Try The Mathematical Tripos

For geometry themed entertainment try Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini.

Isaac Newton entered Cambridge Trinity College in 1661 for it seems a medical degree as a sub-sizar meaning subsidized tuition in exchange for being a domestic servant on campus waiting tables and working in the kitchen, whatever labor that was needed. Cambridge then was just a diploma mill there were seldom any lectures, the oral exams at the end of your degree basically optional, and fellows (graduate students) took jobs as tutors to supplement their income when they weren't getting drunk in nearby taverns. 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 curriculum was a large reading list of classics such as Du Val's four volume collected works of Aristotle like Organon and Metaphysics, Porphyry's Isagoge, Vossius's Rhetoric, Diogenes Laertius, Epicurus, Plato, and many Roman poets like Ovid. This was because of the recovery of Aristotle in the middle ages which shaped 'philosophy of natural sciences' courses for the next few centuries. There didn't exist titles like mathematician, practitioners then considered themselves philosophers. Newton soon stopped taking notes of his official curriculum and must have read Walter Charleton’s Physiologia because he wrote down 37 headings on different pages that were questions to be answered investigating natural science topics, 18 of which came from Charleton's book and some from Aristotle. These were 'Of Atoms, Of a Vacuum, Of Vision' and many others. He gave each topic different sized gaps indicating how much he thought he needed to write about each. Everytime he would learn something about them he'd write notes under the heading like a comet position he once tracked.

According to Newton's notebook and the writings of his close friend De Moivre here is how Newton learned mathematics. Wandering the town fair in 1663 he comes across a book on astrology and out of curiosity of the occult buys a copy. There's a figure he couldn't understand in the book because he didn't know trigonometry. He buys a book on trigonometry but couldn't understand the demonstrations because of a lack of geometry. At Cambridge he goes to the math department and everyone there is deeply immersed in the work of La Geometrie by Descartes. They tell him to read Barrow's Euclide's Elements which is a 'simplified' version of Euclid's 15 books. He skims it and after finding what he needs for the astrology/trig book abandons Euclid as 'too trivial demonstrations' only to later to go back and relearn classic geometry taking Barrow's lectures at Cambridge.

He returns to the math department and is given Oughtred's Clavis mathematicae or the key to mathematics writing he understands it except for the solutions of 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.

Again he returns to the math department and despite being told it was a very difficult book he borrows a 1659 two volume Latin translation of Descartes La Geometrie by van Schooten called Geometria, a Renato des Cartes with appendices and commentary by his students. This book was considered the state of the art of 17th century analytical geometry it would be like reading a graduate text today written by a leading expert that had an appendix full of their student PhD dissertations.

This is the algorithm Newton used to read van Schooten's two volume book. He read a few pages or so, couldn't understand the text, and went back to the beginning. Went a little further then stopped again going back to the beginning. He repeats this loop by himself until he finally 'makes himself whole of Descartes'. Looking at the online copy of this 1659 version the Descartes geometry is about 104 pages and 450 or so pages of additional research/commentary. It is broken up into 3 parts and assuming he didn't have to reread each part everytime he looped, only the relevant part (~34 pages), it probably took him 3 months to finish van Schooten which is what his notebook shows that after a few months he was already doing hiw own research in analytical geometry trying to generalize Descartes. His notebook indicates he learned from the grad student commentaries how to transform a hard problem into a different simpler one. Somewhere around this time he moves dorms complaining his peers were too busy partying so lives with more serious students.

Newton's notes which you can read online show him trying to generalize any math he read which often would lead him into algebraic corners where he would get stuck as there didn't exist at the time methods to solve/simplify. To Newton all this math was just a means to an end of understanding the natural science topics he originally wrote down as 37 headings in his notes.

After Descartes he returned the borrowed copy and bought a different copy of Descartes for himself, and bought van Schooten's Exercitationes mathematicae libri quinque or Five Books of Mathematical Exercises to help him fill in the blanks of his algebra misunderstandings as he had a lot of mistakes in his notebooks regarding negative roots. Newton assumed the cubic parabola was the same shape in all quadrants but soon after corrected these mistakes. Descartes geometry wasn't like today's Cartesian plane it originally consisted of just one positive quadrant but van Schooten and his students had expanded it.

