If you have two GPS coordinates and want the distance between them, it is tempting to treat latitude and longitude like ordinary x/y coordinates and use the Euclidean distance formula.

That shortcut breaks down once the Earth’s curvature matters. Latitude and longitude are positions on the surface of an approximately spherical planet, so the shortest surface path between two points is not a straight line on a flat plane. It is an arc along the sphere.

The haversine formula is a practical way to compute that arc length. Given two lat/lon pairs, it tells you the great-circle distance between them: the shortest distance along the surface of a sphere.

What the haversine formula measures

Imagine two points on the Earth:

• Point 1: latitude \(\phi_1\), longitude \(\lambda_1\)
• Point 2: latitude \(\phi_2\), longitude \(\lambda_2\)

The shortest path constrained to the surface of a sphere lies on a great circle, which is any circle whose center matches the sphere’s center. The equator is a great circle. So is the meridian plane passing through London and New York.

Try dragging the points in the demo below. It shows how the surface arc and the central angle change together.

If we can find the central angle \(\theta\) between the two points, measured from the center of the Earth, then the surface distance is just arc length:

where \(r\) is the radius of the sphere and \(\theta\) is measured in radians.

This is the standard arc length formula. A full circle has circumference \(2\pi r\) and angle \(2\pi\) radians, so an angle of \(\theta\) covers the fraction \(\theta / 2\pi\) of the circle. The corresponding fraction of the circumference is:

So the problem reduces to: given two lat/lon pairs, how do we compute \(\theta\)?

From spherical geometry to haversine

The spherical law of cosines gives the central angle between two points on a sphere:

This is correct, but in practice it can lose precision for very small distances, where \(\theta\) is close to zero and \(\cos(\theta)\) is close to 1.

The haversine form rewrites the same relationship in a numerically friendlier way.

The haversine function is defined as:

To get from the spherical law of cosines to the haversine form, we use the cosine difference identity and the half-angle relationships below:

From the last identity, we can rearrange to get:

Let:

First apply the cosine difference identity to \(\Delta \phi\):

Rewrite the spherical law of cosines using \(\Delta \lambda\):

Now solve the previous identity for \(\sin(\phi_1)\sin(\phi_2)\):

Substitute that into the spherical law of cosines:

Now factor out \(\cos(\phi_1)\cos(\phi_2)\):

Next convert everything into haversines by using \(1 - \cos(x) = 2\operatorname{hav}(x)\):

Divide both sides by 2:

Now replace each haversine term with \(\operatorname{hav}(x) = \sin^2\left(\frac{x}{2}\right)\):

So:

In code, we usually name the right-hand side \(a\):

Then:

Since the central angle on a sphere satisfies \(0 \le \theta \le \pi\), we have \(0 \le \theta/2 \le \pi/2\), so the sine is nonnegative. Taking square roots gives:

Apply inverse sine to recover the central angle:

Many implementations use the equivalent and numerically stable form:

where \(c = \theta\) is the central angle in radians.

Why half-angles appear

The half-angle terms are not arbitrary. They come from the identity:

This matters because for small angles, directly subtracting from 1 can amplify floating-point error. Writing the expression in terms of \(\sin^2\left(\frac{x}{2}\right)\) behaves better numerically.

That is one reason the haversine formula became the standard practical choice for many navigation and mapping problems.

Coordinate units matter

Latitude and longitude are usually stored in degrees, but JavaScript, C, Python, and most math libraries expect trigonometric inputs in radians.

So before applying the formula:

If you forget this conversion, the result will be wrong by a large factor.

A practical JavaScript implementation

Here is a compact implementation:

const EARTH_RADIUS_METERS = 6371008.8;

function toRadians(degrees) {
  return degrees * Math.PI / 180;
}

function haversineDistance(
  lat1,
  lon1,
  lat2,
  lon2,
  radius = EARTH_RADIUS_METERS
) {
  const phi1 = toRadians(lat1);
  const phi2 = toRadians(lat2);
  const dPhi = toRadians(lat2 - lat1);
  const dLambda = toRadians(lon2 - lon1);

  const sinHalfDPhi = Math.sin(dPhi / 2);
  const sinHalfDLambda = Math.sin(dLambda / 2);

  const a =
    sinHalfDPhi * sinHalfDPhi +
    Math.cos(phi1) * Math.cos(phi2) *
    sinHalfDLambda * sinHalfDLambda;

  const c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1 - a));

  return radius * c;
}

This returns the distance in meters when using the default Earth radius \(r = 6371008.8\) meters.

