WHO INVENTED THE EQUALS SIGN?
Professor Robin Wilson
In my last lecture I described the development of Islamic ideas in
southern Europe, up to the founding of the universities in Western
Europe. I’ll now continue the story by looking at mathematics in
the Renaissance, starting with Renaissance art, and continuing with
navigation, the calendar, the invention of printing, and some interesting
developments in algebra.
Renaissance art
One notable feature of Renaissance painting was that, seemingly for the
first time, painters became interested in depicting three-dimensional
objects realistically, giving visual depth to their works, as contrasted
with earlier works such as the Bayeux tapestry where such depth is not
to be found. This soon led to the formal study of geometrical perspective.
The first person to investigate perspective seriously was the
artisan-engineer Filipo Brunelleschi, who had designed the self-supporting
octagonal cupola of the cathedral in Florence. Brunelleschi’s ideas were
developed by his friend Leon Battista Alberti, who presented mathematical
rules for correct perspective painting and stated in his Della pittura
[On painting] that ‘the first duty of a painter is to know geometry’.
Piero della Francesca was another who investigated mathematical
perspective. In particular, he used a perspective grid in his
investigations into solid geometry, and wrote books on the perspective
of painting and the five regular solids. This 1472 picture, his Madonna
and child with saints, shows his mastery of perspective.
Another work of the time was a 1509 book On divine proportion on regular
polygons and polyhedra by Piero’s friend Luca Pacioli, whom we’ll
meet again later. The woodcuts of polyhedra for this book were prepared
by Pacioli’s student Leonardo da Vinci, who explored perspective more
deeply than any other Renaissance painter, and whose notebooks contain
much of mathematical interest. In his treatise on painting, da Vinci
warns ‘Let no one who is not a mathematician read my work’.
Albrecht Dürer was a celebrated German artist and engraver who learned
perspective from the Italians and introduced it to Germany. He produced a
number of drawings showing how to realise perspective, and his famous
engravings, such as St Jerome in his study, show his effective use
of it. His Melencolia is also well known, and features a number of
mathematical items, such as a truncated tetrahedron and a 4 × 4 magic
square in which the date of the engraving (1514) appears in the middle
of the bottom row.
The age of exploration
The Renaissance coincided with the great sea voyages of Vasco da Gama,
Columbus and others. Portuguese explorers sailed south and east, with
Vasco da Gama becoming the first European to sail around the tip of
Africa and reach the west coast of India.
Meanwhile, their rivals the Spanish headed west, hoping to reach India by
circumnavigating the globe. From 1492 Christopher Columbus, a navigator
of genius, led four royal Spanish expeditions to pioneer a western route
to the Indies. His expeditions reached, not India, but the new lands of
North and Central America, the West Indies, and the coast of Venezuela.
Such nautical explorations made necessary the construction of accurate
maps and globes and led to major developments in map-making.
Around 1500 European navigators rediscovered Ptolemy’s Geographia and
his maps came to be used extensively by navigators. Ptolemy’s writings
contained detailed discussions of projections for map-making and included
a ‘world map’ featuring Europe, Africa and Asia as well as many
detailed regional maps. With the invention of printing, woodcut copies
could easily be mass-produced and a number of editions appeared in the
sixteenth century, each one revised to take account of new explorations.
But solving the problem of accurately representing the spherical earth
on a flat sheet of paper was not easy, and this led to new types of map
projection – most notably the Mercator projection, named after Gerard
Mercator. The first ‘modern’ maps of the world were due to him,
and he coined the word ‘atlas’ for his three-volume collection of
maps in 1585–95.
Roughly speaking, the Mercator projection can be obtained by projecting
the sphere outwards on to a vertical cylinder and then stretching the
map in the vertical direction in such a way that the lines of latitude
(horizontal) and longitude (vertical) appear as straight lines, and all
of the angles (compass directions) are correct.
