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I have read through it and can honestly say it should be required reading for all college students. Most students do not start thinking about resumes and the classes they should take to keep them on the track to success until their junior or senior year, when it is almost too late.
Give this book to your freshmen, or buy it, freshmen! It is one of the most useful you will ever own.
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It is rare to see among mass-audience science books illustrations liberally included unless they are there for a "gee whiz" factor. (Case in point: Hawking's Universe in a Nutshell. ) In Mathematics and Optimal Form the illustrations, including graphs, diagrams, and photographs of microorganisms, soap films, and buildings are present on nearly every page and serve to clarify and truly illuminate the text.
The section on minimal surfaces is especially strong. The authors manage to include a basic introduction to the topology of manifolds and give an enlightening history of Plateau's Problem and related work including applications to architecture. (Additionally, this chapter includes some spectacular photographs!)
History is the focus of two early chapters. The approach to the Greek geometers, a few pre-Enlightenment mathematicians, and development of the calculus is in places lacking relevance (and is better treated elsewhere.) There are a few high points of this section, Heron's theorem on the shortest path from a point to a line to another point being one of them; this could have been included in a near-absent section on geodesics, and the brachistochrone (and related) problems--the chronological approach did not work here and ought to have been redone or avoided.
The absence of formal or semi-formal mathematics, or even equations, is frustrating. While the book is written for a lay audience it is written for the mathematically literate. Basic literacy including familiarity with the differential calculus should have been assumed. The authors are capable and clear writers; inclusion of formulae and a few derivations would only have served to enhance the book. At the very least, it is ridiculous to mention things like "differential equation" without supplying one--anyone who knows what a differential equation is must must have some familiarity with them--or at least explaining what it and other mentioned but unused concepts are. The authors do provide notes and references for those seeking more mathematical detail.
Overall, this was a very worthwhile read. Unlike most humans it could benefit from putting on a little weight and an inflammed appendix.
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The final chapter, titled Eigenvalues and Eigenvectors, was a bit too concise, touching too briefly on more complicated topics like diagonalization of matrices, Hamilton-Cayley Theorem, and quadratic forms. I recommend working through the fourth chapter (this is a short book) rather quickly to get a basic understanding, but then moving to more detailed treatments of eigenvalues and related topics elsewhwere. As a follow-up, I recommend another Dover edition - An Introduction to Matrices, Sets, ad Groups for Science Students. However, this text assumes substantial familiarity with manipulation of determinants.
All in all, this book is a good buy and a good starting point for studying matrices.
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Anthony Hecht, one of the very few finest poets of the past 50 years, is also one of the most learned, wide-ranging, perceptive, and engaging critics. Now 80, he has gathered in this new collection, "Melodies Unheard," 18 essays, most of them done in the past five years in response to invitations and assignments of various kinds, from centenary observances to pieces for The New York Review of Books.
As to his subtitle, he remarks in his Introduction, "What, I have asked myself, is the critic trying to do? And there are plenty of answers. But perhaps we might begin with the urge governing Poe's Auguste Dupin: to solve a mystery. Not infrequently this means discovering that there was a mystery to be solved in the first place, because no one had noticed any need for scrutiny."
Though he does not say so explicitly, Mr. Hecht appears also to be concerned with "mystery" in the sense of skill, craft, or art; this older usage appeared often in the indentures of apprentices, bound for a period to learn the mysteries of , say, tailoring. Near the end of his Introduction, he says, "No poet examines someone else's poem, especially a major poem or a large body of poetry, without hoping to learn something from such scrutiny; and, moreover, to learn something he can put to his own personal use."
It is an inspiring and humbling object lesson for any serious reader to behold the thoroughness with which Mr. Hecht opens his powers of perception to the variety of texts he encountered in the course of writing these pieces. He has his preferences, to be sure; he understands that meter and rhyme have been integral to poetry for centuries, and that to dispense with them is to incur serious risks. It must be noted, however, that he is no knee-jerk enemy of free verse; one of the best pieces here is a penetrating and highly favorable consideration of Charles Simic, whose unmetrical surrealism Mr. Hecht praises for its resonance and responsibility.
Collections of separate critical pieces can sometimes seem too miscellaneous, whatever the brilliance or persuasiveness on display in the individual essays. Mr. Hecht has addressed this matter with unusual thoughtfulness and diligence, and the result is a solid book rather than an assemblage of book parts.
First, his Introduction takes up some points not dealt with to his satisfaction in the essays. For example, early in an essay chiefly concerned with the opening of T. S. Eliot's "The Waste Land," Mr. Hecht mentions "the bedeviling topic of Eliot's anti-Semitism, which I must leave for another time." He makes short work of William Empson's attempt to make Eliot out as a decent fellow of his time who had no problem with some of his time's notions, and just slightly longer work of Eliot's own claim that he was not an anti-Semite and never had been.
Anti-Semitism arises again in a discussion of St. Paul's Epistle to the Galatians; Mr. Hecht's treatment of it is notable for its patient use of a remarkable amount of Biblical knowledge, and for its tact with personal reminiscences of direct encounters with wounding prejudice.
Second, the book concludes with an essay called "The Music of Formsþ" which appears to have been written for the collection; it is not mentioned in the acknowledgments of editors and lecture venues. It is here that Mr. Hecht is most clearly distressed at what seems to him the shrinking audience for such delicacies of technique as he explores in the rest of the book. For years, he tells us, it was his habit, following some exposition of metrical terminology and example, to ask his undergraduate students to locate for him the place where the dialogue of Romeo and Juliet shifts from prose into verse. It didn't last:
"After a certain number of years I gave up asking my classes this question, which obviously embarrassed them and discouraged me, for it became transparently clear that the overwhelming majority of my students were quite simply deaf to almost all metrical considerations and that my introductory lecture on the topic was purposeless and wasteful. And I reluctantly concluded that there are many who are not so much mystified by meter as completely oblivious to it."
One who has repeatedly urged students to observe line-ends when quoting poetry in critical papers can but agree with this assessment. However, it seems always to have been the case that those to whom matters of precision and beauty of language are matters of great importance - of life and death, as we sayþ - are greatly outnumbered by those to whom such matters are of no importance. Mr. Hecht strikes a firm balance between taking pessimistic notice of this situation, and considering that Milton's "fit audience though few" is entirely deserving of the best effort he can muster.
Thus Mr. Hecht proceeds, with grace, urbanity, good humor, and vast erudition, to consider certain literary works from the past eight centuries, and to shed light on their techniques and on a few instances of obscurity.
Among the poets treated are Shakespeare, Philip Sidney, A.E. Housman, Gerard Manley Hopkins, Robert Frost, W.H. Auden, Elizabeth Bishop, Richard Wilbur, and Yehuda Amichai. There are also essays on aspects of "Moby-Dick," and on the prose of Seamus Heaney. The approaches range from close explication of text to sweeping historical surveys. Throughout, the style and manner are those of a deeply knowledgeable and polished conversationalist, grateful to be in the presence of the works he understands so well. Care for poetry and its traditions has seldom been so memorably exemplified.