MERRY NEWTONMAS 2025

Sir Isaac Newton was born on Christmas Day 1642. Following the Gregorian calendar, Newtonmas would be on January 4. One advantage to keeping the traditional date is that this coincides the year of Galileo's with the year of Newton's birth. Growing up in Cleveland, Ohio, many of my friends were Ukrainians and others whose churches kept the old Julian calendar, giving some of the kids two dates for presents. (It also explains Tom Clancy’s Red October, even though the Bolshevik phase of the 1917 Revolution began on November 7.)

THE LEGEND OF THE MARTIANS 

György Marx (Eötvös University Department of Atomic Physics)

There is a rumor in America that there are two intelligent races on Earth: humans and Hungarians. (Isaac Asimov)

The story is here from Page 116 forward.
Akadémiai Kiadó, Budapest, 1994, 1997, 2001

- Enrico Fermi was an outstanding talent who was interested in many things besides nuclear physics. He was also known for asking famous questions. Fermi questions have a long introductory text, for example: - "The Universe is vast, with billions of stars, many of them similar to our Sun. Planets may also orbit many stars. A significant proportion of these planets may have liquid water and a gaseous atmosphere on their surfaces. The light from the star may have triggered the synthesis of organic compounds on them, turning the ocean into a thin, warm soup. The carbon compounds linked together to create self-reproducing structures. The simplest living things reproduce, develop through natural selection, become increasingly complex, and eventually develop into active thinking beings. Civilization, science, and technology develop. Longing for new and fresh worlds, they travel to neighboring planets, later to planets of nearby stars, and thus spread throughout the Galaxy. Such highly developed, talented peoples can hardly ignore this beautiful planet, Earth. - And then Fermi he came to his essential question:  - If all this is true, then where are they?

Szilárd Leó had a good sense of humor, and he answered Fermi's rhetoric this way:

- They are here among us, but they call themselves Hungarians.

Online in Hungarian at the HUNGARIAN ELECTRONIC LIBRARY / MAGYAR ELEKTRONIKUS KÖNYVTÁR

https://mek.oszk.hu/03200/03286/html/tudos1/marsl.html 

I cut and pasted it into Google Translate to provide the text above. 

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Still Riding the Gray Planet 

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Monsters from the Id

A Numismatic History of Hungary 

Christmas Stars: Menkar and Keid; Scrooge and Franklin

Every Christmas we wonder why astrologers were compelled to travel a thousand kilometers across the desert. I have no idea. But I do know that there is always something interesting happening in the sky. In 2020, the December conjunction of Saturn and Jupiter rewarded anyone who looked up at night. This year my telescopes will be turned to the constellations Eridanus and Cetus. 

Not being on the zodiac, they do not enjoy the attendant mysticism. It is just that my living now at 30N latitude makes those constellations easier to explore. Also, I am writing about Keid (40 Eridani) and its companion omicron Eridani. (It has two, denoted B and C, but I do not expect to overcome the urban sky here.) I observed the orange giant Menkar alpha Ceti last week. Tonight through Thursday, the sky should be clear and the full Moon not obscuring the view with Thursday night being the best opportunity with a later rising of the Moon.  

It is nice to have the week off as a paid holiday. That is an old tradition, but not the one most people usually think of. The year of 365¼ days was reconciled with the circle of 360 degrees by not counting the five dead days between the winter solstice feast of Saturn and the start of the Julian New Year. It was a direct import from Egypt because of Cleopatra Ptolemy’s alliance with Gaius Julius Caesar.  

The Thebans say that they are the earliest of all men and the first people among whom philosophy and the exact science of the stars were discovered, since their country enables them to observe more distinctly than others the rising and settings of the stars.  Peculiar to them also is their ordering of the months and years. For they do not reckon the days by the moon, but by the sun, making their month of thirty days, and they add five and a quarter days to the twelve months and in this way fill out the cycle of the year. But they do not intercalate months or subtract days, as most of the Greeks do. – Diodorus Siculus, Library of History, Book I, paragraph 50. (Online at U Chicago Penelope here.) 

We think of Christmas as a continuing tradition, uninterrupted for 2000 years. But it was interrupted and abandoned. We do not understand Ebenezer Scrooge. He was a religious conservative of that time, a fundamentalist Protestant. Calvinists of that time had come to accept that you demonstrate your election and bring others to salvation by working hard and being thrifty. Max Weber’s The Protestant Ethic and the Rise of Capitalism begins by republishing Benjamin Franklin’s The Way to Wealth. 

