently solved del Ferro’s type in time for the competition. Thus, he was able to solve all 30 of the del Ferro type problems. Unfortunately for Fiore, the problems set by Tartaglia proved much too challenging and Tartaglia won.

Ten days before contest, Tartaglia found algebraic solutions to cubic equations of del Ferro type

22 February 1535 contest in Venice: Tartaglia winner

All Fior’s challenges reduced to del Ferro type cubics

Tartaglia initially refused to reveal his algebraic solutions to
cubics because was translating the Elements into Italian and hoped to publish his own research on cubics for the whole world to see.

Question 7

Girolamo Cardano

Answer 7

Girolamo Cardano (1501 - 1576) published an important book on algebra, the Ars Magna.

Eventually, tempted by the promise of meeting the governor of Milan as a prospective patron, Tartaglia visited Cardan in Milan

He contacted Tartaglia with the hopes of learning his method. He was able to convince him by promising to keep it a secret and introduce. Tartaglia to the governor of Milan, as he hoped to gain a job at the Milanese court. Tartaglia provided his solution in the form of a poem but began to question his decision. Cardan made the effort to continue the friendship however he rebuffed him.

Cardan traveled to Bologna in 1543 and learned that del Ferro was the first to solve this type. He used this to justify publishing his Ars Magna but still credits Tartaglia, del Ferro and student Ferrari (who was able to solve quartic equations).

Aware of complex but ignored

Question 8

Poem told to Cardan by Tartaglia in 1539

Answer 8

“When the cube and things together
Are equal to some discrete number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing”

Poem told to Cardan by Tartaglia in 1539

25 March 1539: Cardan’s will prevailed. He swore an oath never to make public Tartaglia’s secret, revealed to him in twenty-five lines of rhyming verse.

Question 9

BOOKS IN CUBIC CONTROVERSY

Answer 9

Tartaglia felt betrayed and included solutions on cubics along with the problems from 1535 in Quesiti et Inventioni the following year.

1546 Tartaglia publishes Quesiti et inventioni diverse

In 1548, a competition was held between Ferrari and Tartaglia, however, Tartaglia fled in the night.

Question 10

Rafael Bombelli

Answer 10

In 1572 Rafael Bombelli published parts of Algebra, giving rules for calculating with complex numbers.

Question 11

There is a tree 12 braccia high,….

Example Tree 12 units high is broken in two. Height of tree
remaining is cube root of length cut away. What is height of
tree remaining? [ x3
+ x = 12, x is height remaining]

Answer 11

17th problem set by Fiore to Tartaglia in Venice, 1535.

Tartaglia won

Question 12

CUBIC TIMELINE

Answer 12

1515 ‘del Ferro type’ solved
1526 del Ferro dies
1535 Competition between della Nave & Tartaglia in Venice
1539 Meeting between Cardan & Tartaglia in Milan
1543 Cardan travels to Bologna
1545 Ars Magna published
1546 Tartaglia publishes Quesiti et inventioni diverse
1548 Competition between Ferrari & Tartaglia
1572 Bombelli publishes parts of Algebra

Question 13

NOT IN EXAM

using verses to solve

Answer 13

x^3 + cx =d

Find u and v st u-v=d and uv= (c/3)^3

Then x= cuberoot(u) - cuberoot(v)

Question 14

These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea.

Answer 14

Tartaglia 1539
CITY GIRDLED BY THE SEA
VENICE comp 1534

MILAN told

Question 15

ACT III: Scene I - 1536

CARDAN

Answer 15

Cardan hires 14 year old Ludovico Ferrari
Relationship between them rapidly changed:
master ~ servant, teacher ~ pupil, colleague ~ colleague

Discovered algebraic solutions to cubics & quartics
Wished to publish. Stymied by Cardan’s oath to Tartaglia
Visited Bologna in 1543 to inspect del Ferro’s papers
del Ferro first to find algebraic solutions to cubics

Question 16

ARS MAGNA

DATE***!!

Answer 16

ACT III: Scene II - 1545
Cardan publishes Ars Magna
solutions to all thirteen types of cubic Ferrari’s solution to quartics full attributions given, including credit to Tartaglia

