3 Renaissance mathematics Flashcards
RENAISSANCE MATHEMATICS
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
Dramatis Personae
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)
NUMBER SYSTEM FOR CUBICS
- integers used
- irrationals used
- couldnt cope with negative numbers
- no complex numbers but they became “accepted”
- RHETORICAL ALGREBRA- written in words
Prologue: Sixteenth century
Prologue: Sixteenth century
University positions temporary
Public problem-solving contests
Scipione del Ferro
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.
Niccolò Tartaglia (1500 - 1557),
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.
Girolamo Cardano
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
Poem told to Cardan by Tartaglia in 1539
“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.
BOOKS IN CUBIC CONTROVERSY
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.
Rafael Bombelli
In 1572 Rafael Bombelli published parts of Algebra, giving rules for calculating with complex numbers.
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]
17th problem set by Fiore to Tartaglia in Venice, 1535.
Tartaglia won
CUBIC TIMELINE
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
NOT IN EXAM
using verses to solve
x^3 + cx =d
Find u and v st u-v=d and uv= (c/3)^3
Then x= cuberoot(u) - cuberoot(v)
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.
Tartaglia 1539
CITY GIRDLED BY THE SEA
VENICE comp 1534
MILAN told
ACT III: Scene I - 1536
CARDAN
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
ARS MAGNA
DATE***!!
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
QUESITI ET INVENTIONI AND AFTER
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
COMPLEX IN CUBICS
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
IMPORTANT?
*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.
CARDANS RULE
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?
MATHEMATICS IN RENAISSANCE BRITAIN
TIMELINE
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
The Chaucer astrolabe
Treatise on the Astrolabe
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
ROBERT RECORDE
one sentence: about books he wrote
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
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.
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
WHETSTONE OF WITTE
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
THOMAS HARRIOT
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
Thomas Harriot plaque
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
JOHN NAPIER
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
*
1615: NApier meets Briggs
- 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
ROBERT RECORDE
extra facts
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.
*
his books and their contents ROBERT RECORDE
one sentence: about books he wrote
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
his books and their contents THOMAS HARRIOT
one sentence: about books he wrote
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
his books and their contents JOHN NAPIER
one sentence: about books he wrote
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