In connection with this, one of the first Europeans to apply mathematical
techniques to cartography was Pedro Nunes, Royal cosmographer and
the leading figure in Portuguese nautical science. Nunes constructed
an instrument for measuring fractions of a degree, and showed how to
represent the path of a ship on a fixed bearing as a straight line called
a rhumb line or loxodrome.
Terrestrial and celestial globes were also used to represent the positions
of geographical and astronomical features. During the sixteenth century,
with the new interest in exploration and navigation, these became
increasingly in demand.
Such explorations also necessitated the construction of appropriate
navigational instruments to measure the altitude of heavenly bodies,
such as the sun or the pole star, so as to determine latitude at
sea. We’ve already encountered some of these, such as the astrolabe,
which reached its maturity during the Islamic period and took many
complicated forms. For navigational purposes a more basic and sturdy
version was needed, and this became known as the mariner’s astrolabe.
There were also armillary spheres, usually made of metal circles
representing the main circles of the universe, and used to measure
celestial coordinates or for instructional purposes.
Quadrants were in use in Europe from around the thirteenth century. As
their name suggests, quadrants have the shape of a quarter-circle
(90°); their relations, the sextant and octant, similarly correspond
to a sixth (60°) and an eighth (45°) of a circle. To measure an
object’s altitude, the observer views the object along the top edge
of the instrument and the position of a movable rod on the circular rim
gives the desired altitude.
Some years earlier, the Jewish mathematician and astronomer Levi ben
Gerson had invented the Jacob’s staff, or cross-staff, for measuring
the angular separation between two celestial bodies. Although widely
used, it had a major drawback – to measure the angle between the sun
and the horizon one had to look directly at the sun. The back-staff is
a clever modification in which a navigator can use the instrument with
his back to the sun.
Somewhat more complicated was an attractive astronomical instrument
gilt brass compendium of 1568, designed by Humfrey Cole for the wealthy
collector. Among the towns whose latitude is included is Oxford at 51
degrees, 50 minutes.
Calendars
Before the time of the Romans many different calendars were in use. As
early as 4000 BC the Egyptians used a 365-day solar-based calendar of
twelve 30-day months and five extra days added by the god Thoth. The
Greek, Chinese and Jewish lunar-based calendars consisted of 354 days
with extra days added at intervals, while the early Roman year had just
304 days. In 700 BC this was extended to 355 days, with the addition of
the two new months Januarius and Februarius.
In 45 BC Julius Caesar introduced his ‘Julian calendar’. This
had 365¼ days, the fraction being taken care of by adding an extra
‘leap day’ every four years. The beginning of the year was moved to
January and the lengths of the months alternated between 30 and 31 days
(apart from a 29-day February in leap years); this was later changed
by Augustus Caesar who stole a day from February to add to August and
altered September to December accordingly.
Later writers determined the length of the solar year with increasing
accuracy. In particular, the Islamic scholars Omar Khayyam and Ulugh Beg
independently measured it as about 365 days, 5 hours and 49 minutes –
just a few seconds out.
The Julian year was thus 11 minutes too long, and by 1582 the calendar had
drifted by ten days with respect to the seasons. In that year Pope Gregory
XIII issued an Edict of Reform, removing the extra days. He corrected the
over-length year by omitting three leap days every 400 years, so that
2000 was a leap year, but 1700, 1800 and 1900 were not. The Gregorian
calendar was quickly adopted by the Catholic World and other countries
eventually followed suit: Protestant Germany and Denmark in 1700, Britain
and the American colonies in 1752, Russia in 1917, and China in 1949.
Meanwhile, the line from which time is measured (0° longitude) was
located at the Royal Observatory in Greenwich in 1884, giving rise to an
international date line near Tonga. In 1972, atomic time replaced earth
time as the official standard, and the year was officially measured as
290,091,200,500,000,000 oscillations of atomic caesium.
The invention of printing
Johann Gutenberg’s invention of the printing press (around 1440)
revolutionised mathematics, enabling classic mathematical works to be
widely available for the first time. Previously, scholarly works, such
as the classical texts of Euclid, Archimedes and Apollonius had been
available only in manuscript form, but the printed versions made these
works much more widely available.