WRITTEN IN FIVE YEARS, MAY IT LAST AS
MANY THOUSANDS

*in the verse they give 3 solutions(ie to solve 3 types of cubic equations) he wasn’t given the proofs and for 13 different types he finds the formula and 10 solutions

cubic and quartic

Question 17

QUESITI ET INVENTIONI AND AFTER

Answer 17

ACT III: Scene III - 1546
Tartaglia incensed by publication of Ars Magna
accused Cardan of deceit. Told his side of the story in
Quesiti et Inventioni diverse, which
contains verse he gave Cardan & problems set him by Fior
Volatile letters flew between Tartaglia and Ferrari
Culminated in a public contest, Milan, 10 August 1548
Ferrari seems to have won, Tartaglia’s fortune waned
Rule for solving cubics is known as Cardano’s formula

extra chapter how annoyed he was

Question 18

COMPLEX IN CUBICS

Answer 18

Cardan accepted negative solutions to equations, describing them as fictitious. His procedure for solving the cubic equation

as cubic has three real roots:
Rafael Bombelli
In Ars Magna Ars Magna Cardan considered problem:
Divide 10 into two parts so their product is 40

Quadratic techniques led him to solutions:
described as refined as they are useless. Great book ends

WRITTEN IN FIVE YEARS, MAY IT LAST AS MANY THOUSANDS

Question 19

IMPORTANT?

Answer 19

*CARDAN was illegitimate
for some reason wasn’t allowed to practise as a doctor in Milan wasn’t expected to live and bathed in wine
. predicted the date of his death and starved himself to meet this. Gambled everyday.
Thrown in prison for writing a horoscope of crime but released by the pope?

• TARTAGLIA was injured on his head and face by sabre cuts
  meant that in later life he stuttered , he wore a beard to cover scars, wasn’t expected to survive and licked by a dog

*In Ars Magna dates and comment on including rule
Ars Magna Cardan considered problem:
Divide 10 into two parts (NUMBERS) so their product is 40
answers 5+- sqrt(-15)
describes as REFINED AS THEY ARE USELESS.

Question 20

CARDANS RULE

Answer 20

solving cubic equations?
Cardano’s formula

x^3 + cx =d

Find u and v st u-v=d and uv= (c/3)^3

Then x= cuberoot(u) - cuberoot(v)

from verses?

Question 21

MATHEMATICS IN RENAISSANCE BRITAIN

TIMELINE

Answer 21

670 Archbishop Theodore teaches ecclesiastical computation
(running of church maths calc of easter day involves also astronomy)

720 Venerable Bede writes handbooks on the computus
1391 Chaucer: English treatise on astrolabe (prev books written in latin)
1509 Henry VIII

1543 Recorde: Ground of Artes
1547 Edward VI
1551 Recorde: Pathway to Knowledge
1557 Recorde: Whetstone of Witte
1558 ELIZABETH 1, Record dies in Prison
1585 Harriot: Granville expedition to Virginia
1588 Harriot: Reprt of New Found Land of VIrginia
1603 James 1
1614 Napier: Mirifici logarithmorum canosis DESCRIPTIO
1616 Edward Wright: translates NAPIER’s DESCRIPTIO
1617 Nampier dies. Briggs: common logs 1-1000, fourteen dp
1619 Napier: Mirifici logarithmorum canosis CONSTRUCTIO
1621 Harriot dies Oughtred slide rule?
1624 Briggs extends 1617 table to beyond 20,000 places
1631 Artis analyticae praxis

Question 22

The Chaucer astrolabe

Treatise on the Astrolabe

Answer 22

diagram

circular with markings

Treatise on the Astrolabe 1391
to his son lowis he writes how he is considering to learn the treatise of the Astrolabie and will show him in ENglish as he is too young for latin, but be happy to learn in english.
Geoffrey Chaucer

Question 23

ROBERT RECORDE

one sentence: about books he wrote

Answer 23

1510 -1558 WELSHMAN

PLAQUE AT ST MARYS CHURCH

English Treatises on ALGEBRA ARITHMETIC ASTRONOMY and GEOMETRY

no picture of him only thought of as picture

REMEMBERED FOR WRITING ENGLISH TEXTBOOKS 4 and very successful

*some works are written in dialogue form to aid students (not 2nd), made interesting and added jokes so very successful

(english textbooks, not latin, first examples of books in english)
physician to queen mary and edward 6
INVENTED THE EQUALS SIGN

1510 Tenby, south Wales
1531 BA Oxford Uni
1543 GROUND OF ARTES-~COMMERCIAL/ BASIC ARITHMETIC
(most famous)

1545 MD Cambridge uni

1547 Urinal of Physick- medical treatise

1551- Pathway to Knowledge~ plane geometry
**
(geometry, related to first 4 books of Euclids elements but NO PROOFS, NOT in form of scholar and teacher)

invented STRAIGHT LINE- no markings stretched linen for ruler

1556 Castle of Knowledge ~ spherical geometry/astronomy

1557 Whetstone of Witte ~ ALGEBRA

1558 dies Kings Bench Prison, London

2008 Conference in Wales marks 450th anniversary of death

Question 24

THE PATHWAY TO KNOWLEDGE
the commodities of geometry

Robert recrode 1551

sith Merchauntes by shippes great riches do winne.
I met with good right at their seat beginne.
the ships on the sea with sale and with all. Were first founded and still made by geometries law.

the compass the card the bullets that anchors. Were founded by the skill of witty do you want metres. To settle for the cap Stoke and at you two other part. Would make great show of geometries art.