Example:

const london = { lat: 51.5074, lon: -0.1278 };
const newYork = { lat: 40.7128, lon: -74.0060 };

const distanceMeters = haversineDistance(
  london.lat,
  london.lon,
  newYork.lat,
  newYork.lon
);

console.log(distanceMeters / 1000); // about 5570 km

Interpreting the terms

The formula has a clean geometric interpretation:

• \(\sin^2\left(\frac{\Delta \phi}{2}\right)\) captures north-south separation
• \(\sin^2\left(\frac{\Delta \lambda}{2}\right)\) captures east-west separation
• \(\cos(\phi_1)\cos(\phi_2)\) scales longitude separation by latitude

That last term is important. Lines of longitude converge toward the poles, so one degree of longitude does not represent a fixed physical distance everywhere on Earth. Near the equator it spans a large distance. Near the poles it shrinks dramatically.

Why not use plain Euclidean distance?

Suppose two points differ by 1 degree of longitude.

At the equator, that corresponds to roughly 111 km. Near latitude of 60 degrees, it is roughly half of that. Near the poles, it approaches zero. A flat 2D distance formula ignores this geometry entirely, so errors become larger as distances grow or as you move away from the equator.

Projecting to a local plane

For small regions, a common approach is to approximate the Earth’s surface as flat and work in a local Cartesian coordinate system. The idea is to convert latitude and longitude offsets into metric distances at the reference latitude of the region.

Given a reference point \((\phi_0, \lambda_0)\) and a nearby point \((\phi, \lambda)\), the offsets in meters are approximately:

where \(M \approx 111{,}320\) meters per degree (one degree of arc along a great circle). The \(\cos(\phi_0)\) factor accounts for the fact that longitude lines converge toward the poles: the east-west distance per degree of longitude shrinks as latitude increases.

The Euclidean distance in the local plane is then:

This approximation is called the equirectangular projection (or plate carree projection). It is fast and simple, but the error grows with distance from the reference point and with latitude. The flat-earth assumption breaks down once the angular separation between the points is large enough that curvature matters.

As a rough guideline:

Beyond a few tens of kilometers, or near the poles where the \(\cos(\phi_0)\) factor changes rapidly over the region, haversine is the safer default.

Accuracy and limitations

The haversine formula assumes the Earth is a perfect sphere. It is not. The real Earth is better modeled as an oblate spheroid, slightly flattened at the poles. The polar radius is about 6357 km and the equatorial radius is about 6378 km, a difference of roughly 21 km (0.3 %).

Haversine uses a mean spherical radius of approximately 6371 km. The mismatch between sphere and spheroid introduces a systematic error that varies with the direction of travel.

The worst-case error of haversine against the WGS84 ellipsoid is roughly 0.5 %, which corresponds to about 5 meters per kilometer of distance. The error is largest for long paths that run diagonally between equatorial and polar latitudes; it is near zero for paths along the equator or along a meridian, because those paths happen to lie on circles of the spheroid.

As a practical table:

These are worst-case estimates. Typical real-world paths have errors closer to half of these figures.

For comparison, the equirectangular approximation over the same 1000 km would accumulate errors of 30 km or more. Haversine is therefore a large improvement over flat-Earth approximations, and accurate enough for most practical navigation and mapping work.

Where haversine is not enough:

• High-precision surveying, cadastral mapping, or scientific geodesy require ellipsoidal methods such as Vincenty’s formulae or Karney’s geodesic algorithm (used in GeographicLib).
• For routing over roads, the dominant error usually comes from the fact that roads do not follow great-circle arcs, not from the spherical assumption.

So the key question is not “Is haversine perfect?” but “Is spherical great-circle distance the right approximation for this system?”