A Colonial Christmas

Christmas on December 25 has not much support in Biblical text. After a hiatus, Christmas was re-invented. Charles Dickens’ A Christmas Carol (1843) fit with the invention of the Christmas greeting card (1843) which took advantage of the invention of the postage stamp (1840).  

 

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Merry Newtonmas 2022 

Merry Newtonmas 2011 

The Christmas Star 2017 

Before Email 

Merry Newtonmas 2022

The first recorded Newtonmas is credited to the Newton Kai at Tokyo University, circa 1891. Physics students had been celebrating with a party of their own in their undergraduate years. As they became graduate students, they brought some of their professors into the circle. The report is in Nature, Volume 46 Number 1193 Page 459, published 8 September 1892. So that party had been convened the Christmas before. 

 

I do not remember my flash of inspiration back in 1982. However, I was enrolled in physics at Lansing Community College and was the physics lab aide. Our professor for that class was Dr. Alan Saaf who more than once called a McDonald’s Quarter Pounder a “Newtonburger.”[1] So, Newtonmas could have been a logical leap. I sent out Newtonmas cards several times in the 1980s and recorded a Newtonmas greeting as a “Community Commentary” for WKAR-FM, the public broadcaster of Michigan State University in 1983 or 84.

 

For that, I built up the imagery of a little boy born in a small village across the sea who would grow up to bring light to the world. When I quoted the poet's eulogy – Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light.—I emphasized the word Pope and then announced that Sir Isaac Newton was born on December 25, 1642, the same year that Galileo died.

 

Michael Shermer and the Skeptic Society have been celebrating Newtonmas for at least 25 years by now. Richard Dawkins touted it in 2007. And it was an element in an opening scene of Big Bang Theory Season 3 Episode 11, “Maternal Congruence.” 

 

Why do we not have Galileomas or Keplermas? Sir Isaac Newton stands as perhaps the most accomplished scientist in the modern world. At his level would be only Aristotle or only Archimedes. As much as we owe to everyone who invented something new, not every savant earns the same honors as Newton. 

 

He invented the calculus to prove his physics. He did that with geometry. When Richard P. Feynman attempted to recreate that work from scratch, he found that he could not. We  rely on algebra and calculus and have forgotten much. Newton’s physics demonstrated that the laws on Earth are the same as they are in Heaven. Heaven is not perfect; Earth is not flawed. Force equals mass multiplied by the second change in distances over times and the path of an object acting under central force motion always fits to a curve sliced from a cone. 

 

That would be enough. But Newton did more. 

 

He served two terms in Parliament, representing Cambridge University (1689-1702). He was president of the Royal Society (1703-1727). He found another proof for the Binomial Theorem (also known as Pascal’s Triangle). He found a clever shortcut for rapidly approximating square roots. 

His work in optics was fundamental to understanding light and, in truth, the entire electromagnetic spectrum. That work also led to the development of spectroscopy. Newton’s work in optics led to his invention of the reflecting telescope, which (unlike a refractor) introduces no chromatic aberration. 

 

While he was Warden and Master of the British Royal Mint for 30 years, he had himself sworn as a justice of the peace. He went disguised into the pubs where criminals exchanged wares and gossip. With the information he gathered, he arrested, interrogated, and prosecuted counterfeiters. The execution of the notorious William Chaloner hallmarked Newton’s labors to protect England’s money. Chaloner was a master criminal who even testified before Parliament in his efforts to rob from the Royal Mint with impunity. Clever as he apparently was, Chaloner was no match for Newton. 

 

In fact, no one was. 

 

Newton’s Principial Mathematica was more than the birth of modern astronomy or physics or science or mathematics. The Constitution of the United States with its balances of powers is a translation of the principles of physical science to the social world. You can get an ought from an is.

 

General Biographies of Sir Isaac Newton

Berlinski, David. Newton's Gift: How Sir Isaac Newton Unlocked the System of the World. New York: Free Press, Simon and Schuster, 2000.

Calligas, Elini (editor), Coincraft's 1998 Standard Catalogue of English and UK Coins 1066 to Date.London: Coincraft, 1998.

Keynes, Milo. “The Personality of Isaac Newton,” Notes and Records of the Royal Society (49), London: The Royal Society, 1995.

