While it is quite true Dirk Struik's work is on classical differential geometry, the older methods and treatment do not necesarily imply obsolescence or mediocrity as some readers or reviewers suggest in their evaluations. Classical Analysis is still an important branch of Mathematical Analysis. So classical approaches and topics should not be dismissed as a waste of time, useless, outdated or even invalid. Remember Andrew Wiles' recent attack on Fermat's Last Theorem and his ultimate proof of its validity, an event that made headline news. That is a quintessential classical problem in mathematics (i.e., in number theory), only recently resolved. So remember: CLASSICAL Differential Geometry is part of the title.
First of all, this book is very readable, being that it requires no more than 2 years of calculus (with analytic geometry and vector analysis) and linear algebra as prerequisites. Exposure to elementary ordinary and partial differential equations and calculus of variations are highly desirable, but not absolutely necessary. There are numerous carefully drawn diagrams of geometric figures incorporated throughout the book for illustration and, of course, better understanding. Topological methods are not used in the book, and the concept of manifolds not mentioned, much less treated. So this is an older work that bridges the very foundational and applied aspects of differential geometry with vector analysis, a field and body of knowledge widely used nowadays in the sciences and engineering and exploited in applications such as geodesy. For those insisting on modern approaches and want to omit studying foundations and historical development, please read up on other books such as O'Neill and Spivak. These are essential to approaching the subject of differential geometry from a more modern and global perspective with heavy emphasis on rigor in proofs and derivations, mathematically speaking. (Also, there are tons of other newer works, i.e., on "modern differential geometry", I am unfamiliar with. They are probably available for browsing in college bookstores.)
The author begins by leading the reader from analytic geometry in 3-dimensions into theory of surfaces, done the old fashion or classical way, i.e., utilizing vector calculus and not much more. Along the way, he takes the reader through subjects such as Euler's theorem, Dupin's indicatrix and various methods for surfaces. Then he continues with developing important fundamental equations underlying surfaces, e.g., Gauss-Weingarten equations, looks at Gauss and Codazzi equations, and proceeds to geodesics and variational methods. He includes a somewhat detailed treatment of the Gauss-Bonnet theorem as he progresses. He ends up with introducing concepts in conformal mapping, which plays an important role in differential geometry, minimal surfaces and various applications, one of which is geodesic mapping useful in geodesy, surveys and map-making. He does all of it with clarity and focus, including problems or "exercises" as he calls it, in under 240 pages - brevity that is rare in many mathematical books and works these days.
For those with a mind for or bent on applications, e.g., applied physics (geophysics), applied mathematics, astronomy, geodesy and aerospace engineering, this book would be an excellent introduction to differential geometry and the classical theories of surfaces - being that one need not worry about abstract analysis and topological aspects of mathematics. Perhaps the title should be "Topics in Classical Differential Geometry" or "Introduction to the Theory of Surfaces in Classical Differential Geometry". But one must keep in mind that Dirk Struik is an old MIT hand and contemporary of Norbert Wiener, also at MIT, and Richard Courant (and many great German-educated mathematicians) who lived and worked in the early to mid-20th century, a long time ago and before computers became commonplace, an era in which total abstraction in mathematics and physics was not quite widely emphasized, but clear concrete thinking was important. A good friend of mine and co-worker who studied at the University of California, Berkeley, told me he had great respect for the classical geometers such as Struik and Eisenhart, understanding that they built ideas from a scratch and wrote in such a way that readers can discern the physical origins of geometry, in particular of differential geometry, a subject that supposedly started with Gauss during the early or mid-19th century when he performed survey work for his government in Germany. (The term "torsion" introduced and sed by Struik in the first few chapters of the book comes from classical mechanics, and is commonly employed in mechanical structures/structural engineering nowadays.)
I for one am an aerospace engineer. There were one or more occasions where I consulted the book for formulas and expressions of curved surfaces and spheroids in my work of flight navigation (flying over the ellipsoidal Earth, as one example). I am sure that are other areas, e.g., space engineering and relativity, where classical methods of differential geometry embodied in Struik's book can come in handy.
The only problem I have with the book is that the "exercises" do not come with solutions, but I do not think that is a major drawback unless one uses it as a textbook for a course that requires assignments and drill exercises.
Judge for yourself by borrowing this book to read, i.e., if you are interested, can tell whether you like or dislike it on the first pass, and for what reasons one way or another. Find out for yourself.