Numerical stability notes

The atan2 form is preferred:

rather than:

Both are mathematically equivalent, but atan2 tends to behave better near the edges of floating-point precision.

Also note that due to rounding, a can occasionally drift slightly above 1 for nearly antipodal points. In defensive production code, it is reasonable to clamp:

const safeA = Math.min(1, Math.max(0, a));

before evaluating the final angle.

In 1627, a year after the death of the philosopher and statesman Francis Bacon, a short, evocative tale of his was published. The New Atlantis describes how a ship blown off course arrives at an unknown island called Bensalem. At its heart stands Salomon’s House, an institution devoted to “the knowledge of causes, and secret motions of things” and to “the effecting of all things possible.” The novel captured Bacon’s vision of a science built on skepticism and empiricism and his belief that understanding and creating were one and the same pursuit.

No mere scholar’s study filled with curiosities, Salomon’s House had deep-sunk caves for refrigeration, towering structures for astronomy, sound-houses for acoustics, engine-houses, and optical perspective-houses. Its inhabitants bore titles that still sound futuristic: Merchants of Light, Pioneers, Compilers, and Interpreters of Nature.

Engraved title page of The Advancement and Proficience of LearningPublic Domain

Bacon didn’t conjure his story from nothing. Engineers he likely had met or observed firsthand gave him reason to believe such an institution could actually exist. Two in particular stand out: the Dutch engineer Cornelis Drebbel and the French engineer Salomon de Caus. Their bold creations suggested that disciplined making and testing could transform what we know.

Engineers show the way

Drebbel came to England around 1604 at the invitation of King James I. His audacious inventions quickly drew notice. By the early 1620s, he unveiled a contraption that bordered on fantasy: a boat that could dive beneath the Thames and resurface hours later, ferrying passengers from Westminster to Greenwich. Contemporary descriptions mention tubes reaching the surface to supply air, while later accounts claim Drebbel had found chemical means to replenish it. He refined the underwater craft through iterative builds, each informed by test dives and adjustments. His other creations included a perpetual-motion device driven by heat and air-pressure changes, a mercury regulator for egg incubation, and advanced microscopes.

De Caus, who arrived in England around 1611, created ingenious fountains that transformed royal gardens into animated spectacles. Visitors marveled as statues moved and birds sang in water-driven automatons, while hidden pipes and pumps powered elaborate fountains and mythic scenes. In 1615, de Caus published The Reasons for Moving Forces, an illustrated manual on water- and air-driven devices like spouts, hydraulic organs, and mechanical figures. What set him apart was scale and spectacle: He pressed ancient physical principles into the service of courtly theater.

Drebbel’s airtight submersibles and methodical trials echo in the motion studies and environmental chambers of Salomon’s House. De Caus’s melodic fountains and hidden mechanisms parallel its acoustic trials and optical illusions. From such hands-on workshops, Bacon drew the lesson that trustworthy knowledge comes from working within material constraints, through gritty making and testing. On the island of Bensalem, he imagines an entire society organized around it.

Beyond inspiring Bacon’s fiction, figures like Drebbel and de Caus honed his emerging philosophy. In 1620, Bacon published Novum Organum, which critiqued traditional philosophical methods and advocated a fresh way to investigate nature. He pointed to printing, gunpowder, and the compass as practical inventions that had transformed the world far more than abstract debates ever could. Nature reveals its secrets, Bacon argued, when probed through ingenious tools and stringent tests. Novum Organum laid out the rationale, while New Atlantis gave it a vivid setting.

A final legacy to science

Engraved title page of Bacon’s Novum OrganumPublic Domain

That devotion to inquiry followed Bacon to the roadside one day in March 1626. In a biting late-winter chill, he halted his carriage for an impromptu trial. He bought a hen and helped pack its gutted body with fresh snow to test whether freezing alone could prevent decay. Unfortunately, the cold seeped through Bacon’s own body, and within weeks pneumonia claimed him. Bacon’s life ended with an experiment—and set in motion a larger one. In 1660, a group of London thinkers hailed Bacon as their inspiration in founding the Royal Society. Their motto, Nullius in verba (“take no one’s word for it”), committed them to evidence over authority, and their ambition was nothing less than to create a Salomon’s House for England.