Marotta, Michael. “Sir Isaac Newton: Warden and Master of the Mint,” The Numismatist, Vol. 114, no. 11 (November 2001), p. 1302-1308, 1363 : ill., port. (George Heath Literary Award, Second Place, 2002)

Newman, E. G. V. “The Gold Metallurgy of Isaac Newton.” The Gold Bulletin Vol 8. No. 3, London: The World Gold Council, 1975.

Trowbridge, Richard J. Queen Anne, 1702-1714 Mystery Farthings. Long Beach: Coins of the British World, 1970.

Westfall, Richard S. Never at Rest: a Biography of Isaac Newton. Cambridge: Cambridge University Press, 1980.

White, Michael. Isaac Newton: The Last Sorceror. Reading, Mass.: Helix Books, Perseus Books, 1997.

www.newtonproject.sussex.ac.uk/, The Newton Project, Professor Rob Iliffe Director, University of Sussex, East Sussex - BN1 9SH

 

Newton at the Mint

(Sir John Craig was Deputy Master and Comptroller of the British Royal Mint.)

Craig, Sir John. Newton at the Mint. Cambridge: University Press, 1946.

Craig, Sit John. “Isaac Newton - Crime Investigator,” Nature 182, (19 July 1958), pages 149-152. 

Craig, Sir John. “Isaac Newton and the Counterfeiters.”  Notes and Records of the Royal Society (18;2), London: 1963, pages 136-145. 

www.royalmint.com/museum/newton Web site pages of the British Royal Mint. 

Shaw, W. A., F. B. A., Select Tracts and Documents illustrative of English Monetary History 1626-1730.  London: Clement Wilson, 1896; reprinted, London: George Harding, 1935.

 

Newton versus the Counterfeiters

Levenson, Thomas. 2009. Newton and the Counterfeiter: The Unknown Detective Career of the World's Greatest Scientist. Boston and New York: Houghton Mifflin, Harcourt.

Sources cited by Sir John Craig, Notes and Records of the Royal Society. (18;2), London: 1963, pages 136-145. 

Chaloner, William: Proposals.. ... to prevent clipping and counterfeiting, II.2.I694/5: Against raising £1,000,000 for the great recoinage, c. 3.1695: The defects.. ... of the Mint, 1697: Appeal after arrestc . I8.2.I698. (Four pamphlets.)

Middlesex Sessions Roll for 1699, containing original indictments at the Old Bailey, in the County of Middlesex Record Office, London. now in the London Metropolitan Archives.

Newton MSS. in the library of the Royal Mint; 5 large volumes of those papers from Newton's residence which were sorted by Conduitt as 'Mint'. 

Depositions about and letters from criminals, 1697-1704; one volume in the Royal Mint Library. 

Anon. GuzmanR edivivus, a short view of the life of William Chaloner, the notorious Coyner, who was executed at Tyburn on Wednesday the 22nd of March 1698/9; with a brief acount of his trial, behaviour and last speech. London: J. Hayns, 1699. 

Reasons Humbly Offered Against Pass an Act for Raising Ten Hundred Thousand Pounds by William Chaloner, 1694.

Proposals Humbly Offered, for Passing an Act to Prevent Clipping and Counterfeiting of Money, by William Chaloner, London, 1694.

 

[1] Just to note, 1 N = 0.2248 lbf and 0.25 lbf = 1.112 N. Dr. Saaf came to physics at LCC via astronomy at the University of Chicago. So, he probably would have accepted an order of magnitude approximation if the theory were useful.

 

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Newton versus the Counterfeiter 

Merry Newtonmas 2021 

Of Watches and Beaches and Atheists 

 

Merry Newtonmas 2021

Nature and nature's laws lay hid in night;

God said "Let Newton be" and all was light.

 

If your neighborhood is like ours, then, it started with Halloween which began about Labor Day: lights; people lit up their yards and houses with displays of icons and arrays of sparkles. As soon as Halloween passed, Christmas was launched. I believe that this is a reaction to the continuing Covid crisis. People are resisting despair and fighting back against both biological and social malaise with symbols of hope and good cheer. 

 

Around our neighborhood;
the unicorn is snorting shifting colors.

In Newton’s time, it was the Great Plague of 1665. That was followed by the great London fire of 1666. When the plague struck, Cambridge University was closed and Newton went home to Woolsthorpe. There, isolated with his own thoughts, he expanded his work in mathematics, optics, and gravitation. 

