General request for a book on mathematical history, for a VERY advanced reader. I am aware that there are answered similar questions on here, however I am specifically after a text that would be engaging for a professor of mathematics, also Fellow of the Royal Society (FRS).
He is unwell and in the hospital, and I would like to get him something to pass the time. However anything aimed at undergraduate / postgraduate level is going to be far too patronising. Honestly, I'm not sure if there exists such a book, but if anyone has any recommendations, I would be extremely grateful. 
 A: not sure why you emphasize history... There are now translations of Fricke and Klein into English. Favorite of mine. The preface is pretty funny. 
https://bookstore.ams.org/ctm-3/
Your ailing friend can try to find where the theorems are. A  major one I have used with considerable enjoyment is on pages 507-508 of the German original, page 412 in the translation. I guess we know it is a theorem or lemma or the like as it is in italics. There is also one in the middle of the page where they admit everything, "obtain the following theorem:" 
I bought a cheap reprint of the 1897 German original. Cheap because it was scanned online by a major library, but I wanted a paper version  

A: A classic is the four-volume set The World of Mathematics, edited by James R. Newman (GoodReads, Amazon). The original 1956 hardcover edition has 2500 super-thin pages, but there also exists a more recent, and more affordable, paperback Dover edition. The collection contains essays by mathematicians including Von Neumann, Russell, Descartes, Galileo... There's bound to be material in there of interest even to an FRS.
A: Not a book on history, but if he hasn't read it already, Gödel, Escher, Bach is a good option. 
A: I suggest


*

*Mathematical Thought from Ancient to Modern Times, Vol. 1&2, by Morris Kline

*Mathematics and Its History, by John Stillwell

A: Another really good read is Constance Ried's book Hilbert, which (as the title suggests) covers the history of one mathematician in depth, but also touches upon many other mathematically "significant figures" as it tells its tale.
A: Stanislaw Ulam's autobiography Adventures of a Mathematician is a very uplifting personal history.
A: Probably, any comprehensive book about history of mathematics is intended for undergraduates (almost all subjects are presented, but none of them is deeply discussed). So, here are two suggestions that tell very specific stories (about fairly sophisticated results):


*

*A Mathematician Grappling with His Century by Laurent Schwartz. (Translated from the French Un mathématicien aux prises avec le siècle.)

*Birth of a Theorem: A Mathematical Adventure. (Translated from the French Théorème vivant.)
Both authors are fields medalists and, in these books, they talk about their important discoveries.
A: An alternative to a history may be a serious introduction to an advanced field of mathematics that FRS has never touched. Mathematics is so broad now that no one can cover it all. A stay in a hospital is a chance not to recapitulate the old but to explore something genuinely new. That spirit of discovery can be more healing, and who knows what serendipity is possible.
A: For an engaging read, you could try Cauchy, infinitesimals, and ghosts of departed quantifiers.
A: Here I'd like to propose four books which might at least partly met also more sophisticated demands. The first two cover the Hilbert problems, one is about the history of a specific mathematical subject and the last one is about one of the great masters.

  
*
  
*The Hilbert Challenge by Jeremy J. Gray - See this AMS review
  
*The Honors Class by Ben H. Yandell - See this AMS review
  
*History of Algebraic Geometry by Jean Dieudonne
See this MO post which provides a link to an online pdf version and the reference list (p. 2) of this presentation
  
*The Legacy of Leonhard Euler - A Tricentennial Tribute by Lokenath Debnath - See this AMS review

A: I suppose that a very good choice is A History of Mathematics, by Victor J. Katz.
A: I have not read it myself, but I have heard excellent things about The Princeton Companion to Mathematics.  It is not specifically a history book, but apparently has a decent amount of history in it, with many pages devoted to mini-biographies of mathematicians.  It is written by, and primarily for, mathematicians.
A: In the comments, Kevin Long suggested

Hopefully this isn't getting too off topic, but I've heard that when Stan Ulam was in the hospital for encephalitis, Paul Erdos went to meet him when he was discharged and spent a few weeks at his house, plying him with math questions and playing chess with him. At the time, Ulam was afraid that the incident would have affected his mathematical ability, but Erdos helped him build his confidence back up. So if the professor in question is feeling suboptimal, some (small) math problems might be good.

Personally, when I had to stay for longer periods at the hospital, the most difficult part for me was to overcome the boredom. There aren't that many books I would enjoy reading for 8 hours a day, 7 days a week... from that perspective, math problems might make sense, because mathematicians like to spend sheer endless amounts of time on problems that they find engaging. In that spirit, I'd like to suggest The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás. It contains many interesting problems, all with elegant solutions, some due to famous mathematicians such as Erdos. 
A: Another engaging read:

Bair, J.; Błaszczyk, P.; Heinig, P.; Katz, M.; Schäfermeyer, J.; Sherry, D. "Klein vs Mehrtens: restoring the reputation of a great modern." Mat. Stud. 48 (2017), no. 2. See arxiv.

The abstract reads:

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). 
  We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. 
  Klein's Goettingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. 
  Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. 
Mehrtens' claim that [the future SS-Brigadefuehrer] "Theodor Vahlen ... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position. 