At first the new books were printed in Latin or Greek for the scholar,
and many scholarly editions appeared. The earliest printed version
of Euclid’s Elements, published in Venice in 1482, and there is an
attractive 1492 edition of Ptolemy’s Almagest. Apollonius’s Conics
appeared in 1537, and seven years later the works of Archimedes were
published in both Latin and Greek, and there was a celebrated edition
of Diophantus’s Arithmetic in 1621, reissued in 1670, with the Greek
text, a Latin translation by Bachet, and comments by Fermat, including
his famous marginal comment on the ‘last theorem’.
No doubt because of all these translations, there was a resurgence of
interest in Greek mathematics in the sixteenth century, stimulated in
particular by the massive publishing programmes of two mathematicians
in Italy – Federigo Commandino and Francesco Maurolico. Maurolico
translated and reconstructed works of Apollonius, Archimedes, Aristotle,
Ptolemy and others, while Commandino edited Latin versions of all these
and also Euclid, Aristarchus and Pappus.
These editions were all in Latin or Greek, for the scholar. But
increasingly, vernacular works began to appear at a price accessible
to all:
If cunning latin books were translate
Into english well correct and approbate
All subtle science in english might be learned
As well as other people in their own tongues did.
The new printed vernacular works included introductory texts in
arithmetic, algebra and geometry, as well as practical works designed to
prepare young men for a commercial career. Important among the former was
the 1494 Summa de arithmetica, geometrica, proportioni et proportionalita
of Luca Pacioli, a 600-page vernacular compilation of the arithmetic,
algebra and geometry known at the time; it is remembered for containing
the first published account of double-entry bookkeeping.
Particularly important for commerce at the time were the vernacular
commercial arithmetics, cont
aining computational rules and tables to help with financial
transactions. In Germany the most influential of these was by Adam Riese;
it proved so reputable that the phrase ‘nach Adam Riese’ [after Adam
Riese] came to indicate a correct calculation.
In Oxford the earliest books with any mathematical content to
be published were in Latin. The first was the attractive Compotus
of 1520, which included rules for calculating the date of Easter on
one’s fingers. Another book with Oxford connections was by Cuthbert
Tonstall, an Oxford scholar who migrated to Cambridge to develop what
would soon become a thriving mathematical community there. His 1522 De
arte supputandi was the first major arithmetic text to be published in
England, and was the best of its time.
The invention of printing also led to the gradual standardisation of
mathematical notation. In particular, the arithmetical symbols + and –
first appeared in a 1489 arithmetic text by Johann Widmann. Surprisingly,
the symbols × and ÷ were not in general use until the seventeenth
century – we’ll see how × developed shortly; the division sign ÷
was introduced by John Pell.
Needless to say, the quality of the mathematical printing in those days
was very variable. Here we see two version of Pascal’s arithmetical
triangle from the same year, 1545: Stifel’s publisher was having a
good day, while Scheubelius was less fortunate.
Tunstall was not the only migrant from Oxford to Cambridge – such
migrations were common in both directions. Most well known of these
was Robert Record, probably the most important writer of textbooks in
English. He studied at All Souls in Oxford, studied mathematics and
medicine in Cambridge, and later became physician to Edward VI and Queen
Mary in London before being thrown into jail for debt.
Record was such a fine lecturer that his audience regularly applauded
his lectures. We don’t know what he looked like. For a long time, there
was only one known picture of him, but recently severe doubts have been
raised as to its authenticity. One might well ask: ‘Is this a Record?’
Record’s books were written in English, and ran to many editions. The
ground of artes of 1543 was an arithmetic book explaining the various
rules so simply that ‘everie child can do it’. As with all his books,
it was written in the form of a Socratic dialogue between a scholar and
his master.