The Royal Society and its successors realized fragments of Bacon’s dream, institutionalizing experimental inquiry. Over the following centuries, though, a distorting story took root: Scientists discover nature’s truths, and the rest is just engineering. Nineteenth-century “men of science” pressed for greater recognition and invented the title of “scientist,” creating a new professional hierarchy. Across the Atlantic, U.S. engineers adopted the rigorous science-based curricula of French and German technical schools and recast engineering as “applied science” to gain institutional legitimacy.

We still call engineering “applied science,” a label that retrofits and reverses history. Alongside it stands “technology,” a catchall word that obscures as much as it describes. And we speak of “development” as if ideas cascade neatly from theory to practice. But creation and comprehension have been partners from the start. Yes, theory does equip engineers with tools to push for further insights. But knowing often follows making, arising from things that someone made work.

Bacon’s imaginary academy offered only fleeting glimpses of its inventions and methods. Yet he had seen the real thing: engineers like Drebbel and de Caus who tested, erred, iterated, and pushed their contraptions past the edge of known theory. From his observations of those muddy, noisy endeavors, Bacon forged his blueprint for organized inquiry. Later generations of scientists would reduce Bacon’s ideas to the clean, orderly “scientific method.” But in the process, they lost sight of its inventive roots.

Student accommodations in elite lecture halls are reshaping the ways that privilege manifests in America. At Stanford, 38% of students are registered as disabled. At Harvard and Brown, more than 20% of students are registered as disabled. Better testing and comfort talking about disability has helped spur this rise in ways that reflect social and medical progress, but this statistical anomaly is not just about medicine, it reflects a two-tiered system where we “accommodate” the elite but often abandon the poor and those who need our support most.

The proliferation in accommodation plans, known as 504 plans (after a section of federal law that prohibits discrimination based on disability), has made even the most academically rigorous universities more welcoming to students with disabilities. Those 504 accommodations include extended test time, note-taking services, and special testing rooms, but many students have reported that it also has given them priority housing, ability to live in a single, and meal plan benefits. The most common diagnoses in these 504 plans are ADHD, neurodivergence, and mental health conditions including severe anxiety.

There’s a real tension at the heart of this data - on one hand, critics say that students are gaming the system. No formal evaluation is required for a 504 accommodation, only a documented physical or mental impairment from a doctor’s note. Parents submit a request to a school administrator and most schools get to decide what types of materials they request, according to the Department of Education. On the other hand, advocates say that the rise is a sign of progress. They say that the underlying mental and physical disabilities were always there for students, but only now are medical professionals, schools, and parents doing a better job of actually diagnosing the disabilities.

At the University of Chicago, the number has more than tripled over the past eight years; at UC Berkeley, it has nearly quintupled over the past 15 years. Only 14% of Americans below 35 have a disability, but at schools like Amherst, 34% of students report a disability.

Who Really Benefits from the ADA?

What is clear is that wealthy families tend to benefit most from this system. The rates of students claiming disabilities seems to be far higher in places with higher median household incomes. In Weston, Connecticut, where the median household income is $220,000 nearly 1 in 5 students claims a disability. Just 30 minutes north in Danbury, Connecticut where the median household income is $83,000, the rate is 8-times lower.

Still, the system remains open to abuse by those who know how to game it. The Varsity Blues college-admissions scandal showed that there are wealthy parents who are willing to pay unscrupulous doctors to provide disability diagnoses to their non-disabled children, securing them extra time on standardized tests. Studies have found that students exaggerate symptoms when they go in for these tests, making it hard for doctors to evaluate the children.

The wealthiest students across America claim the most disabilities.

Students in every ZIP code are dealing with anxiety, stress and depression as academic competition grows ever more cutthroat. But the sharp disparity in accommodations raises the question of whether families in moneyed communities are taking advantage of the system, or whether they simply have the means to address a problem that less affluent families cannot. While experts say that known cases of outright fraud are rare, wealthy parents who want to give their children every advantage in life are spending tens of thousands of dollars on tests and finding doctors who may offer the neuropsychological diagnosis they think their child needs.

