At its most basic, mathematics is the field of scientific study that deals with calculations and logical reasoning. It’s one of the fundamental parts of our understanding of the universe – without math, it’s impossible to advance in other areas of science and technology. Despite all that, though, some of the most important mathematicians in history – from Leonhard Euler to Brahmagupta to Euclid – remain largely forgotten and unacknowledged. 

10. Girolamo Cardano

Girolamo – or Hieronimo – Cardano was an Italian polymath born in 1501. He was a physician, mathematician, astrologer, and philosopher, with important contributions in the fields of algebra, probability, physics, and medicine. Some of his discoveries were so influential that he’s still referred to as one of the most important mathematicians of the Renaissance era. 

Cardano was born in Pavia in the Duchy of Milan, now Italy, as an illegitimate child of a lawyer and a widow. He studied at the universities of Pavia and Padua, earning his medical degree in 1526. He’d go on to become a lecturer in mathematics, as well as an accomplished physician. More than 200 books on various topics – ranging from medicine, mathematics, physics, philosophy, religion, and music – could be credited to him, including his autobiography. Cardano’s most important contribution was in the field of probability, making him one of the early pioneers of probability theory.

9. Carl Friedrich Gauss

Carl Friedrich Gauss was a German mathematician and physicist who lived in the 18th and 19th centuries. He is also called one of the greatest mathematicians of all time, thanks to his  contributions to fields like number theory, geometry, probability theory, astronomy, theory of functions, and electromagnetism.

Gauss was born in Brunswick in 1777 as a mathematical prodigy, receiving his doctorate from the University of Helmstedt in 1799 with an initial proof of the fundamental theorem of algebra – a topic he’d work and improve on throughout his life. He made many important discoveries while he was studying – in 1796, Gauss proved that it was geometrically possible to construct a 17-sided polygon, or a heptadecagon. 

He is best known for his book Disquisitiones Arithmeticae – or Arithmetical Investigations – along with important contributions like the development of various surveying instruments and an early form of a magnetometer. Gauss’s works describing magnetic and electric fluxes later formed the basis for James Clerk Maxwell’s famous electromagnetic theory. Many mathematical concepts and equations are still named after him, like Gaussian distribution, Gaussian elimination method, Gaussian gravitational constant, Gauss-Bonnet theorem, and many others.

8. Srinivasa Ramanujan

Born on December 22, 1887 in Erode – a small town in Tamil Nadu, India – Srinivasa Ramanujan would go on to become an important figure in the history of mathematics, especially in the area of number theory. Unlike the other names on this list, Ramanujan taught himself advanced mathematics by reading books and journals, and developed his own formulas from an early age. His work caught the attention of Godfrey H. Hardy – a famous British mathematician – who invited him to Cambridge University in 1914. 

There, Ramanujan (who you might recall being name-checked in the film Good Will Hunting) collaborated with Hardy and other mathematicians on various topics, making important contributions in number theory, infinite series, partition function, and the Riemann zeta function, among others. He published multiple papers in prestigious journals, getting elected to the Royal Society of London in 1918. Even today, mathematical constants like the Landau-Ramanujan constant and the Ramanujan-Soldner constant are named after him. 

7. Bernhard Riemann

Bernhard Riemann was a German mathematician and physicist born in the Hanover region in 1917. He studied mathematics at the universities of Göttingen and Berlin, where he was influenced by established mathematicians of the time like Carl Friedrich Gauss. He obtained his doctorate in 1851 with a dissertation on complex analysis, and became a professor at Göttingen in 1859 after delivering his famous lecture on the foundations of geometry. Throughout his life, Reimann collaborated with other mathematicians like Bernhard Bolzano, Richard Dedekind, and Gustav Roch to improve his work.

Riemann’s work revolutionized many fields of mathematics and physics. He introduced new concepts like Riemann surfaces, Riemann integrals, Riemann zeta function, Riemann hypothesis, Riemannian geometry, and Riemann-Roch theorem. His ideas laid the foundation for the development of concepts like complex and real analysis, number theory, algebraic and differential geometry, and even Einstein’s general relativity. Reimann’s discoveries remain useful across fields like quantum mechanics, cryptography, and string theory even today. 