It also explains how to carry out multiplication. To multiply 8 by 7,
for example, we write them on the left, and opposite we subtract each
from 10 to give 2 and 3. Now 8 – 3 (or 7 – 2) is 5 and 3 ´ 2 = 6,
so we get 56. The cross eventually shrank and became the multiplication
sign we use today.
Record’s other books included the Castle of knowledge (on astronomy),
the Pathway to knowledge (on geometry), the Whetstone of witte (on
algebra), and my favourite, his delightfully-named book on medicine,
the Urinal of physic.
As I said before, the production of books was rapidly leading to a
standardisation in terminology and notation. Record introduced several
entertaining terminologies that didn’t catch on, such as sharp and
blunt corners for acute and obtuse angles, touch line for a tangent,
and threelike for an equilateral triangle, but he also introduced the
term straight line, which is still used.
Record’s most celebrated piece of notation made its first appearance
in the Whetstone of witte of 1557. Here we find the first appearance of
our equals sign:
And to avoide the tediouse repetition of these woordes: is equalle to:
I will sette as I doe often in
woorde use, a parre of paralleles, o: Gemowe lines of one lengthe, thus:
== because noe 2 thynges can be moare equalle.
These improvements in notation went hand in hand with developments
in calculation. Decimal fractions had taken many centuries to become
established throughout Europe. In the late fifteenth century the Flemish
mathematician Simon Stevin wrote a popular book De thiende [The tenth]
that explained decimal fractions, advocated their widespread use for
everyday calculation, and proposed a decimal system of weights and
measures. This work and its translations into other languages really
seemed to do the trick at last. Stevin also wrote an important treatise
on statics that included the first explicit use of the triangle of forces.
The first English edition of Euclid’s Elements was published in 1570 by
Henry Billingsley, a former Oxford student who managed to combine being
a translator of Euclid with being a prosperous London merchant. He later
became Lord Mayor of London and Member of Parliament for the City. His
book owes its success partly to the fact that it later became adopted
as a manual at Gresham College.
Billingsley’s Euclid opened with a ‘very fruitfull Praeface,
specifying the chiefe Mathematicall Sciences’, written by the alchemist,
astrologer and mathematician John Dee. In his far-reaching and influential
preface Dee classified the mathematical arts and sciences, particularly
arithmetic and geometry, into nineteen categories which he then discussed.
However, the science of this period was increasingly that of merchants
and craftsmen, rather than of Euclid and the ancient texts. As we’ve
seen, many of the new books were commercial arithmetics, containing
computational rules and tables to help with financial transactions,
while others involved practical skills, such as surveying. And by the
late sixteenth century, books on navigation appeared regularly, such
as Thomas Blundeville’s A new and necessarie treatise of navigation
containing all the chiefest principles of the Arte.
Around this time, the mathematical practitioner Thomas Harriot appeared on
the scene, possibly the greatest English mathematician that ever lived,
with extensive writings on geometry and exciting new work on algebra. He
is also remembered for helping Walter Ralegh to survey and colonise the
part of America now called Virginia. Harriot busied himself with every
aspect of navigational theory and practice, and his success is described
in a letter sent to Sir Walter Ralegh:
Ever since you perceived that skill in the navigator’s art might
attain its splendour amongst us if the aid of the mathematical sciences
were enlisted, you have maintained in your household Thomas Harriot,
a man pre-eminent in those studies, in order that by his aid your
own sea-captains might link theory and practice, not without almost
incredible results.
Harriot worked extensively in geometry, trigonometry, algebra and
combinatorics, and has been called the founder of the English algebra
school. To him we owe the symbols for < and >, a 2 and a 3, and the cube
root sign. Almost all his work is in manuscript, which is still being
worked through. But although he published very little, his posthumous
algebra book Artis analyticae praxis was very influential.