The Cleveland Metropolitan School District, one of the poorest in the country, had a 504 rate of less than 1 percent. One mother in Montgomery County, Maryland transferred her son, who has A.D.H.D. and a reading disability, from a public high school to a private one that charges $45,000 per year in tuition. When the boy arrived, the staff at the new school told his mother about ACT accommodations she had not known about. Her son scored a 33 after taking the exam over multiple days, and is now considering applying to Ivy League schools.

Buying Time and Abandoning the Meritocracy

Many companies are now making millions off this side-door cottage industry that both gives students more time on tests and helps them do better once they are enrolled in elite universities. “Get ACT Extra Time,” reads one blunt web advertisement from the Cognitive Assessment Group. Dr. Wilfred van Gorp, who runs the company, says he assesses 24 patients a month and charges $6,000 per patient (amounting to $1.73M per year). About 70 percent of the patients he sees leave with a diagnosis. But Dr. Gorp has a checkered past himself. Dr. Gorp once testified that the Genovese crime boss Vincent Gigante was mentally impaired, though years later Mr. Gigante admitted that he had feigned his mental illness . Dr. Gorp says that he was tricked.

The shift began in 2008 when Congress amended the Americans with Disabilities Act (ADA) to restore the law’s original intent. The government broadened the definition of disability, effectively expanding the number of people the law covered. In response to the 2008 amendments, the Association on Higher Education and Disability (AHEAD), an organization of disability-services staff, released guidance urging universities to give greater weight to students’ own accounts of how their disability affected them, rather than relying solely on a medical diagnosis. Schools began relaxing their requirements. A 2013 analysis of disability offices at 200 postsecondary institutions found that most “required little” from a student besides a doctor’s note in order to grant accommodations for ADHD.

Read more

Stop me if you’ve heard this one before:

A crack team of hardware and software engineers, inspired by breakthroughs in computer science and electrical engineering, are driven to work 18-hour days, seven days a week, on a revolutionary new system. The system’s capabilities and speed will usher in a new era, one that will bring transformative computing to every workplace. The long hours are necessary: the team knows that every major computer company sees what they see on the horizon, and they too are working around the clock to take advantage of powerful new chips and innovative information architectures.

The team is almost entirely men, men whose affect and social skills cluster in a rather narrow band, although they are led by a charismatic figure who knows how to persuade both computer engineers and capitalists. This is a helpful skill. Money, big money, is flowing into the sector; soon it will overflow. Engineers are constantly poached by rival companies. Hundreds of new competitors arise to build variations on the same system, or to write software or build hardware that can take advantage of this next wave of computing power. Some just want to repackage what the computer vendors produce, or act as consultants to the companies that adopt these new machines.

The team solves one problem after the next, day and night, until the machine is complete. They focus, overfocus, block out the other concerns of the world. Their wives are ignored, as are the kids. The work is too important.

* * *

Such is the story of Data General and the group that built the computer system code-named “Eagle,” which would be successfully marketed as the Eclipse MV/8000. My summary above comes from Tracy Kidder's wonderful book The Soul of a New Machine, published in 1981. It’s about the rise of minicomputers, a now-amusing name for machines the size of double-wide refrigerators, which were considered a major advance during the 1970s, when gargantuan IBM mainframes still roamed the earth and were possessed only by the largest companies and bureaucracies. Minicomputers used new CPUs and memory that made computing accessible to a much wider range of applications and locations, and were relatively cheap. They flourished.

The Soul of a New Machine has much to recommend it — it won the Pulitzer Prize for its propulsive narrative and crisp explanations of complex technology — but I’m writing about it now, following Kidder’s recent passing, because the book helpfully dislodges you from your presentist perspective and asks, “Look what happened before — sound familiar?”