Two Newtonian reflectors:
my first Celestron Equatorial 130-mm and
my current Besser 208-mm. 
Reflectors have no chromatic aberration
and are easy to make very large.

We can only wonder if forty years from now we will have time travel and warp drive (perhaps the same phenomenon) because some genius was isolated by Covid-19. 

 

We accept today that Leibniz and Newton
developed the calculus independently.
We generally use the Leibniz notation 
though Newton's can be a convenient shorthand.

The American Numismatic Association granted a George Heath Literary Award (2002) to a feature article that I wrote about Sir Isaac Newton’s 30-year career at the Royal Mint, published in November 2001. Historian David Berlinski called that time “uninteresting” ending his biography with the publication of the Principia. 

Not many people read Philosophiae Naturalis Principia Mathematica today in any language and surely not in Latin. I have two editions, one translated by I. Bernard Cohen whose work I already knew from reading ancient sources when writing about numismatics. 

Newton’s ideas are easy to explain today, especially using algebra and the Leibniz notation for calculus. The proofs in the Principia are—and could only have been—delivered via geometry. We do not know it so well today. Richard P. Feynman intended to develop and present his own proofs to a class as a treat to relieve the stress of up-coming semester final examinations. He could not do it. He did not know enough geometry. 

 

The book is the Principia, opened to the page proving
Kepler's laws and central force motion.

The core of that work is Newton’s mathematical proof that Kepler’s laws of motion are required by logic and determined by the nature of central force motion as measured for timekeeping, navigation, and astronomy. They are necessary factual truths. 

 

Previously on Necessary Facts

Newton versus the Counterfeiter 

Measuring Your Universe: Alan Hirshfeld’s Astronomy Activity Manual 

Meteorites

Robert Heinlein's Starship Troopers 

Reminders of Newtonmas Past 

Merry Newtonmas 2020

For about 1500 years, the story of the Star of Bethlehem was accepted as historically accurate because it was divine truth. Miracles were not questioned. With the Renaissance, a new way of looking at the world evolved. 

The scholarly tradition of explaining the Star of Bethlehem with scientific evidence apparently began with Johannes Kepler who identified a triple conjunction as the likely event.

13 December 2020 scale drawing

In 1604, he published The New Star in the Foot of the Serpent (De stella nova in pede serpentarii: et qui sub ejus exortum de novo iniit, trigono igneo…). In that tract, he examined a triple conjunction, as well as a nova, which he identified as the cause of the conjunction. He was not alone in that kind of a belief. Others expected the conjunction to cause a comet. Reviewing the facts in 1614, Kepler said that the Star of Bethlehem was a nova in 4 BCE caused by a triple conjunction in 7 BCE. (See “Common Errors in ‘Star of Bethlehem’ Planetarium Shows,” by John Mosley, The Planetarian, Third Quarter 1981.)

19 December 2020
Scale drawing

Over the centuries, the Christmas Star has been explained as a comet, a meteor or meteor shower, but the conjunction theory has been the most popular. 

In science, a good problem takes us far beyond the results of a single observation. The Christmas Star has been debated on many levels. The International Planetarium Society website (www.ips-planetarium.org) lists over 100 citations to the Star of Bethlehem. Some of those articles and letters were part of a multifaceted decades-long argument among at least five astronomers and one editor. Writing in Archaeology Vol. 51, No. 6 (Nov/Dec 1998), Anthony F. Aveni cited 250 “major scholarly articles” about the Star of Bethlehem. 

The sciences here are psychology and sociology, not astronomy. People want to believe. The need for religion derives from a more basic need for certainty. I like it when my own plans come together. Beyond that, I observe that most other people do not find freedom in the unknown, the unplanned, or the spontaneous response to unforeseen events. As with a spontaenous choice, an effective response is based on a rational understanding of experiential circumstances. For myself, an invention is a kind of discovery. Right now, my instrument is a telescope. But I also own two microscopes. For myself, the Christmas conjunction is no more or less special than the structure of a poinsettia, something I have not investigated at all, but which is pretty easy to find. Conjunctions happen all the time. And they happen at all only because of our inertial frame of reference. Standing on the spinning Earth orbiting a focus of an ellipse, we see arrangements come and go with -- I confess -- comforting predictability.

