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.
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.
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.
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