* * *

A half-century after it was published, The Soul of a New Machine does a better job challenging AI hype than most current criticism. (Also, there are probably writers working on books about AI who are shaking their fists at Kidder for beating them to that memorable title.) It's hard to read The Soul of a New Machine in 2026 without wondering whether all this AI hype is really so new. Is AI truly more revolutionary than a previous wave of computer technology that offered, for the first time, to put screens on every desk of every company? The Data General team helped to bring about a transition not from existing software and hardware to incredibly intelligent software and hardware, or from powerful computers to superpowerful computers, but literally from paper to digital files and high-speed processing. Now that is a transition. The millions of companies that could not afford an IBM mainframe could afford a Data General Eclipse or a DEC VAX system or a minicomputer from another competitor. They could, for the first time, give every employee the power of computers. Is having Microsoft Copilot help your accountants with their spreadsheets more revolutionary than moving those accountants from physical spreadsheets to electronic ones?

Amazingly, the final chapters of The Soul of a New Machine tackle exactly the same profound questions we are struggling with today regarding the impact of artificial intelligence, and Kidder records Data General engineers expressing concerns that sound straight out of the mouths of engineers working at OpenAI or Anthropic. The team’s excitement upon the completion of the Eagle leads to reflections and bigger worries than beating their competitors. What if the Pentagon wants to use the Eagle for war or other destructive purposes? Should the team object or build back doors into the machine? What will the new computer system mean for employment, since it will replace many functions of work with software, and do those tasks faster than anyone can imagine, in nanoseconds? What if their work culminates in true artificial intelligence, and the machines take over and destroy us?

Spending so much time with the team, Kidder begins to ponder these questions himself — and has his own unsettling encounter with the technology. An engineer introduces Kidder to Adventure, one of the engrossing text games of early computing. He is sucked into its digital world, playing nonstop for hours. The computer suddenly feels alive, intelligent.

But Kidder pulls back.

It was the time of night when the odd feeling of not being quite in focus comes and goes, and all things are mysterious. I resisted this feeling. It seemed worth remembering that Adventure is just a program, a series of step-by-step commands stored in electrical code inside the computer.

How can the machine perform its tricks? The general answer lies in the fact that computers can follow conditional instructions.

Kidder turns to one of the Data General engineers, Carl Alsing:

I asked Alsing how he felt about the question — twenty years old now and really unresolved — of whether or not it's theoretically possible to imbue a computer with intelligence — to create in a machine, as they say, artificial intelligence.

Alsing stepped around the question. “Artificial intelligence takes you away from your own trip. What you want to do is look at the wheels of the machine and if you like them, have fun.”

Alsing’s focus on the role of the new machine in your life or work, rather than its purported soul, instantly dispels the mythology surrounding this emerging technology. Even after an all-nighter building a revolutionary computer, Alsing is lucid about what he is making: a tool that might be helpful for some people and some purposes, but not for others.

* * *

In the 1980s, most of the minicomputer companies, launched with such excitement in the late 1970s, failed. Data General was acquired for a fraction of the billions it was once worth. The minicomputer, however, was broadly adopted, was transformative, became routine, and then was surpassed by a new new machine, the personal computer.

Later, Data General’s domain name, DG.com, was sold to a chain of discount stores, Dollar General.

For The New Yorker, Hanif Abdurraqib considers the cost of what he calls “relentless convenience,” our ability to stream content of every kind, be it movies, music, and even potential mates on dating apps, and what we lose when we live in this frictionless existence. There’s so much to learn and to savor in friction, he suggests, be it waiting for a song on the radio to complete a prized mixtape or the deep connection we can find if we go through the time and trouble to ditch electronic communication and actually meet others, face to face.

Maybe what my pal who insists on finding love the old-fashioned way is saying is that it shouldn’t be as frictionless as browsing Amazon from your couch. If you believe, as she does, that the next person you fall in love with could be the last partner you ever pursue, and the last who ever pursues you, then that pursuit should find you thrown fully into the world, eager for the beauty and discomfort of spontaneous human interaction. And I tell her that I mostly agree, though I generally just avoid dating apps because the onslaught of visual information overwhelms me. Still, I understand her desire, because so many of my own desires are detached from the reality of the times we live in. I am still inventing inconvenience in order to bolster my desire to feel alive.