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Eclipses 

Merry Newtonmas 2019 

Reminders of Newtonmas Past 

Atheists in Foxholes 

Merry Newtonmas 2019

Born on December 25 (OS), 1642, Sir Issac Newton was more than the most brilliant man of his time. He was the first modern scientist. He discovered the physics and the mathematics that made possible the industrial age and the electronic age. He invented a new kind of telescope, based on his research into optics. He was an accomplished lawyer, both a prosecutor for the state, and later an effective political scientist. His achievements in theology are unappreciated today. He was a skilled chemist. He served in Parliament as the representative of Cambridge College. He was president of the Royal Society. In addition, he served as Warden and Master of the Royal Mint for 30 years, rescuing the economy from imminent disaster. Had Sir Isaac Newton done any one of these, his place in history would have been assured. He did all of them. 

Newton's famous “three laws of motion” were only introductory propositions to his Principia Mathematica. The purpose of the Principia was to demonstrate that the force which at once moves the planets and holds them in their orbits is the same force that pulls apples from their trees. He achieved this proof by creating a new kind of mathematics, called the calculus. 

Newtonian Reflector
https://spaceplace.nasa.gov/telescopes/en/

Many consider Newton to have been the first modern scientist. John Maynard Keynes dubbed him “the last sorcerer.” We commonly believe that alchemy is a superstition or a fraud, like witchcraft or theater magic. To Newton, alchemy was science. We see two magnets drawn together or repulsed. We play with static electricity. We do not doubt that the Sun attracts the planets. Newton believed these things as well specifically because of his understanding of alchemical principles. To Newton, the world operated according to basic principles of natural law that were the same everywhere for everyone. 

 

Feynman's Lecture on Newton's Proof of Kepler's Laws 
https://upload.wikimedia.org/ from energy dot gov

Few people have read the Principia Mathematica in its entirety. The very popular Nobel laureate physicist, Richard P. Feynman, attempted to delight a class with a demonstration of Newton's proof of Kepler's Laws of Planetary Motion. He could not produce the mathematics as Newton had done it and was forced to present his own geometric proof.

PREVIOUSLY ON NECESSARY FACTS

Reminders of Newtonmas Past

Newton Versus the Counterfeiter

The Big Whimper of Modern Philosophy

Physics for Astronomers: The Works of Steven Weinberg

Reminders of Newtonmas Past

Newtonmas 2011

https://necessaryfacts.blogspot.com/2011/12/merry-newtonmas.html

Today, we ignore his religious writings, the extent of which actually eclipsed his scientific production.  His heretical Arian beliefs foreshadowed modern Unitarianism, but he swore under oath to be a Trinitarian so that he could teach at Cambridge. 

Newton’s colleagues called him fearful, cautious, suspicious, insidious, ambitious, excessively covetous of praise, and impatient of contradiction. Even his relatives and his true friends were modest in their praise of Newton. Physically sound in his life, he died at 84. He had lost only one tooth, still had much of his hair, and read without glasses. Yet, he was a hypochondriac, suffering from illnesses and diseases that he treated with medicines he made for himself.  

Newtonmas 2012

https://necessaryfacts.blogspot.com/2012/12/merry-newtonmas-2012.html

The English crown turned to him to save the Royal Mint. Even when they were not corrupt - which they usually were - the Mint officials were unable to solve the basic problem of creating and maintaining a system of money that worked. A stern Protestant, deeply religious, and moralistic in the extreme, Newton cleared out the criminal element and gave England a reliable monetary system.

Newton and the Counterfeiter: The Unknown Detective Career of the World’s Greatest Scientist by Thomas Levenson (Houghton Mifflin Harcourt, 2009) is intended for a general readership, yet rests on an extraordinary foundation of careful scholarship.  Thomas Levenson teaches science journalism at MIT.  He has been granted several awards for his PBS documentaries. Levenson delivers to print the videographer’s impact of sight and sound.  You walk down the alleys and into the pubs where Isaac Newton investigated crimes against the Mint of which he served as warden and later master.

Newtonmas 2013

https://necessaryfacts.blogspot.com/2013/12/merry-newtonmas-2013.html

Nature and nature's laws lay hid in night;

God said, 'Let Newton be'  and all was light.