Descartes' La Geometrie was an appendix of his larger book Discourse on the Method. That book is about how you would conduct yourself when seeking to make discoveries in the natural sciences and it seems many philosophers then had their own books on how to reason except that Descartes had filled the appendix with applied examples of discoveries he made using his own method. If you buy a copy of the Discourse today none of the appendices are included unfortunately. There is this unusually massive preface he wrote trying to explain what exactly this book is for. The contents of his book tell you how to organize your mind to solve the hardest problems ever and the appendix is multiple examples where he applied this method the most famous being his analytical geometry a complete change from Euclid. He even wrote a book I haven't read that is a complete rewrite of Aristotle's philosophy that Newton apparently used an idea from there in his Principia.

The geometry appendix is filled with little comments of encouragement as he was afraid nobody would read his work if it was too long so everywhere there is these reassurences like 'don't worry reader the following isn't too hard' or insisting the reader figure out a problem themselves to not deny them of the satisfaction he found figuring it out himself. The different copy Newton buys has all these comments left in tact and Newton misunderstands, or the translation is incorrect, one of these comments by Descartes about the equation of curves stating it is easy to find everything you want to know about a curve from it's equation and the reader need not be bothered by a lengthy demonstration when in fact this is an unsolved problem. Newton can't figure out this supposedly simple exercise in Descartes' book so breaks down the equation of a curve into many headings in his notebook and begins to generalize which led to him creating his own advanced analysis well beyond any other mathematician at the time or any mathematician for the next 200 years actually. A misprint invented modern calculus because Newton was annoyed he couldn't figure it out. Edit: maybe Descartes knew the power of forcing the reader to figure out everything for themselves and lied in his book hoping someone would solve it.

After he reads Arithmetica Infinitorum by Wallis which is the arithmetic of infintesimals. Newton recognizes many of the sums are similar to what he read in Oughtred's book, known today as Pascal's triangle and in typical fashion seeks to generalize and invents the binomial theorem. Newton then reads Viete's Opera Mathematica which was another textbook by van Schooten compiling all the work of Viete such as Diophantine equations. 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 and begins self-directed research writing out 22 headlines of 'problems' in his notebook and classifying them into groups regarding integration, analytic geometry and mechanics.

In April 1664 Newton has been a subsizar for three years and has to apply for a new scholarship at Trinity College where he first meets the Lucasian Professor Isaac Barrow who examines him on Euclid and finds he knows nothing about classical geometry. Barrow had also worked on infinitesimals and apparently had invented some of calculus himself but not noticed if you read the book The Geometrical Lectures of Isaac Barrow. The author found in Barrow's notes he definitely had a kind of proto-calculus worked out. Newton attends his lectures on Euclid and then laments that he didn't learn geometry before analysis as geometry is a way of doing math where you can reason without needing to do calculations. An example of this is in his published Cambridge lectures Universal Arithmetic where he shows with geometry how imaginary roots must occur in pairs.

The plague shuts down the school from the summer of 1665 to spring of 1667 and Newton returns home, makes himself an office by building bookshelves for his now large library and spends all his time doing research with his new analytical tools building them into modern calculus. 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. This is basically the end of story for Newton's analysis research, sometime during the pandemic years he is satisfied with his binomial theorem and calculus thus abandons research in math to pursue lens making for optics research. There is a lot of notes about how tedious it was to do calculations before he came up with his analysis.

Newton refused to publish any work and circulated his analysis tools privately. Leibnitz wrote Newton in 1676 asking about his methods for infinite series and if he knew the solution of 'the inverse method of tangents' which we know today as solutions to ordinary differential equations. The response from Newton is lengthy giving a full account on how he had found some of his results then Leibniz replies explaining his differential calculus how he does integrations. Newton gives a description of his calculus in one of the appendices in his book Optics. When quantity A (volume, acceleration, time) is related to quantity B where changing one then changes the other this means A is a function of B. The ratio of their rate of change is today called a derivative, then called a 'differential coefficient' or 'fluxion' by Newton. If you know the derivative and corresponding values of the two quantities then you can use summation to determine the original functions which is often difficult but if we are able to reverse the process of differentiation we can obtain the same result directly. This undo method of derivatives is called integration today and called 'the method of quadrature' by Newton.