6. Leonhard Euler

Leonhard Euler was a Swiss mathematician, physicist, astronomer, and engineer, among many other things. He was one of the early founders of the studies of graph theory and topology, along with pioneering discoveries in branches like analytic number theory, complex analysis, and infinitesimal calculus. 

Euler was born in 1707, in Basel, Switzerland, with an aptitude for advanced mathematics throughout his early years. After being tutored by Johan Bernoulli – an important figure in early calculus – he’d go on to attend the University of Basel, earning his master’s degree at the age of 20 in 1726.

Euler made important discoveries that would have a huge impact on modern mathematics and science, particularly in graph theory and analytic number theory, including concepts named after him like the Euler’s formula for complex analysis and the Euler characteristic for topology. 

5. Euclid

Living in Alexandria during the reign of Ptolemy I, Euclid was one of the most influential of all the early Greco-Roman mathematicians. He is best remembered for his magnum opus – Elements – which served as the foundation of all geometrical studies until the 19th century.

While we don’t know much about Euclid’s early life, his mathematical legacy has been recorded and shared many times throughout history. Elements, consisting of thirteen books – established the foundations of axiomatic geometry – an important area of study even today. Euclid’s logical approach of rigorous proofs revolutionized the field at the time, providing a codified system of geometric concepts for other Greek mathematicians and scientists. Apart from advancing the understanding of geometry, his work laid the groundwork for mathematical reasoning and proof in general.

4. Brahmagupta

Born in ancient India during the sixth century AD, Brahmagupta was one of the most influential mathematicians of the time, thanks to important contributions like the concept of zero. He was the first thinker to define rules for negative and positive numbers, allowing complex practical calculations like debt. 

Brahmagupta’s treatise Brahmasphutasiddhanta was the first work in history to introduce zero as a calculable number, with strict rules around the arithmetic operations it can be used with. This  proved to be a revolutionary idea, influencing important mathematicians and astronomers during the later Arabic and Byzantine ages. He also significantly contributed to the study of algebra, developing some of the earliest solutions for quadratic equations. 

3. David Hilbert

David Hilbert was a renowned German mathematician living during the late 19th and early 20th centuries. Born in 1862 in Königsberg, Prussia – now Kaliningrad, Russia – he began his career as a mathematician at the University of Königsberg. He’d later become a professor at the University of Göttingen, where he’d spent most of his professional life. 

Hilbert is considered one of the most influential mathematicians in history due to his contributions to fields like algebraic geometry, number theory, and physics. 

He is famous for his work on axiomatic systems and formal logic that laid the foundation for many modern mathematical concepts – his work on integral equations in 1999, for example, was the basis for all 20th-century research in functional analysis. In addition to mathematics, Hilbert also made important contributions to physics and philosophy. 

2. Archimedes

Archimedes – also known as Archimedes of Syracuse – was a polymath living in Syracuse, Sicily between 287 and 212 AD. Also known as one of the most important mathematicians in history, Archimedes is still remembered for his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder. Additionally, he’s known for the formulation of his famous Archimedes’ principle, and for the invention of a device to raise water called the Archimedes screw. 

Archimedes was instrumental in advancing our understanding of simple levers, which were then used to construct impressive parts of the infrastructure of Syracuse. Sadly, he was killed during the infamous siege of the city by the Roman general, Marcus Claudius Marcellus, which also destroyed a large part of that infrastructure. 

1. Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi – popularly remembered as al-Khw?rizm? – was a Persian polymath during the Islamic Golden Age. He was born in Khw?rizm, now Khiva, Uzbekistan, around 780 AD, working mainly as a mathematician, astronomer, and geographer in the city known for its universities and scholars.

Most importantly, Al-Khwarizmi is known as the ‘Father of algebra’. The word, algebra, in fact, derives from one of his works known as The Compendious Book on Calculation by Completion and Balancing, as it was the first book to present workable solutions to known linear and quadratic equations. He was also influential in the propagation and acceptance of Arabic numerals and the decimal system across the Islamic empire, which remains in popular use around the world today.