Alexander Pope

Sir Isaac Newton was born on Christmas Day 1642, the year that Galileo died.  For most people, Newton is famous for his Three Laws of Motion.  Beyond that, those with additional education know him for creating the Calculus to prove his theories of celestial and terrestrial mechanics. In addition, Newton invented the reflecting telescope as a result of his experiments with light.  And he also proved the general case for the Binomial Theorem (“Pascal's Triangle”). He served in Parliament, representing Cambridge, where he had been a professor of mathematics.  He served as president of the Royal Society of scientists. Few people except numismatists know him to have been the Warden and Master of the British Royal Mint for thirty years.  He had himself sworn as a justice of the peace so that he could pursue and prosecute counterfeiters.  Any one of those achievements would have made important to us today. That he accomplished all of that - and more - set Sir Isaac Newton apart even from the geniuses and polymaths recorded by history.

Newtonmas 2014

https://necessaryfacts.blogspot.com/2014/12/merry-newtonmas.html

I offered Newtonmas in a radio script for WKAR-FM East Lansing in the early 1980s (1982-1984).  In that script, I built up the imagery of a little boy born in a small village across the sea who would grow up to bring light to the world.  When I cited the poet, I emphasized the word Pope – and then announced that Sir Isaac Newton was born on December 25, 1642, the same year that Galileo died. 

Over the years, I sometimes sent out "Newtonmas cards" to our friends, most of whom we knew from college classes in computer science and related fields.  Our physics professor, Alan Saaf, occasionally called the McDonald’s Quarter Pounder a “Newton Burger” (1.0 N ~ .224 lb-f). I do not recall Dr. Saaf ever saying “Newtonmas” but he could have.

Newtonmas 2015

https://necessaryfacts.blogspot.com/2015/12/the-first-newtonmas.html

We commonly call the Nativity scene “the first Christmas” though it was not. The first Christ Mass could not have been celebrated before the Church existed. So, too, did the first Newtonmas not come until 248 years after his birth.

 

It seems that three students at Tokyo University started Newtonmas in their dormitory sometime before 1890.  As the undergraduates developed into graduates and assistants, their professors were drawn into the celebration, and a more suitable assembly hall was found in the University Observatory.  By 1890, they called themselves the Newtonkai  (Newton Association; 皆 = kai = “all”) and moved to the Physical Laboratory. There, they played games symbolic of great mathematicians, physicists, and astronomers: Newton’s apple, Franklin’s kite, a naked doll for Archimedes … 

That story comes from “A New Sect of Hero-Worshippers” published in Nature, Vol. 46, No. 1193, p. 459, 8 September 1892. It available from the publisher for $18 if you are not a member, or it can be found online at Google Books.

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Merry Newtonmas 2017

Celebrating Newtonmas off and on since 1984, this year I honored Sir Isaac by buying myself one of the commemorative 50p coins from the British Royal Mint. 

"Near the end of  his life, Newton described  himself to  his nephew and biographer,  John Conduitt, in these pleasant words: 'I do  not know what I may appear to the  world, but to myself I seem  to have been  only a  boy, playing on  the seashore, and  diverting myself in  now and then finding a smoother pebble or a prettier shell  than ordinary, whilst the great ocean of truth lay all undiscovered before me.'

"Two hundred years later, biographer Milo Keynes wrote: 'This  life of apparent serenity was, however, far  from the  truth, for Newton is known to have  had a  most complex and difficult  personality.' His colleagues described  him variously as  fearful, cautious, suspicious, insidious, ambitious, excessively covetous of  praise and impatient of  contradiction. Even his closest relatives and true friends were modest  in their praise." -- "Sir Isaac Newton: Warden and Master of the Mint," by Michael E. Marotta, The Numismatist, November 2001. (George Heath Literary Award, 2nd Place, 2002.)

10 REM MIKE MAROTTA. FEB 5, 1987. NEWTON'S ALGORITHM FOR SQUARE ROOTS
19 LIMIT= .0001
20 PRINT "ENTER A NUMBER"
21 INPUT X
56 XN = X/2
60 R1 = (XN + X/XN)/2
70 IF ABS(XN-R1) < LIMIT THEN GOTO 80
75 XN = R1
76 GOTO 60
80 PRINT "THE SQUARE ROOT OF ";X;" = ";R1

PREVIOUSLY ON NECESSARY FACTS
Newton versus the Counterfeiter
Fig Newtons and Leibniz Biscuits
The Structure of Scientific Revolutions
Feynman's Rainbow