Newton wrote up papers for all this research and circulated them privately but refused to publish until 20 years later when he is offered to write the Principia. Wanting to prove infintesimal calculus he tries uncovering the ancient analysis used in classical geometry. Pappus' book 7 contains a commentary about the tools and propositions that Euclid, Apollonius, Eratosthenes and other geometers of the day used that Pappus referred to as the 'treasury of analysis'. These were said to be contained in books by Euclid but those were never recovered. These lost Porisms (corollaries) are speculated by Newton to be projective geometry and he believed the ancients had already figured everything out and this information was lost over time based from his own experience living during the plague, the great fire of London and political upheaval. Of course Newton also solved the problem of Pappus using classical geometry when reading Collection book 7 which was the problem Descartes was working on which led him to invent cartesian analytical geometry in the first place.

This belief is also what led him to alchemy where he assumed their knowledge was encoded in myths written in Roman and Greek literature writing in his notes that every myth was real but that their lives were embellished through story telling. A common alchemist practice for example was to interpret Ovid's Metamorphoses where the god of the forge/metalwork Vulcan catches his wife Venus and Mars locked in an embrace so traps them in a fine metallic net. The alchemists of the Royal Society that Newton belonged to frequently used the names of planets for metals so obtaining an alchemists manuscript by George Starkey he recreated this myth using it as a recipe and ended up with an alloy with a strided net like surface. Newton also decoded Cadmus and the founding of Thebes from Ovid into practical lab instructions.

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. 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 geometric demonstrations 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 other words if he didn't rely on geometry he could not have found most of the critical results of the Principia. He didn't want his laws of the universe being undermined by criticism of his infintesimal analysis used for many of the results so cast the entire book into Euclidean geometry. His limits are very similar to what everyone today is using in modern calculus courses. Newton even used an equivalent method to how we would solve differential equations today which he used to revise the Principia editions calculating moon orbit irregularities something that Euler could not solve 70 years later.

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/elipses via the movement of a compass, or via translation like in Proposition 4 of Book 1 of Euclid's Elements where one triangle form is moved to compare to another. This is where he demonstrated that rotation of rulers were in fact transformation of the plane (linear algebra). Here is how Newton used his rotating ruler to create a power series.

"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

Another example of geometry was the appendices in his book Optics where he shows the shadow of a circle cast by a luminous point on a plane generates all the conics, then gives an equation for the shadow of all curves that generate all cubics. He translates these into infinite series for which he has analytical tools and states the importance of determining if these series are convergent which wasn't formalized until the 1800s with Cauchy and Gauss. In the same appendix is the first known use of indices being given to variables.

• YouTube - Arithmetic and Geometry Math Foundations 2

The successor function s(n) = n + 1 for natural numbers:

• s(0) = 1
• s(s(0)) = 2
• s(s(s(0))) = 3

Let's prove the laws of multiplication using his definitions

Prove n * 1 = n

The definition of multiplication: mn is n + n repeated m times so n1 is 1 + 1 repeated n times so 1₁ + 1₂ + … + 1_n = n

We will need this result for the other proofs.

Prove the distributive law: (k + n)m = km + nm

km is m + m repeated k times or m₁ + … + m_k and (k + n)m means m + m repeated (k + n) times. Distributing m in (k + n)m we get: ((m₁ + m +…m_k) + (m₁ + m +…m_n)) and now the left side matches the right side since km + nm is the same when expanded.

Prove the associative law: (kn)m = k(nm)

(kn)m is m + m repeated kn times and kn is n + n repeated k times:

(n₁ + … + n_k)m or (mn₁ + … + mn_k) using distributive law.

Right hand side: k(nm) is nm + nm repeated k times (nm₁ + … + nm_k) and factor out m, both sides of the equation are the same since we haven't proven the commutative law yet so can't claim nm sequence is the same as mn sequence.

Prove the commutative law: nm = mn

First let's prove n1 = 1n or 1n = n. We proved n1 = n already, so we can use substitution to replace n in 1n with 1(1₁ + .. + 1_n) and using the distributive law this is (1₁ + .. + 1_n) or n. Now we have n1 = 1n.

Looking at the right side: mn is (n₁ + … + n_m)

Factor out n: n(1₁ + .. + 1_m) and a sum of 1's up to m is m, we have nm = mn.