Coming up with an idea for a new business is never easy. While most people take the traditional route and stick to tried-and-tested concepts, a rare few dare to think out of the box. That may not always be a good idea, as some of the most unusual business ideas in history have also been massive commercial failures, though quite a few have succeeded, too, creating their own market niches where none existed before. 

10. iSmell

Back in 1999 when the dotcom bubble was at its peak, two Stanford graduates – Joel Bellenson and Dexster Smith – attempted to bring smell to the digital world. By breaking down scents into digital files that could be transmitted online and reproduced by a specialized, USB-powered device on the other end, their product – iSmell – promised to change the way we consume things online. 

It might sound weird today, though it was the next big thing in Silicon Valley at the time, raising over $20 million in venture funds from multiple investors. They envisioned a whole ecosystem built around iSmell and its parent, DigiScents Inc., which could be further incorporated into other industries like gaming, movies, and porn. 

While the device itself worked well enough, it was poorly marketed and eventually shut down in late 2001. There was also the whole question of whether anyone wanted such a technology, as being able to smell things over the Internet no longer sounds as pleasing as it likely did in 1999.

9. Washboard

Washboard has also been called one of the worst startup ideas of all time, and for good reason. Launched in June 2014, the whole idea was based on the assumption that people are willing to pay extra to get change for laundromats. For $15, Washboard would send you quarters worth $10, using the markup for delivery costs, operational expenses, and – obviously – a bit of profit. You could even go for their saver package, getting change worth $20 for a low sum of $26.99.

Of course, it was a ridiculous idea. Apart from the fact that you can just walk into a bank and get loose change whenever you want, a majority of people now have washing machines at home. Washboard was declared shut down by its founder in July of the same year, as they could barely get any lasting customers to keep it running.

8. NYC Garbage

Justin Gignac, a New York-based artist, came up with the idea for NYC Garbage back in 2001, following an argument with someone over the importance of package design. While most of us would think nothing of such a trivial topic, Gignac decided to prove them wrong by finding the most unsellable thing in New York City – garbage – and started repackaging it in fancy, limited-edition garbage cubes. Each cube comes signed, dated, and numbered by the artist, presumably indicating exactly when the garbage was collected.

While it sounds absurd, the idea surprisingly took off. According to NYC Garbage’s own website, more than 1,400 people around the world have already ordered one, priced anywhere between $50-$100. The collection also includes special editions, like the World Series at Yankee Stadium and New Year’s Eve at Times Square.  

7. Fashism

Launched in 2009, Fashism was a fashion-based startup based out of New York. Despite the horrible naming choice, it wasn’t an inherently bad business idea. Fashism was a kind of social media for fashion, where you could post pictures of your outfits and get feedback from a community of users. At its peak, the service had over 80,000 unique visits per month, with over 15,000 user accounts. It was even backed by several high profile investors, including actors Ashton Kutcher and Demi Moore.

We’re not sure if the name had anything to do with its downfall, or if it just wasn’t a viable business idea. Fashism eventually shut down in 2014, owing to a lack of growth and low user count. According to the numbers on Crunchbase, the outlet could only raise about $1 million in total funding. 

6. Blippy

Blippy was launched as a private beta app in 2009, quickly grabbing the attention of venture capital investors around the world. In essence, it was like a Twitter feed for your credit card transactions, allowing users to see and comment on other people’s purchases. Surprisingly, most people didn’t see it in terms of privacy, and the idea itself found many investors. Overall, they were able to raise over $13 million in funding, and the company was valued at around $46 million at its peak. Apple even launched their own service, Ping, to compete with Blippy’s business model. 

Unfortunately, Blippy suffered from a classic drawback: no one wanted it. At the end of the day, there was no real use for a social media service for credit card purchases, even if it was integrated right into your phone. Blippy soon ran out of money and shut down their primary service in 2010, eventually turning into a user reviews app. 

5. I Am Rich

While I Am Rich was mostly made as a novelty joke app, the fact that some people actually bought it makes it count as a business idea. Made by a German developer, Armin Heinrich, it was officially launched on the App Store in August, 2008. At $999.99, it was one of the first few apps on Apple’s newly-launched store, as well as possibly one of the most expensive apps ever. 

