Choose Your Own Math Adventure

For geometry-themed entertainment try reading Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini as you go through the geometry problems here, as Newton claimed geometric synthesis helped him make mathematical discoveries when writing Principia by performing geometric synthesis on the cause to prove the effects he observed through analysis.

Isaac Newton entered Trinity College in 1661 as a sub-sizar meaning subsidized tuition in exchange for being an unpaid servant on campus. Cambridge then was just a diploma mill there were seldom any lectures and fellows (graduate students) took jobs as P/T tutors to supplement their income. Undergrad degrees consisted of one of these tutors giving you a list of books to read. According to his Trinity College notebook which Cambridge still has, this was a large reading list of Greek and Roman classics in Latin: Aristotle's Organon and Metaphysics, Porphyry's Isagoge, Plato and manuscripts of commentary on Aristotle usually written by other fellows to sell to undergrads. This was because of the recovery of Aristotle in the middle ages which shaped logic courses for the next few centuries. Newton didn't read much of Aristotle or any of the books assigned to him, but he later claimed he read enough to have learned how to properly categorize a problem, how to think logically, and later when he wrote the Principia how to use a cause to prove the effect. His algebra lectures that later became the book Arithmetica Universalis described different 'species' ie: Aristotelian categories of math objects that can be combined together, essentially he was describing types.

According to Newton's notebooks and the writings of his close friend DeMoivre, here is how Newton learned mathematics. Wandering the town fair in 1663 he comes across a book on astrology, and out of curiosity buys a copy as Newton was always into alchemy and other occult topics. It features trigonometry he doesn't understand so he goes to the mathematics department of Cambridge to find books on trig only to discover everyone there is deeply immersed in the work of Descartes. They give him Euclid and Oughtred's Clavis. He retries Euclid's Elements after having given up reading it before, and sees the proof of the Pythagorean theorem which makes him want to read the rest of the text. Newton's college notebooks show he was influenced by books II (geometrical algebra), V (proportions), VII (number theory) and X (irrationals) and that Euclid taught him how to write a proof while Aristotle's system of logic he studied gave him the rigorous thought to be able to follow the proofs.

Since his Cambridge degree is essentially a joke, with the school so lax that nobody cares what you're doing, he abandons the official curriculum for the rest of his undergrad to hang around the math department. He pursues Kepler and Oughtred's book, writing he understands it except for the quadratic and cubic equations however his takeaway in his notes is that algebra can be used for exploration, which he starts doing by writing out hundreds of examples. He gets a B.A. in 1665 despite never finishing any of the official curriculum and is offered to be elected as a minor fellow pursuing a master's degree but the school is closed because of the plague in the summer of 1665.

He goes home to his family farm and takes with him a 1659 two volume Latin translation of Descartes La Geometrie by van Schooten with appendices and commentary by other Cartesian research mathematicians. Newton later writes in his school notebook how a proof by Hudde to estimate the slope with a tangent in the appendix taught him how to transform one type of problem into another to study it in a different way. This book was considered a very hard text and the state of the art of 17th century analysis which uses almost the same notation we use today. He read 10 pages or so, couldn't understand the text, and reread from the beginning. Went a little further then stopped again going back to the beginning. He repeats this algorithm for all summer and autumn until he finally finishes the whole text. He misunderstands a technique of Descartes to his own benefit where he thinks Descartes is hinting the reader can figure out any properties of a curve and while trying to do so as an exercise develops his own sophisticated analysis well beyond the book or any other book of that time. During this period is when he started working 18 hours a day, 7 days a week. He didn't stop this work ethic until he died, even after becoming wealthy and living in London.

In the winter of 1665 he reads Arithmetica Infinitorum by Wallis which is the arithmetic of infintesimals and Viete's Opera Mathematica another large book. In less than a full year Newton managed to bring himself up to date with the entire achievement of mid 17th century mathematics by himself. He discovers the generalized binomial theorem and infinite series analysis which he claimed to have invented to find short cuts to calculations. He returns briefly in 1666 to Cambridge before it's shut down by the plague again, his notes revealing he finished his differential and integral calculus during these few months back at school.

Describing his activities during the second plague year: "I am ashamed to tell to how many places I carried these computations, having no other business at the time, for then I took really too much delight in these inventions". There is notes he kept of calculating a logarithm to it's 52nd decimal point. He returns to Cambridge in 1667 where he is obsessed with uncovering the ancient analysis used in classical geometry. Pappus' book 7 contains a commentary about a lost group of tools and propositions that Euclid, Apollonius, Eratosthenes and other geometers of the day used that Pappus referred to as the 'treasury of analysis'. These lost Porisms (corollaries) are speculated by Newton to be projective geometry.

Newton in his Lucasian chair lectures said that the ancients would never bother to introduce the algebra of curves with geometry because you lose the simplicity of working within geometry as its whole point was to escape the tediousness of calculations by simply drawing lines and circles. He also claims books like Pappus's Collectio deliberately hid the analysis, as it was considered an inelegant tool, and that ancient synthesis where they deduced a consequence from a given premise (a corollary) using visual means was a superior method as the analysis could not be reversed in steps like geometric synthesis could. He went further and claimed if you wanted to discover seemingly unrelated corollaries you had to use synthesis, describing the analysis of his day as a 'tedious pile of probabilities used by bunglers'.

In Arithmetica Universalis which is an unauthorized book released by his successor at Cambridge who discovered Newton's Lucasian lectures in the library and published them as is, Newton writes how Descartes' attempts to include into standard plane geometry various curves like parabolas more than just a waste of time, they are 'ruining the simplicity of geometry'. In figure 108 he shows how he would solve this problem, by reinterpreting an ellipse, arguing the Cartesian methods fail to understand the point of doing everything with just a ruler and compass.

He later wrote in a manuscript on geometry that mechanics of motion was what generated all geometry, and that the ancients had understood this as well conceiving geometrical objects as generated by moving along a straight edge, circles via the movement of a compass, or translation like in Proposition 4 of Book 1 of Euclid's Elements where one triangle form is moved to compare to another. He then demonstrated that rotation of rulers were in fact transformation of the plane something that wasn't formally developed by other mathematicians for another 200 years. Here is how Newton used his rotating ruler to create a power series.

Newton invented the concept of modern limits using epsilons (very small quantities) when writing the Principia during the geometric synthesis stage of his infinitesimal analysis. He decided limits demonstrated as geometric vanishing augments to be 'more in harmony with the geometry of the ancients in that there should be no need to introduce infinitely small figures into geometry'. Since infinitesimal analysis was merely a heuristic tool to him, he downgraded infinitesimals (which had no established theory or proofs anyway) to prefer his new limit proof method he derived from geometry synthesis, and is very similar to what everyone today is using in modern calculus courses.

"Comparing today the texts of Newton with the comments of his successors, it is striking how Newton’s original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibnitz" -Vladimir Arnold 1990

There's a book Cut The Knot that has a chapter on solving probabilities in a continuous sample space by recognizing the problem or solution can be represented using geometry.

What is the probability that three randomly drawn real numbers in the interval between 0 and 1 have a sum less than 1? The solution recognizes that the three numbers can be points in 3D space, and form a pyramid. Dropping a volume 1 cube over the interval provides the distribution needed where by definition it must sum to 1. Now take the volume of the pyramid, that's the answer. Easy and intuitive.

Some entertainment

• A brief history of geometry I
  • Mesolithic times to 1400 A.D.
• A brief history of geometry II
  • European/Cartesian geometry
• A brief history of geometry III
  • 19th century Complex/Non-Euclidean geometry
• A brief history of geometry IV
  • Higher dimensional geometry