John Venn

Numbers, and the underlying rationality, have long fascinated us. While the precise origins are obscure, it’s easy to envisage our hunter-gatherer forebears apportioning fruit or root vegetables. The transition to more stable lives, centred upon agriculture, occurred in the Holocene, so a relatively recent development: in the past twelve thousand years. Communities became dependent on farmers and harvests, making the recording of time, solstices, and seasons vital, and implying planning for the coming year. This requires numbers.

Another source of nutrition swims in the planet’s oceans. While the hunter-gatherer might have sought a catch in rivers or lakes, later coastal settlements were more ambitious, taking to the sea in search of more plentiful prey. Boats were built, requiring rudimentary plans and raw materials. Quantification arises. How big? How many? Numbers again.

Numbers again.

The first evidence of a written script, from around 3400 BCE, relates to the Sumerians whose clay tablets served a dual purpose: the birth of writing and the earliest recorded mathematical system. Their pictographic script is termed cuneiform. The numbering format was sexagesimal, therefore based around 60. Today we use a base-10 standard, termed denary, from the Latin denarius, meaning ‘containing ten’.

Sexagesimal concepts were later embraced and enhanced by the Babylonians. Notably, they introduced a place-value convention, the world’s first positional numeral scheme, analogous to our modern decimal point. A massive leap. Cultures flourished and receded, but the legacy of these mathematicians remains, reflected in our measurement of time and direction: we have sixty seconds in a minute, sixty minutes in one hour, and 360° in a circle. Customs we seldom consider.

Progressing further, we witness the Ancient Greek era, a period of both abstractions and practicalities. Pythagoras was a controversial figure from circa 500 BCE. His assertion that the cosmos and its contents could be described by numbers was a radical and disruptive view. To some, heretical. Slightly later, from about 400 BCE, we encounter Plato, founder of the Academy, regarded as the first Western university and a source of mathematical erudition, associated with the five polyhedra, termed Platonic solids. Euclid, active around 300 BCE, was a mathematician and logician. His masterpiece, Elements, captured the geometry of earlier researchers and his own insights. A basis for textbooks and teaching over two millennia. Trigonometry had emerged by 100 BCE, credited to Hipparchus, but with a lengthier lineage.

If geometry was founded by the Greeks, then algebra can best be assigned to the Islamic scholars of the Golden Age. In Baghdad, the House of Wisdom was a hub. An academic working in 800 CE was al-Khwarizmi, whose treatise named Al-Jabr led to the English term, algebra. A man influential in mathematics, astronomy, and geography, later remembered through another term: algorithm. Remarkably, he and his peers addressed familiar algebraic scenarios without modern notation, posing questions orally.

A century or so later, we meet Alhazen, who came from Basra but worked in Cairo. He is lauded for his optical studies but was also an accomplished mathematician and astronomer. His legacy includes insights linking algebra and geometry. Also, an early exponent of perfect numbers; six is the first. Remembered primarily for his optics.

Proceeding rapidly, now in Europe, from circa 1200 CE, we find Fibonacci, perhaps best known for his eponymous sequence, present in nature and even deployed in musical composition. Worth a read. Related is the Golden Ratio and a constant: approximately 1.618, famously applied to violin design. More fundamental was his introduction to Europe of Hindu-Arabic numbering practice, a decimal place-value paradigm that includes zero, used until today, replacing its unwieldy Roman predecessor. A step change for scientists and merchants.

A 15th century thinker, an associate of Leonardo da Vinci, was Luca Pacioli, who wrote about the Golden Ratio’s use in architecture and artistic proportionality. Divina proportione was his signature work. Far less stylish, but immensely useful was his subsequent description of bookkeeping routines used by Venetian traders. A Renaissance man, Pacioli formalised the essence of double-entry accounting that spread and endured, still recognisable after 500 years. An energetic individual who lived for seventy years, he even found time to write about chess. The related sketches have provoked wonder: was da Vinci the source?

Mathematical erudition was now growing at an explosive rate. A 16th century Edinburgh sage, John Napier, is best known for his discovery of logarithms, easing the reckonings of astronomers and navigators: addition and subtraction are simpler than multiplication and division. Henry Briggs, later Professor of Geometry at Oxford, helped advance these notions into valuable tools. This arrived just in time for the era of Copernicus and Galileo: both relied on trigonometry, but only Galileo lived to use logarithms. Hot news for scientists at the time.

A 17th century milestone was the advent of calculus, concerning continuous change. Revealed independently but concurrently by Isaac Newton and Gottfried Leibniz. Disputatious at the time, the modern appraisal is that Newton unearthed the principles earlier, but Leibniz was the first to publish. These methods, essential for determining rates of change (derivatives) and accumulated effects (integrals), are vital tools for physics, astronomy, and engineering.

A giant of the 18th century, a rich philosophical era, was Leonhard Euler, a Swiss active in St Petersburg and Berlin, regarded by some as the most productive mathematician in history and a proponent of calculus. Moreover, a physicist, astronomer, geographer, and logician. His multifarious legacy exceeds the scope of this modest piece but includes harmonised notation we still employ today: sigma (∑) to express summations, and pi (π) for the ratio of a circle’s circumference to its diameter are examples. His solution of the Basel problem, posed by Pietro Mengoli in the previous century, established his reputation. Earlier, he had successfully tackled the Seven Bridges of Königsberg challenge. Both are worthwhile reading. Another dimension to his exploits was logic, framing diagrammatic forms representing sets and their links to explore syllogistic reasoning. These closed curves were termed Euler diagrams. The history of syllogisms is lengthy, dating back to Plato’s pupil, the renowned Aristotle.

Euler’s career was hampered by failing eyesight, but an exceptional memory combined with apparently limitless imagination made this a minor constraint. Pierre Simon Laplace, the esteemed French polymath opined: “Read Euler, he is master of us all.”

The 19th century brought further evidence of Homo sapiens’ cleverness, but time is pressing, so just two are noted, both Cambridge University alumni. Charles Babbage, a star of the age, a mathematician, astronomer, and engineer, who is commonly deemed an instigator of computer science, abetted by the celebrated Ada Lovelace.

The second, less widely remembered, is John Venn. Like Babbage before him, Venn graduated in mathematics but was similarly unhappy with the syllabus. He resolved to depart from the field and instead followed the family tradition, becoming an Anglican priest. By 1862 CE, then aged twenty-eight, he had returned to Cambridge to teach logic, probability, philosophy, and economics. Logic was a mainstay; he wrote textbooks and gained respect as a leader in the field. The Royal Society elected him a Fellow. He adopted Eulerian circles, as he termed them, as a teaching tool, used to depict relationships between populations and subsets. Unusually in this era, he taught students across several Cambridge colleges. Later, he was elected president of Gonville & Caius College, a post he held for the rest of his life.

‘Venn diagram’ is a 20th century label for a toolset soon applied to probability, languages, computer science, and demographics. A stained-glass window in the college dining hall portrays a Venn diagram, consisting of three overlapping circles or sets.

John Venn was a versatile man: an ally of female emancipation, alongside his wife, a champion gardener, an antiquarian, and inventor of a bowling machine to test top cricketers.