The Four Pillars of Geometry (Undergraduate Texts in Mathematics) 2005th Edition by John Stillwell (PDF)

Description

This book is unique in that it looks at geometry from 4 different viewpoints – Euclid-style axioms, linear algebra, projective geometry, and groups and their invariantsApproach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraicAbundantly supplemented with figures and exercises

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Around 1900, David Hilbert published his “Foundation of Geometry.” It was the first book to really make Euclid’s Elements obsolete. Humans had developed mathematics for thousands of years – from bone markings to the Summerian/Babylonian algebra. They certainly left some record of their mathematical activities. But, for one, their mathematics was arbitrary; it kind of worked. Only with the Greek discovery of deductive reasoning(as far as everybody can tell) did mathematics become firmly established. It’s a subtle point. But, a major point is that with Hippocrates of Chios(not the guy from cos who did the Hippocratic oath) discovery of axiomatics which was an advance beyond Thales and Pythagoras’s deductive reasoning discovery(not to be systematized till hundreds of years later with Aristotle; Aristotle’s only real lasting contribution), and eventually Euclid’s effort at axiomatisation, all that knowledge before and Greek mathematical knowledge was systematized, put in a bottle figuratively speaking, and sent out to the world(most people back then never heard of it or saw a copy). Euclid’s “Elements”, the works of Archimedes, and Appolloniuses Conics were the only real systematic deductive mathematics all the way up to David Hilbert’s work.That last part isn’t strictly true. There was some beginning efforts to axiomatise algebra in the 1800s. There’s also Ptolemy’s work; i’m really not sure how much of an axiomatic effort that book is. The point is that David Hilbert’s “Foundations of Geometry” was the first axiomatic effort of geometry to make Euclid’s Elements more or less obsolete(see Thomas Heath’s translation of Euclid’s Elements; in it, Thomas Heath shows the mathematical, and historical significance of just about every theorem in there.But, this book is more of a easier version. John Stillwell tries to make an easier approach by relating advanced with ancient mathematics. The truth is that advanced mathematics solves problems the ancient or previous mathematics couldn’t solve or did so in a less elegant way. Still, John finds modern fresh versions of those ancient theorems like Pythagoras and Thales. As with new deductive theories that are suppose to be able to derive the old theories and deduce new theorems the old couldn’t, one certainly should try to make connections between old and new. One must realize that John Stillwell can only fit so much in one book(even with his putting extra stuff sometimes more advanced stuff as exercises). As far as I can tell, nowhere does John Stillwell feel the need to show Archimedes theorems about pie, and all the great Greek mathematics involved and relations to the new modern mathematics. This is just one example. See Thomas Heath “History of Greek Mathematics” not to mention his translation of Euclid’s Elements, and Van Der Waerden’s “Science Awakening for a proper technical history of Mathematics(certainly more about ancient mathematics).I’d like to note that John Stillwell does some things in Geometry that I never got into in two geometry courses(one in High school, and then when I went to college after the Navy, they made me take it again). In those two geometry courses, I never proved the Pythagorean theorem once, much less delt with transformations. Transformations overcome a famous logical bottleneck of Euclid’s Elements. So yes, John Stillwell’s book is like the geometry class you never got to have.I wouldn’t worry about figuring everything out. I’d worry if you can’t figure your way through the main text though.A big point is that John Stillwell tries to show some great connections between ancient mathematics and modern. He tries to show what little one can understand of advanced mathematics. The stress is on geometry of course, but if you read his other books which do similar things, you’ll see that his point is that one shouldn’t disregard geometry. So, in some ways this book is a good place to start. He has other geometry books, but I wouldn’t get into them till you get through the number theory/abstract algebra, and at least one semester of calculus. Overall, John Stillwell has succeeded in showing people that they too can learn mathematics. I would read the majority of these books before reading his “Mathematics and its History” as well. He’s able to show some more stuff there, but his accounts of abstract algebra and number theory are even sketchier there.

⭐This book is written in the typical fashion of many mathematics textbooks: very short and concise and dense. It is understandable and written well-enough if one devotes enough time and energy to it, but at the same time, it could benefit from more details. Finally, it whizzes through each of the main topic areas so quickly that it is hard to get a good understanding of a specific area of geometry.

⭐Professor Stillwell was my History of Mathematics Professor at the University of San Francisco. He studied at the Massachusetts Institute of Technology with the PhD in Mathematics. He is an engaging lecturer and has written many textbooks on mathematics, especially important in the history of mathematics. His writing is clear prose and well revised and published. This textbook on the four pillars of geometry provides a gentle introduction to the geometries studied by professional mathematicians, without delving too deeply into advanced topics. He provides exercises for interested readers. This book is well organized and counts a significant contribution to the literature of mathematics in the modern century.

⭐The book is okay. Right off the bat the author makes a major error, claiming that Euclid requires us to “carry” a measurement by compass from one line segment to another in the third theorem of Book One. Good grief. The whole point of The Elements is to prove things without doing that! There is an order to the theorems, and in the immediate my previous one Euclid instructs the reader how to reproduce a line segment without using the compass as a measuring device. It’s pretty odd that Stillwell did not grasp this.

⭐The book is extremely interesting, but there is a major problem with the book binding. Initial copy received missing text on dozens of pages scattered throughout the book–i.e. blank pages. Returned first copy asking for a replacement, second copy promptly shipped, but received with the same problem. Second book returned asking for full refund.

⭐Usually I find math books, especially geometry, to be nothing short of terrible. I’ve purchased this for a class and it’s actually very well written. Includes diagrams and a good set of problems, though I’m only a few chapters in at the time of writing. It seems clear enough without spending pages and pages describing one concept. Except to reread paragraphs if you’re not well versed in geometry already. It’s a fairly small book, easy to toss in a bag and take with you.

⭐As a textbook in general, it’s ok at best. It’s far shorter than it needs to be, lacking explanation where it would be helpful/needed. As a result it can be frustrating and hard to understand. Sometimes even basic asumptions (that are key to problems/explanations but not given) are left out.On Kindle in particular, this book is unusable. It is ridden with basic typos that make an already hard to understand and needlessly concise book even harder to read. Is this problem actually hard, or is it just full of fundamental typos that make it actually incorrect/impossible? No way to know without having the physical book as well.It has been a frustrating semester using this book on Kindle.

⭐I found this text to be of excellent quality. Dr. Stillwell’s writing is easy to read, yet very informative. He creates many examples that helps in uderstaning his main thesis. I found his reason why ‘The Slope is consistent throughout a line’ by using similar traingles quite interesting. I highly recommend this book for all who enjoy mathematics, regardless of your degree or interest. I am anxious to read and study more of his books.

⭐Perfect quality, quick shipping.

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