Carpenters carvers joiners and Masons, paintings and limners with such occupations. Brothers, goldsmiths if they be coming. Must year 2 geometry thanks for their learning.

The card and the plough who does makes them well. I made by good geometry. And so in the wake of Taylor’s and shoemakers in all shapes and fashion. The work is not praise if it wanted proportion.

So Weavers by geometry Hade their foundation
the Loom is a frame of strange imagination. The wheel that dustbin, the stone that. Grind. The mill that is driven by the water or the wind
are works of geometries strange and their trade .
Few could them devise, if they were unmade.

Answer 24

THE PATHWAY TO KNOWLEDGE
the commodities of geometry

you need geometry to build /make ships, naval, carpenters, painters, embroiderers, millers, farmers, shoemakers

GEOMETRY is important
used and copied

Question 25

WHETSTONE OF WITTE

Answer 25

Robert Recorde Whetstone of Witte introduces MODERN EQUALS SIGN A pair of paralleles, because no 2 things can be more equal Recordes farewell Master: WHat meaneth that hastie knocking at the door? Scholar: It is a messenger Master: My fortune is not so good to have time to teach Scholar: my harte is so oppressed i can not express my grief Master: Amen and amen ROBERTE RECORDE 1557 last book last passage taken to the debtors prison in london

Question 26

THOMAS HARRIOT

Answer 26

1560-1621 ENGLISHMAN 3 pictures but not sure which one if any right *pure, applied, *LIVED COMFORTABLY SO NEVER HAD TO PUBLISH WORK sent on expeditions and employed most of his life being employed by wealthy people *famous as student and thus timeline *greatest ENglish mathematician before Isaac newton 1560 Born Oxford 1577 ENters Oxford University 1580 Graduates 1584 Employed by Raleigh 1585 Surveyor on Grenvilles Virginian expedition (FIRST MATHEMATICIAN in AMERICA) 1588 Briefe and true report of the new found land of Virginia 1591 Hexagonal nature of snowflakes (FIRST TO WITNESS THAT SNOWFLAKES ARE HEXAGONAL) 1598 Employed by Earl of Northumberland 1601 Law of refraction (unpublished) 1605 Imprisoned following Gunpowder Plot 1607 Observes Halleys Comet 1609 Uses telescope in lunar observations 1621 Dies London from nostril cancer (taught how to inhale tobacco and first reported death by cancer) 1631 Artis Analyticae praxis 1971 Harriot plaque unvieled at Bank of England great fire of london had destroyed gravestone (looked at cardans work on cubics, quartics, artis analticae praxis published and FIRST TO FACTORISE) INVENTED INEQUALITY SIGN

Question 27

Thomas Harriot plaque

Answer 27

Bank of england Thomas Harriot was buried in the church of St Christopher le Stocks in Threadneedle Street, near where he died. The church was subsequently damaged in the Great Fire of London, and demolished in 1781 to enable expansion of the Bank of England. Stay, traveller, lightly tread; Near this spot lies all that was mortal Of that most celebrated man THOMAS HARRIOT He was that most learned Harriot Of Syon on the RIver Thames; By birth and education An Oxonian Who cultivated all the sciences And excelled in all In all- in Mathematics, Natural Philosophy, Theology A most studious searcher after truth, A most devout worshipper of the Triune GOd, At the age of sixty, ot thereabouts He bade farewell to mortality not to life The year of our Lord 1621, July 2 he was a celebrated astronomer observed Halley's Comet survived the heavens with a newly invented telescope his map of the Moon drawing of sunspots and satellites of Jupiter survive he studied Optics refraction by prism snowflakes and Rainbows however he wrote no mathematical tree ties in his lifetime and he's been dof the founder of the English School of algebraic on account of his work on the theory of equations and tension to mathematical notation. He found relationships between roots polynomials... He introduced inequality signs employed lower case files 4 unknowns represented powers he improve the theory and practice of maritime navigation interesting mercator map projection