As for its functionality, the app itself just consisted of one large red button on the screen. When pressed, it would display a bunch of inspiring sentences like – you guessed it – ‘I am rich’ or ‘I deserve it’, which should prove to anyone sitting around you that you are, indeed, rich. While Heinrich has since admitted that he made the app as a joke, it was purchased by eight people, and only two of them asked Apple to reverse the sale. 

4. Potato Parcel

On paper, Potato Parcel sounds like another wacky business idea with no long-term prospects. For $9.99, someone at the company would handwrite a message on a potato and parcel it to anyone you want. It was a pitch on Shark Tank back in 2016, asking for $50,000 in exchange for 10% of the company. The offer was accepted, in addition to a royalty sharing agreement with one of the judges.

As it turns out, personalized messages on potatoes is actually a viable startup idea. Until 2018, Potato Parcel had sold more than 70,000 potatoes, with six figures in yearly revenues. The service has now expanded to include messages on other items like socks and pillows, as well as more advanced potato messages like pictures. 

3. DiedInHouse.com

Founded in 2013 by software engineer Roy Condrey, DiedInHouse.com is perhaps the only business of its kind. It tells you whether someone has died in the house you’re about to rent or purchase, as long as it’s a US address. It’s a paid service, using information from death certificates, news reports, and police records to provide an accurate measure of whether your house is haunted. The website also displays any other serious criminal cases at a particular property, making it rather useful for potential homeowners and ghost hunters. 

According to a Forbes story from 2016, DiedInHouse.com had sold over 40,000 reports until 2016. Each report contains information like deaths, meth operations, nearby cemeteries, registered sex offenders in the area, previous fire incidents, and other details about the property. 

2. Bottled Air

Vitality Air started out as a prank, when two Canadians – Moses Lam and Troy Paquette – filled a ziploc bag with fresh Canadian air and posted it on eBay. It quickly grabbed media attention and a frantic bidding war, and the bag ended up selling for $130. Sensing a business opportunity, Lam and Paquette decided to look into bottled air as a potential startup idea.

As it turns out, it was, and Vitality Air was born in early 2015. They now export bottles of fresh air to countries around the world, including Mexico, India, Vietnam, and China. The operation has vastly expanded from just driving around with a ziploc bag and sealing it shut at the right time, as it now includes massive air-collection machines and a fully-automated bottling plant. As per a 2019 report, Vitality Air made more than $300,000 in annual sales for two consecutive years, making it a rather successful business idea.

1. No More Woof

No More Woof was a promising – even if a bit ambitious – crowdfunding pitch posted by a few Scandinavian engineers on Indiegogo. They claimed that the device could accurately translate your pet’s brainwaves into intelligible speech, using EEG brain-scanning technology that’s currently used in many brain-related medical procedures. No More Woof was successfully crowdfunded by over 200%, and the story was picked up by multiple news outlets including Mashable, CNET, and Engadget. 

Sadly, while it was a good idea, it was too good to be true, as the technology to make such a device simply doesn’t exist. While there have been major advances in brain-reading tech in the past few years, it’s still impossible to accurately translate thoughts into speech. No More Woof was shelved after the founders publicly-admitted that they don’t have the technology to make it work.

Engineering is the practice of applying scientific and mathematical principles to real-life situations. While scientists usually get all the credit for all of humanity’s scientific achievements, it’s actually engineers who build the things that make all that possible – from intricate lab equipment to large-scale city projects. Some of the best engineers ever also happened to be prodigious scientists, even if their names have now been largely lost to history.

10. Martha Coston

Night flares were an important method of communication in early modern warfare, especially before the advent of radio and other electronic forms of communication. Developed and patented by Martha Coston in 1859, they were deployed to great effect during the American Civil War. The flare and code system was eventually adopted by military forces around the world, paving the way for the elaborate flare systems still in use today. 

Surprisingly, Coston never intended on becoming an engineer or inventor, as it was her deceased husband – Benjamin Franklin Coston – who originally came up with the blueprint. His version didn’t work, however, and Martha largely came up with a working model on her own. According to some historians, her flares were crucial during the civil war, as the Union Army was the first major client to purchase them in bulk.

