Customer Reviews

Mathematics for the Nonmathematician (Dover books explaining science)

5 star:		(20)
4 star:		(9)
3 star:		(3)
2 star:		(0)
1 star:		(0)

 

 

This book was originally written as a textbook (for a math-for-the-non-mathematician type course). It can be used as one (though as a textbook it's a bit dated), read cover-to-cover for edification and pleasure (the style is a bit more instructional than the average popular math book), or dipped into here and there for the topics the reader personally finds interesting. With well over 500 pages of fairly small print, there's a lot here, covering a wide variety of topics, with (it seems to me) particular emphasis on history, geometry (of various kinds), and applications of math to physics. If you leaf through the book, you'll find some pages of nothing but text, some pages full of geometrical diagrams, some of equations and formulas, and even a few Renaissance paintings (in the discussion on mathematical perspective). With so much here, readers will probably find some parts more interesting than others--though which parts are the interesting ones may be a matter of personal opinion.

I am not a mathematician. I just happen to use math everyday in my trade. Just the usual stuff you use in finance. But I am interested in math, maybe just for fun (??!!). And I found this book to be very well written, very interesting and also it has a lot of history so you get to learn a lot and maybe in doing so, you also get to understand more clearly things that just pass you by when you were at school but you didn't really grasp them. This is math pageturner..I'm not kidding !!!

A must have for the mathmatically curious. The subject seaquence is laid out in a logical order. Beginning with the premises of inductive vs. deductive reasoning, basic algebra, geometry, and the Calculus. This is not a good book for becoming proficient in sepcific areas of mathematics, but offered for me at least, a logical reference point for approaching the core sujects. I highly recommend this book for self-study.

This book is truly an achievement. While not intended for true practitioners, the book is entertaining while though provoking at the same time. I take it with me to my favorite coffee shop sometimes just to open it randomly and read a few pages at a time. Not only does the author weave great historical moments with the progression of mathematical thought, he covers areas such as physics, art, music, and astronomy. He has also renewed my interest in taking the subject up again after many years. I have enrolled in a course in the Calculus based on this as well as other great math books.

Kline, a noted historian and educator of mathematics, wrote a book that stands the test of time. This isn't of much use to anyone with high-school math who doesn't care to know why math is the way it is. For everyone else, this is a good book. Solutions to problems at the end of the book are very handy. I recommend this book along with Timothy Gowers's "Mathematics: A Very Short Introduction".

This is the classic text (available in Dover hardbound and paperback editions) for a college course on mathematics for those who are not science majors and probably hated math in high school. Making math interesting for such a group is a tall order, but one that Professor Kline fills admirably. This book is well written, quite informative and a great choice for the target audience and for many who are much more well versed in mathematics. The book starts with numbers and goes through Euclidian geometry, algebra, calculus, statistics, probability theory and some topics in non-Euclidian geometry and advanced algebra. All this is all done from the prospective of the historical development of mathematics and why it is useful to everyone. This is all done in somewhat simplified manner and in a non-condescending tone. I found that this approach enriched my understanding of many mathematical concepts. For instance, I learned the general solution for a quadratic equation as just a formula to be memorized. Professor Kline derives it. He does this by first solving quadratic equations by factoring them. He then shows how an equation that cannot be factored can be solved by a transformation of variables and that the general solution to the quadratic equation is nothing more than a generalized form of this latter approach. In school I learned about the ellipse and parabola from the standpoint of Cartesian geometry. This book first introduces these curves from the standpoint of how they would be drawn with string and a straight edge. This approach eliminates algebraic notation and I think more clearly shows the nature of these curves. There are many other gems in this book. For instance, there is a chapter on oscillatory motion that describes the motion of a spring-weighted bob, leading to the development of mechanical watches and clocks.

While I thought that I had a reasonably good mathematical education, I found that I was largely ignorant of topics such as non-Euclidian geometry and advanced algebra (for instance symbolic logic and set theory). While this book provides only a fragmentary introduction to these topics, it was enough to let me get a glimpse of the boarder world of mathematics. The breadth of topics that are covered should not put off potential readers. The aim is to survey mathematics, rather than provide an in-depth presentation of any of the topics that are covered. As such, it is not a substitute for a good geometry, algebra or any other text on a specific mathematical subject. For the most part, nothing beyond high school mathematics is required and those more advanced topics such as calculus are covered in chapters that were prepared so that they can be skipped without destroying the subsequent continuity of the book. The presentation is clear and focuses on the readers understanding, rather than on being as rigorous as possible. Much of the treatment is based on simple geometrical arguments and straightforward high school algebra. The book concentrates more on how the mathematics was developed than on calculating for its own sake, but there are many numerical problems, whose solutions are provided in the 1967 Dover edition, which contains the Instructors Manual. The previous edition, published under the title of Mathematics for Liberal Arts, contains only the solutions to selected problems.