Cubic equations
This discussion of Harriot brings us to our last topic for today – the
solution of equations. Last time we saw how Islamic scholars, such as
al-Khwarizmi and Omar Khayyam, gave geometrical versions of the ancient
Mesopotamian method for solving particular quadratic equations. However,
very little progress had been made on solving cubic equations, even though
these arise in two of the ancient Greek classical problems – doubling
the cube and trisecting the angle. Omar Khayyam discussed equations in
general, going from roots and squares to cubes, and proceeding to square
squares, square cubes, and so on. He then systematically classified
cubic equations and attempted to solve one of the form a solid cube plus
squares plus edges equal to a number (x 3 + ax 2 + bx = c) by intersecting
a conic with a hyperbola. However, little progress was made, and even
around 1500 Pacioli and others were pessimistic about solving cubics.
There then follows one of the most celebrated stories in the history of
mathematics. The context is Italian mathematics of the early sixteenth
century, at a time when academics in the universities had no job security,
frequently having to renew their positions on a yearly basis. To do so
they resorted to public problem-solving contests in which they proved
their superiority over other possible contenders – often, the winner
would have to provide thirty dinners for the loser and several of his
friends – a sizeable sum.
In the early sixteenth century, Scipione del Ferro, a mathematics
professor at the University of Bologna, found a general method for solving
cubics of the form a cube and things equal to numbers – that is, x 3 +
cx = d. Much later he revealed his method to his pupil Antonio Fior.
After del Ferro’s death in 1526, Fior felt free to exploit his secret,
and challenged Niccolo of Brescia, known as Tartaglia (the stammerer),
to a contest, presenting him with thirty cubics of this form, giving a
moth to solve them. Tartaglia, who had solved equations of the form cubes
and squares equal to numbers (ax 3 + bx 2 = d), in turn presented Fior
with thirty of these. Fior lost the contest – he was not a good enough
mathematician to solve Tartaglia’s type of problem, while Tartaglia,
ten days before the contest, during a sleepless night, found a method
for solving all Fior’s problems.
Meanwhile, Gerolamo Cardano, wrote extensively about a range of topics,
from medicine, probability (especially its applications to gambling),
arithmetic and algebra. On hearing about the contest, he determined
to get Tartaglia’s method from him, which he did one evening in 1539
after promising to give him an introduction to Spanish Governor of the
city. Tartaglia hoped that the Governor would fund his researches, and
in turn extracted from Cardano a solemn oath not to reveal the solution:
I swear to you, by God’s holy Gospels, and as a true man of honour, not
only never to publish your discoveries, if you teach me them, but I also
promise you, and I pledge my faith as a true Christian, to note them down
in code, so that after my death, no-one will be able to understand them.
Tartaglia’s method for x 3 + cx = d was as follows:
… to enable me to remember the method in any unforeseen circumstance,
I have arranged it as a verse in rhyme …
When the cube and the thing together
Are equal to some discrete number [x 3 + cx = d]
Find two other numbers differing in this one. [u – v = d]
Then you will keep this as a habit
That their product shall always be equal
Exactly to the cube of a third of the things. [uv = (c/3) 3]
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to this principal thing. [x = u 1/3 + v 1/3]
In the event, Cardano came to learn in 1542 that the original discovery
of the method was due to del Ferro, rather than to Tartaglia, and felt
free to break the oath, publishing in his Ars magna of 1545 the method
for solving cubics – and also, incidentally, quartics (equations of
degree 4), while had been solved in the meantime. The Ars magna became
one of the most important algebra books of all time, but the hard-done
Tartaglia was outraged and spent the remaining ten years of his life
writing increasingly vitriolic letters and pamphlets to Cardano and
his secretary.
Thus, after a struggle lasting many centuries, cubic equations had at
last been solved, together with quartic equations. Over the next few
years, simplifications were made, and there was some useful discussion by
Rafael Bombelli about ‘imaginary numbers’ (square roots of negative
numbers), which had arisen from the solution of cubic equations but were
not to be fully understood for many years. Such discussions, along with
other developments in algebra, continued into the seventeenth century,
starting a gradual swing from algebra towards geometry. We shall chronicle
this in the next lecture, and discuss developments in gravitation and
the calculus.
© Professor Robin Wilson, Gresham College, 26 October 2005