9. Filippo Brunelleschi

Filippo Brunelleschi was one of the pioneers of architecture during the earliest phases of the Italian Renaissance, and is even sometimes referred to as the first modern engineer. Brunelleschi had a huge role to play in the modern scientific approach to infrastructure design, breaking away from the more decorative kind of architecture common across Europe before. 

Filippo Brunelleschi’s most renowned masterpiece was the dome of the Cathedral of Santa Maria del Fiore in Florence, built by machines he invented specifically for that purpose. His geometric, proportion-based approach to architecture laid the foundation for many Italian engineers and inventors after him. Apart from engineering, Brunelleschi was also inducted into the Arte della Seta – or the Silk Guild of Florence – as a master, as he was also an accomplished goldsmith and sculptor.

8. Hedy Lamarr

Hedy Lamarr was an Austrian-born American actress still remembered for her roles in classics like Algiers, Samson and Delilah, and Come Live with Me, among countless others. She’s often called one of the most beautiful actresses ever, inspiring iconic cartoon characters like Snow White and Catwoman. What’s much less known, though, is her contribution to modern science, as Lamarr was also a gifted engineer and inventor of the technology used in many communications systems today.

During the Second World War, Lamarr came up with a system called frequency hopping, which could be used to block enemy jammers from interfering with radio-controlled torpedoes. While it wasn’t used during the war, it was widely implemented by the US Navy in the 1950s, even if she was never credited for it back then. The technology has since been used to develop innovations like Bluetooth and GPS, as well as early WiFi systems. For her contributions, Hedy Lamarr was posthumously inducted into the National Inventors Hall of Fame in 2014. 

7. Jacques de Vaucanson

The list of things invented and designed by Jacques de Vaucanson is quite long. Born in 1709, he was a pioneer in automatons, and many machines that made the Industrial age possible would’ve never been possible without his contributions. In 1737, Vaucanson built a life-size flute player that could play a total of twelve songs on a traverse flute, which was unheard-of for that time. In 1738, he designed something called The Digesting Duck – a rather impressive replica of a duck with hundreds of parts that could flap its wings, eat and drink, and even defecate, which is still considered to be a masterpiece of the time. 

His most important invention, however, was the all-metal slide rest lathe in 1750. While lathes – a kind of machine tool based on an axis used to complete various industrial tasks – had existed since ancient times, he massively improved on the design for his age, providing a base machine to build other machines. Nearly every automatic machine used during the Industrial age was based on Vaucanson’s improvements, including the Jacquard loom. 

6. Imhotep

Yes, the guy from The Mummy took his name from a real guy. Let’s get that out of the way first.

The real Imhotep lived and died so long ago that it’s difficult to ascertain precise details about his life, though we do know a few things. Archeological evidence from later Egyptian sources place his achievements somewhere around 2600 BC during the reign of Djoser of the Third Dynasty, or the first Old Kingdom dynasty. He was an immensely important figure across Egypt, and may even have been the first individual to be deified and worshiped as a god… ever.

Mythology aside, Imhotep was also the first architect – or at least the first architect to show up in historical records. We can see his imprint in some of the most impressive ancient Egyptian structures found in and around Cairo, especially the step pyramid complex dedicated to Djoser at Saqqara, called the ‘The Refreshment of the Gods’. It was an unprecedented creation undertaken entirely under his supervision, using a stone-dressed style of building and columns for the first time in history. Not just that, ancient Egyptian structures started to get more complex and elaborate around exactly this time, suggesting that Imhotep may have had a larger role to play in ancient Egyptian architecture than we may ever know. 

Imhotep was also a gifted physician and, well, a God, so it’s difficult to say exactly how influential his works really were, as archeological evidence tends to overplay his achievements. Still, Imhotep’s tomb has never been found, and it may hold some clues to the precise scope of his engineering and scientific achievements.

5. Archimedes

Born in 287 BC in Syracuse, Sicily, Archimedes was easily one of the greatest minds of classical antiquity. While he’s mostly remembered for his contributions to mathematics – especially early geometry – he was also one of the greatest engineers of his time, even if he was always more interested in theorems and proofs rather than actually building anything. 