I purchased this book because it had good discussions of many of the classical problems that had concerned man for millennia. For instance, what is the diameter of the earth, how far away from the earth are the moon and sun, how does one measure distance on the spherical earth? Most books discussing Newton mention that he developed this theory of gravity in part based upon knowledge of the size of the moon and its distance from the earth. In some books this is just stated as a fact, in others it is stated as something worked out by Greek mathematicians more than 1500 years previously. Professor Kline goes through these calculations, showing that they are very simple and straightforward. The book also goes into some topics that are not even touched on in freshmen physics courses. For instance, there is an excellent discussion of how the weight of an object varies with latitude. This book correctly accounts for the rotation of the earth and how the force that this produces varies with latitude (an approach developed by Christian Huygens) and shows that this is much more important than the effect due non spherical nature of the earth (i.e., the earth being slightly flattened at the poles means that the distance to the center of the earth is smaller at the poles, thereby slightly increasing the force of gravity there.)

The book includes some special topics of interest to liberal arts majors. There is a nice description of the development of prospective geometry and the influence of renaissance painters on the development of mathematics. There is also a discussion of sound waves and the mathematics of music. There is also a good discussion of the influence of classical Greek philosophy and how this shaped the development of mathematics and how modern mathematics has overturned the idea that there is such a thing as absolute truth in mathematics (i.e., there are many different geometries and algebras beyond that envisioned by the Greek and renaissance mathematicians).

The historical development of mathematics is included in the discussion of individual mathematical topics and in a few special chapters devoted entirely to historical development. I found that these latter chapters were the weakest part of the book. They present an entirely Euro-centric approach, largely leaving out the contributions of the Babylonians, Indians and Chinese. I recommend Derbyshire's "Unknown Quantity" for a more modern and more balanced approach. This is a small quibble that I do not feel measurably detracts from the very high quality of the rest of the book.

All in all, this is a great book for those who are interested in how mathematics was developed and for those who want a general survey of mathematics, be it for a course, for a review, or for a deeper understanding of the subject. I found that it greatly enriched my understanding of math fundamentals and the thought process leading to the development of mathematical concepts.

I think the author not only knows mathematics, but also knows a lot about other fields, like physics, music, and painting. He is an expert in the history of mathematics and explains well how each important mathematics concept was developed over time. However, I would like to stress that Kline knows how to teach. The structure and the helpful hits in the books are valuble resources for any instructor who wishes to teach a course using this book as the main textbook.

A spectrum exists in the books written by or about mathematicians and mathematics, and this spectrum is heavily weighted at the extremes. At one end, we have books purely verbal, descriptive and, more often than not, hagiographic in their descriptions of great mathematicians and their discoveries. - I suppose one could denominate these as mathematics groupie books. At the other end of the spectrum, we have straightforward mathematics textbooks, rather dry and boring, which roll off the presses with the regularity of the academic terms. To split the difference, to write a literate, narrative book on mathematics, is to attempt something akin to the labours of Hercules and, in the end, impossible to do perfectly, rather like trisecting an angle. Kline, however, does a bang-up job here, and approaches the limit - so to speak - of what is possible. Good show, Morris! But, as many other reviewers have made clear, this book should have printed across its cover the words above the entrance to Plato's Academy: "Let no man ignorant of mathematics enter here." It is a bit of a catch-22 for someone who wants to learn mathematics but is put off by textbooks; you really have to already possess a fluency in the mathematical tongue, to possess a flair for the subject, in order to appreciate the sweep of this really quite grand book. Some pages of the book are simply equation followed by equation leading to a satisfying simplification or representing a derivation. If you can't follow these steps - sometimes quite involved - then you simply won't arrive at the "Aha!" moments which are the chief delight of this book. To quote Kline, "In his wisdom, Thales perceived what we shall perceive as we follow the story of mathematics, that the obvious is far more suspect than the abstruse." The abstruse insights in which this book abounds can only be reached by a competent grasp of mathematical methodology.

That being said, Kline is a whimsical and almost lyrical writer - a bit of a shocker for a book on mathematics! He has a wry wit which he gives full rein to herein. In describing the Greeks and their obsession with astronomy, he observes, "The problem of finding the design of planetary motions continued to engage the minds of the Greeks, possibly because they were not distracted by the `heavenly' stars of stage, screen, and radio with whom many modern minds seem preoccupied." His prose is rich in such quips. The book is actually a lark to read at points.