A: Since you haven't mentioned the professor's field of study, though I am sure you must know their specialized interests,  I'd suggest you make sure they have access to the the current (top) journal(s) in their favorite field of study, which will help them stay abreast in the field.  
If the professor's field of study is history in mathematics, or they simply stay finely attuned to the subject, e.g., may I suggest keeping the professor supplied with recent copies of the journal Historia Mathematica?
Also look into the most recent issue of BSHM Bulletin: Journal of the British Society of the History of Mathematics.
A: It is on one (two?!) specific topic, but still... maybe Jean Dieudonné's "A History of Algebraic and Differential Topology, 1900 - 1960" could be an interesting choice that fits OP needs (it is actually rather advanced).
A: History is also in the great original works, or in a book from the personal library of someone famous - ideally with margin side-notes in the latter case. Or all of these at once. Sometimes such books are obtainable today, for a reasonable amount of money. Not too long ago I saw a signed copy of a math book from the library of Fields Medalist Lars Ahlfors come up in a search engine. Since buying a book that would appeal to the professional interest of such a person as you describe is inherently difficult, you might do better in seeking something to be surely treasured.
A: Littlewoods’s Miscellany


*

*It is a classic.

*Littlewood does not write for the general public.

*There are wonderful anecdotes about the british academic life in the 1st half of the 20th Century.

*There is plenty of hard-core mathematical content.

A: The World of Mathematics by James R. Newman, 1956, 4 volumes. This is a collection of 133 essays written by the pantheon of mathematical thinkers: Descartes, Archimedes, Newton, Euler, Galileo, Bernoulli, Malthus, Laplace, Poincare, Mach, Einstein, Boole, Turing and dozens more.
Chapters average about 20 pages and need not be read in order, so the book is ideal to pick up for a brief diversion and then put aside for later. 
For more detail, see the review by David E.H. Jones in Nature, 337 (February 2 1989), p. 420. (Link here.)
A: I took History of Mathematics as an optional subject in my final year. There is one book really. "A brief history of mathematics" by Howard Eves. Super book, both for ordinary readers and academia. However, I do not know if it is still in print :D as I graduated in 1997. cheers!!
A: Carl Boyer's A History of Mathematics, might be an enjoyable read.  I've read the 1991 revised edition, ed. Uta C. Metzbach (New York: John Wiley, 1991).  It's about 700 pages.  The first 260 pages are dedicated to mathematics before the Renaissance, starting with ancient Egypt and dwelling at length on the ancient Greeks.  The author narrates the history of pre-modern mathematics in an engaging style.  For a specialist in the field, the history might be interesting.
A: Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World
The mathematics in this book is pretty light, but nevertheless, the book presents an interesting case study in the ways that non-mathematical concerns affect the history of mathematics.
A: Jeremy Gray's "Plato's Ghost: The Modernist Transformation of Mathematics" tells the story of how mathematics became modern during several decades before and after 1900. Worth reading to understand how mathematics got to be so far from common intuition.
Amir Alexander's "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World" is a cultural fresco of the 17th century, the one when Europe became modern, the one that ends with the birth of real analysis as conceiver by Newton and Leibniz.
Not mathematics, but very close: Arthur Miller's (not that one!) "Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)". A deeply intellectual book, not content with the just the physical theory but also presenting the general cultural context in pre-war Vienna.
A: Number Theory: An approach through history from Hammurapi to Legendre by Andre Weil is a classic. In it Weil covers some thirty-six centuries of arithmetical progress, with a close account of the founding fathers of modern number theory: Fermat, Euler, Lagrange and Legendre.
The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
edited by Catherine Goldstein, Norbert Schappacher, Joachim Schwermer. In this book eighteen authors - mathematicians, historians, philosophers - discuss the impact C.F. Gauss's Disquisitiones Arithmeticae (1801) has had on mathematics.
From Kant to Hilbert: A Source Book in the Foundations of Mathematics Volume I and Volume II
by William Bragg Ewald. An historical overview starting from Kant's Critique of Pure Reason, widely taken to be the starting point of the modern mathematics, up to the end of the nineteenth century with Hilbert. Ewald's two-volumes contain translations of works by Bolzano, Cantor, Dedekind, Gauss, Hamilton, Kronecker, Riemann, Poincare, and Zermelo  showing the links between algebra, geometry, number theory, analysis, logic, and set theory.
Geometry by Its History by Alexander Ostermann and Gerhard Wanner. This is an undergraduate book, but that's not the point: it's one of the most beautiful historical books on geometry out there.
The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore by Karen Hunger Parshall and David E. Rowe. In this book we see what the title suggests: Sylvester at John Hopkins, Moore at the University of Chicago, with Klein popping over to tour the mathematical scene.
The Mathematics of Plato's Academy: A New Reconstruction by David H. Fowler. In this book Fowler examines what we really know of the mathematics done in Plato's Academy, and before Euclid: not much! A brilliant book written by a classicist for fans of Euclid.
Get well soon.
A: Perhaps an older book if you can find it.
(The chances are higher he hasnt read it...) 
So I recommend E T Bell The Development of Mathematics.
A: Two popular science books by 
Simon Singh:
Fermat's Last Theorem
and 
The Code Book
about cryptography and its history.
A: Men of Math E.T Bell is a great book 
A: I would suggest Mathematics Emerging by Jacqueline Stedall. While not a complete history, it does cover a rather important period from the late 16th century to the 20th century.
I believe that he may like Godël, Escher, Bach by Douglas Hofstadter, a brilliant read not so much about mathematical history as it is about discussions pertaining to knowledge and such metaphysical aspects of mathematics, and full of the most beautiful, sophisticated and interesting puzzles.
Hope you find a good book.
All the best!
A: Hermann Weyl wrote a 43 page essay on the life and work of David Hilbert that I think is quite nice.