Many of Archimedes’s greatest inventions happened under the reign of King Hieron II of Syracuse. He built a lever and pulley system that could easily launch large, newly-constructed ships into the water with a simple mechanism. Archimedes is also credited with building the first ever working odometer – an idea that was eventually adopted and improved upon by other inventors of the time. His creations also contributed to the war effort, as he built a giant Iron Claw to thwart invading ships during the Punic wars between Ancient Carthage and the Roman Republic, among other effective war machines. 

4. Nikolaus August Otto

Nikolaus August Otto was born in 1832 in Holzhausen, Germany, and remains one of the most important inventors of the Industrial Age, even if most people today have likely never heard of him. While many engineers and businessmen aided in the invention of the modern automobile in their own way, Otto could be singularly credited with developing the technology that made it all possible – the four-stroke internal combustion engine. 

Built in 1876, Otto’s engine improved upon Étienne Lenoir’s two-stroke system in many ways, providing the first working replacement for the widely-used steam engine at the time. In his honor, the four-stroke internal combustion cycle – comprising the four stages of intake, compression, combustion, and exhaust – is still called the Otto Cycle. His invention was used in the design of the first motorcycle by Gottlieb Daimler in August, 1885, as well as nearly every type of autmobile engine built since that time. 

3. Fazlur Rahman Khan

Fazlur Rahman Khan is also sometimes referred to as the ‘father of tubular designs’ for high-rise buildings, as well as a pioneer of computer-aided designs – or CAD – in modern architecture. Easily one of the greatest engineers of the 20th century, Khan’s innovations are visible in modern skyscrapers around the world, even the ones he wasn’t directly involved with, such as the first World Trade Center in New York. 

Born in Dhaka, Bangladesh in 1929, Khan joined the renowned Chicago-based architecture firm Skidmore, Owings & Merrill in 1955. His tubular designs ushered in a kind of a revolution in skyscraper design during the second half of the century, especially the ‘bundled tube’ structural system, which minimized the amount of material required for construction. Fazlur Rahman Khan’s most well-known works include Sears Tower (now Willis Tower) and John Hancock Center in Chicago, and the Haj Terminal at the King Abdul Aziz International Airport in Jeddah, Saudi Arabia, among many others. 

2. Ismail al-Jazari

Ismait al-Jazari was a prolific polymath from the Islamic Golden Age – a nearly-five-centuries-long period of scientific, cultural and social renaissance across the Islamic world, beginning with the establishment of the Abbasid Caliphate in the 8th century. Born in 1136, al-Jazari built upon the works of other notable scientists and inventors from the region, and his inventions have since been recognized as a major influence on European inventors during the European Renaissance, like Leonardo da Vinci.

While we don’t know much about his early years – as many records from that time have since been lost – we know that over the course of his life, al-Jazari invented over 50 types of unique devices, including an elephant water clock and a mechanism for raising water for irrigation and other purposes, along with multiple other types of automatons. His inventions are illustrated in his magnum opus, The Book of Knowledge of Ingenious Mechanical Devices, which remains an important treatise in early engineering to this day.   

1. Charles Babbage

It’s unclear exactly when or where Charles Babbage was born, though according to some sources, it was probably in London some time in 1791. Unlike many other inventors on this list, Babbage never went to school and mostly taught himself during his early years. He was exceptionally-good at algebra and calculus, though, earning him a place at the prestigious Trinity College at Cambridge by 1811.

Over the course of his life, Charles Babbage would prove himself to be an important figure in early computing. Out of his countless contributions to the fields of mathematics and engineering, perhaps the most important was the Difference Engine, and later the Analytical Engine, which are now recognized as the earliest types of mechanical computers. His designs were so far ahead of his time that they were impossible to build during his lifetime, as they far exceeded the technology available to him. Later attempts to recreate the Difference Engine would prove successful, however, even if many of his other prototype machines remain incomplete to this day. While he’s now called the ‘father of computers’ due to his contributions to early computation, much of Babbage’s work was never supported by his peers or the British government at the time.