But the heart of the book is unvarnished mathematics - with which I have a slight problem. He covers the derivation of trigonometric functions splendidly, as well as their uses, as well as the kinetics, or mathematics of motion. But his account of the calculus is rather sparse for my taste. Any treatment of derivatives and integrals without mention, account or derivation of the fundamental theorem is, to my mind, lacking. Further, his derivation of the quadratic formula is absurdly complex and takes pages. It is true, as one reviewer noted, that most students are only presented with the equation and not given its derivation. But the derivation is - comparatively - simple. I worked it out on my own when I was in school, and it need only cover one page, at maximum.

The book is really quite an achievement. But I'm only giving it four stars because it is extremely uneven: No fundamental theorem of the calculus but reams about projectile motion etc. As a bonus, I'll tack on as a comment, once the review is posted, Kline's mischievous proof that 2=1 for the prospective reader to see if he can suss out the flaw in it. If the mathematics leaves you stymied this is not the book for you. On the other hand, if you follow the mathematics quite easily, but are left baffled and staring at the page, determined to suss out the mathematical solecism embedded in it, then you poseess just the sort of questing mind that will delight in this tome!

As a person with a background in Math and Physics, I found this Tome (over 600 pages), to be a great review of the basics of Mathematics, including the derivative and its application to motion problems in physics. There is also a lot of relevant and interesting history peppered throughout the book that is covered in greater detail than most other math history books. I cant imagine someone who doesnt like math, or does not have a math background reading this work. This would be a great textbook in one of the classes teachers take who are embarking on teaching mathematics in high school or up to the sophomore year in college. A "Math Appreciation" course? Other writers have done a much better job of making math seem digestable, but few have the depth of presentation, and sheer information contained in this volume. If you like math, this is a very enjoyable and informative read;otherwise I think you will put it down after thumbing through it.

From Kenneth Ellman, ke@kennethellman.com


Morris Kline the prolific author of "Mathematics for the Nonmathematician" (previously known as "Mathematics for the Liberal Arts" ) was one of those rare teachers who not only knew their subject throughly and intimately but also loved the manner and way mathematics integrated with all aspects of human life and activity. I have owned his books and given away copies, for many years. This is an "old" book having been published in 1967, so you would think it is far better known that it actually is. Kline bridged the gap for so many people who felt or feel somewhat estranged from the language of math. His books are wonderful, warm and comfortable as they take the "Nonmathematician" into the world of symbols and numbers and equations and they kind of hold your hand during this excursion into the world of the language and culture of math. You may find yourself learning this math without even realizing it as the ideas are so intertwined with the culture and history of man.

The current Dover edition contains the Instructors Manuel with answer keys; a very valuable resource. This particular book successfully integrates human history and scientific accomplishments with mathematics. A thread runs through the Kline book keeping the connection of the basic human search for knowledge and adaptation with the development of mathematics as a language and expression. As a survey course and introduction it leaves little to be desired. Any work will be subject to criticism, but if you measure the book by the standard of what the student will have learned after completion of a course with this as the textbook, you will be very, very, pleased. Particularly the extensive "Recommended Reading".

Unfortunately this text is not normally used in high school nor college and so many students who could truly benefit will not get exposure to Professor Kline. Perhaps some of the instructors find it difficult to teach as this book does, but then that is the fault of the instructor not the book. It is my opinion that books such as this greatly aid the interest in math and will keep many students of any age attached to the use of math as a language and a part of the culture in which they live.

Inevitably death came to Kline in 1992 so the time has long passed to take his classes at New York University where he was a Professor of Mathematics. However he is well known and left an extensive legacy of articles and books to allow this deceased Professor to continue teaching. If only the Teaching Company has made a video with him!

As reported in the New York Times Obituary for Morris Kline he wrote a 1986 editorial in Focus, a Journal of the Mathematical Association of America and is quoted: "On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things which are of no concern at all to man".

He is also quoted as stating: "the greatest contribution mathematics has made and should continue to make was to help man understand the world about him."
This Kline book can teach math, science, history, and civilization from this unusual prospective. It would be quite refreshing for students to be able to grab onto this type of portrayal and learn about history with numbers while learning how numbers were used, needed and applied in history.
The text consists of 24 chapters with everything from the Classical Greek Period, Arithmetics, Logic, Algebra, Geometry, Gravitation, Calculus and etc.
This is a book for continual enjoyment and review. If you children learn what is in this book you will be happy.
Kenneth Ellman, ke@kennethellman.com