• YouTube - Arithmetic and Geometry Math Foundations 4

Try the subtraction laws

• n - (m + k) = (n - m) - k
  • The left side is n + (-1)(m + k)
  • (-1)(m + k) via distributive law is (-1)m + (-1)k
  • (-1)m is m + m groups at -1 times which makes no sense so we can commute
  • m(-1) now it's m copies of -1
  • n + (-1₁ + -1₂ .. + -1_m) is -1 added up m times or -m
  • n - m + (-1₁ + -1₂ .. + -1_k) is -1 added up k times or -k
    • n - m - k = n - m - k

In Wildberger's natural number lectures he doesn't define subtraction for integers (negative numbers) so my proof is wrong. Can you prove subtraction where n > m or whatever similar because it can be done.

• YouTube - Arithmetic and Geometry Math Foundations 6

A simplified Roman numeral system is introduced

• Uses powers of 10
  • 327 = 3 x 10² + 2 x 10¹ + 7 x 10⁰

• YouTube - Arithmetic and Geometry Math Foundations 7

A new way of hand multiplication is shown, taking advantage of the fact the notation is using the distributive law. Borrowing while doing hand subtraction now makes sense. Wildberger rants how his daughter was taught a ridiculous way in the questionable Australian 'modernized' school system.

• YouTube - Arithmetic and Geometry Math Foundations 8

Wildberger says he didn't find any long division in his kid's curriculum because schools removed it claiming it was too difficult. Another way to do long division:

7)203

How many groups of 7 x 10² fit into 203? 0 so reduce to how many groups of 7 x 10¹ fit into 203? 20

   20  
7)203
 -140
   63  

How many groups of 7 x 10¹ fit into 63? 0 so reduce to how many groups of 7 x 10⁰ fit into 63? 9

   29
7)203 
 -140
   63
  -63

• YouTube - Arithmetic and Geometry Math Foundations 9

Interesting how he defines all these as types.

• YouTube - Arithmetic and Geometry Math Foundations 10

The prime notation used in the proof represents another variable with the same type.

• YouTube - Arithmetic and Geometry Math Foundations 11

More simple proofs.

• YouTube - Arithmetic and Geometry Math Foundations 12

The point in watching these is to see how you can start with just the natural natural numbers and keep going defining more and more complexity with new types of numbers like integers.

• YouTube - Arithmetic and Geometry Math Foundations 13

We learn what a field is. If you're curious in the introduction here he shows how the rational number line has been raised on the y-axis showing how all numbers on a line through the origin (0,0) are equal on the rational strip. It explains why a/1 is a and why a/0 is not defined.

If for whatever reasons you want even more arithmetic try Arithmetic by Herb Gross. This is vocational school style math that was originally taught to adults at community colleges and in prisons in North Carolina. The book is here archived and the YouTube playlist has exercise explanation videos you don't have to watch unless needed but they're very good he teaches everything using bounds and inequalities or relating concepts to English grammar. A free resource for practice is expii it was designed by the US olympiad national coach Poh-Shen Loh. We will do so many problems however in the tripos that you will pick this all up anyway.

The Tripos consists of Part IA, IB and II and is supposed to take 3 years to achieve a BA honors degree. It is a fully immersive degree where all you do is math 6 days a week there is no other electives. Each year has 3 terms that are 80 days long: Michaelmas, Lent and Easter. Each term is 4 courses of 24 lectures each except Easter where computational projects are done and the exams. This is why it's considered a very hard degree because imagine taking 8 courses (192 hours of lectures) and having to know them so well that you can pass a series of 3 hour tests sometimes many months later.

During the term you work through 4 example sheets per course. These are usually due every 6 lectures and attending lectures is optional some just read the notes and work through the recommended books. Each course has around 4-6 recommended books. There's no TAs, no recitations, no midterms, no office hours, the only time you get personalized instruction is during supervision where your example sheet is reviewed and you can ask questions.

Here is the schedule for 2023/2024s. Every year the tripos publishes their curriculum and sometimes professors post lecture notes on their personal sites or by students like here or Neel Nanda's notes. Here is a sample recorded lecture.

We take these all at the same time:

• Numbers and Sets
• Groups
• Vectors and Matrices
• Differential Equations
• Mechanics

• Dynamics and Relativity
  • Prof David Tong's page

TODO

• Variational Principles (examined next year)

Very good lecture notes and this is what Sussman uses for his book SICM which we'll do once we get to Part 1B and take classical mechanics. This material is examined the next year in Part IB.

• Computational Projects
• Exams
  • All are here and they test both Michaelmas and Lent courses.