Question 28

JOHN NAPIER

Answer 28

1550-1617 SCOTSMAN * Famous for decimal point and inventing logarithms * by inventing logarithms tables easing calculations doubles the life of the astronomer * first table of logarithms 1550 Born Merchiston Castle, Edinburgh 1563 Matriculates at St Andrews 1594 A plaine discovery of the whole revelation of Saint John (anti catholic, predicted world would end in 1700, written in English) 1614 Mirifici logarithmorum canonis DESCRIPTIO (DECIMAL POINT) 1615 Henry Briggs visits Napier 1616 Edward Wright translates Mirifici into ENglish (latin then translated into english) 1617 DIes Merchiston Rabdologiae (Napiers Bones) -------- Briggs common logarithms from 1 to 1000 to 14dp 1619 Mirifici logarithmorum canonis CONSTRUCTIO tells you how the tables were constructed 1621? William Oughted invents logarithm SLIDE RULE (used the logarithm scales to invent the slide rule) applied mathematicians use a slide rule to calculate 1624 Briggs extends 1617 table to 20,000 places 1914 Tercentenary celebrations: Royal Society of Edinburgh 2014 Quatercentenary celebrations: Royal Society of Edinburgh *many children *

Question 29

1615: NApier meets Briggs

Answer 29

* when Lord merchiston made public has logarithms Mr Briggs of Gresham College, London, was so surprised with admiration of them, he could have no quietness and himself, until he had seen that no woman, who's invention they were. * he brings Mr Briggs up to Milos chamber, well almost one-quarter of an hour was spent, each be holding the other with admiration before one word with spoke, at last Mr Briggs began

Question 30

ROBERT RECORDE | extra facts

Answer 30

his four mathematical published works including the urinal of physic the medical tract display his teaching skills, and ability to present clear and logical arguments in an engaging style. * All but his geometry text, the pathway to knowledge, or written as dialogues between master and scholar with some lapses into verse * he we thought how mathematics was and should be taught, in the circumstances of that time, and increasing readership for basic mathematical text, growing trade and prosperity, and developing technological capabilities for peace and War. His wish to spread mathematical knowledge as widely as possible lead him to write in English, does opening up learning to those that did not read Latin. He was a humanist scholar who stress the importance of teaching style that arose out of practical usefulness, and his textbooks admirably Woodfield the needs of the day, enjoying an immediate and enjoying success. *

Question 31

his books and their contents ROBERT RECORDE one sentence: about books he wrote

Answer 31

REMEMBERED FOR WRITING ENGLISH TEXTBOOKS 4 and very successful 1543 GROUND OF ARTES-~COMMERCIAL/ BASIC ARITHMETIC (most famous) it was the first mathematical book to be published in Englishand was an unprecedented popular commercial arithmetic running through 50 printing in the last 1699 1547 Urinal of Physick- medical treatise 1551- Pathway to Knowledge~ plane geometry **** (geometry, related to first 4 books of Euclids elements but NO PROOFS, NOT in form of scholar and teacher) invented STRAIGHT LINE- no markings stretched linen for ruler in the pathway to knowledge he gives a discursive treatment of the first four books of the elements, giving explanations but omitting proofs. Because the language of the time was English it was and was deficient in technical geometrical terms he invented his one and only one of straight line which survives 1556 Castle of Knowledge ~ spherical geometry/astronomy was the first English astronomer text to accept copper canices theory that the Earth revolves around the sun 1557 Whetstone of Witte ~ ALGEBRA his most cited work, which introduces the modern equals sign, reaches a dramatic climax when the Masters teaching is interrupted by a knock at the door, signalling both the end of the algebra lesson and records distinguished career. He takes leave as a student and is Led away to prison where he died the following year. A conference to mark the 450th anniversary of his death was organised in Wales in 2008

Question 32

his books and their contents THOMAS HARRIOT one sentence: about books he wrote

Answer 32

1588 Briefe and true report of the new found land of Virginia he was appointed as a surveyor to Greenhills expedition to the new world and he busied himself in all aspects of navigational theory and practice and wrote 1591 Hexagonal nature of snowflakes (FIRST TO WITNESS THAT SNOWFLAKES ARE HEXAGONAL) (unpublished) 1601 Law of refraction (unpublished) 1631 Artis Analyticae praxis he wrote no mathematical tree ties in his lifetime, although ten years after his death someone has algebraic work was published in Latin translated into English in 2007 the manuscript in the British Museum

Question 33

his books and their contents JOHN NAPIER one sentence: about books he wrote

Answer 33

List of works (1593) A Plaine Discovery of the Whole Revelation of St. John (1614) Mirifici logarithmorum canonis descriptio (Edward Wright's English translation was published in 1616). he discovered logarithms and in this book its own table of logarithms (1617) Rabdologiæ seu Numerationis per Virgulas libri duo (published posthumously) Rabdology (Wikipedia) (1619) Mirifici logarithmorum canonis constructio (written before the Descriptio, but published posthumously by his son Robert) book detail the construction of the logarithms the definition of logarithms that he gave made no mention of excellence, but invite ideas of Motion. his books were popular and had to be translated into English (1839) De arte logistica wasn't mentioned in the set book