# Is Plane Trigonometry by S. L. Loney still good as a textbook today?

I am considering using S. L. Loney's Plane Trigonometry as the textbook for my course in trigonometry and would like to ask for opinions about the book. This books is very odd and there might be pros and cons because of its oldness.

Possible Pros:

• It is a classical textbook and stands the test of time.
• The copyright has expired yet the book is still available. Students can obtain the book at a very cheap price.
• No more newer editions. So I do not need to adjust my syllabus every few years.

Possible Cons:

• Some notation may be obsolete, such as the old notation for factorials:

• The author probably assumed no calculators (or computation software), which is now widely used in this course.

• The language used in late 1800s may sound archaic to some students.

Overall do you consider this book still as a reasonable option for a course on Trigonometry today?

Edit: The course is an undergraduate course with full name Pre-Calculus Trigonometry. Its description is:

Introduction to the elementary trigonometric functions using the functional approach, simple identities, identities using the summation, half arc, and double arc formulas, inverse and composite functions, sketching of the elementary functions emphasizing phase shift, period, and amplitude, and the solution of right and obtuse angles.

The prerequisite for this course is called College Algebra with the following description:

This course provides students an opportunity to gain algebraic knowledge needed in many different fields such as engineering, business, education, science, computer technology, and mathematics. Graphical, numerical, symbolic, and verbal methods support the study of functions and their corresponding equations and inequalities. Students will study linear, quadratic, rational, exponential, logarithmic, inverse, composite, radical, and absolute value functions; systems of equations and inequalities modeling applied problems; and curve fitting techniques. There will be extensive use of graphing calculators.

So the students in this course may or may not have any knowledge of trigonometry.

• Students who have to take a university level course on trigonometry are unlikely to have the reading and thinking skills necessary to cope with a nineteenth century text. It is best to avoid potentially creating unnecessary or unproductive obstacles. – Dan Fox Sep 24 '19 at 9:26
• @DanFox I flipped through the book and found it very readable. You seem to be of very low opinion of today's high schoolers. OTOH, I think that college-bound students should have good comprehension skills and a decent grasp of language, so this book can serve as a natural filter. – Rusty Core Sep 24 '19 at 19:01
• @RustyCore, are you suggesting that using this book as the textbook is reasonable? – Zuriel Sep 24 '19 at 20:12
• @RustyCore: my experience with entering university students is that they would rather watch a video tutorial than read a book. Many are not well schooled in reading texts, even those who like math. Students taking trig in college haven't been successful students in high school, and one should minimize the obstacles they face. Potential issues reading a text because of old—fashioned language are best avoided. – Dan Fox Sep 25 '19 at 5:12
• Students who have to take a university level course on trigonometry are unlikely to have the reading and thinking skills necessary to cope with a nineteenth century text. This seems inaccurate. First, my experience with historical textbooks is that often they're at a lower intellectually than modern ones. They are often more concrete and omit topics that modern texts include. Also, a student taking pre-collegiate math in college is sometimes doing it because they're dumb, but in many cases it's because they are reentering school and need review, or were immature in high school. – Ben Crowell Nov 9 '19 at 1:26

I started my career as an archaeologist before I ended up in mathematics. I say this in order to emphasize that I am interested in trying to understand how people thought in the past—the usual "text" that an archaeologist reads is the collection of artifacts which are left behind, but there is also a very active field called Historical Archaeology which seeks to associate historical records with a "ground truth". From the point of view an "archaeologist" (or historian) of mathematics, I think that texts such as Loney may be interesting, and well worth reading. However, I would recommend against using such a text in an introductory class.

The cons which you mention are significant, but there are another couple of issues which should give you further pause:

1. Looking over the table of contents, it appears that much of the focus is on computation. For example, starting around page 106, there are many pages spent on computing the sines and cosines of angles using the angle sum and half-angle formulae. My experience is that modern exposition concerns itself more with the formulae themselves (as these recur in calculus), and largely elides explicit computation. Such computation is, perhaps, useful as an exercise, but a CAS can typically do the job faster and more accurately. In general, my preference would be to use a book which places much less emphasis on computation or which emphasizes the way in which modern computers can aid computation.

2. There are a lot of topics in that text which are kind of archaic, or which are wholly inappropriate for a modern precalculus class. For example, most of Chapters X and XI are not relevant in a modern classroom (there is no reason for students to be taught how to read a log table, for example). Much of Chapter XV seems to focus on aspects of geometry which, for better or worse, are not typically part of the standard US precalculus curriculum (some of it might show up in the high school curriculum, but a lot of it isn't part any standard curriculum prior to upper division courses in geometry (or, perhaps, math competition prep courses)). These topics certainly have some interest, but if the goal is to prepare students for a standard calculus curriculum, then they don't do any favors to the students.

And then there is Part II. Almost nothing in Part II is part of the precalculus curriculum (and nothing in Part II is mentioned in the course description in the question). It starts with series representations of the logarithm and exponential (though using notation which is difficult to parse by modern standards—the funky factorial and the lack of Sigma notation, for example), moves on to a couple of limits, then jumps into complex analysis. The entire second half of the book is, in most modern classrooms, covered in courses on calculus and complex analysis. It doesn't belong in a precalculus class.

The fact that Part II is inappropriate for a precalculus class isn't a big deal—Part I certainly contains enough material for a semester—but I think that it might be better to select a book which is more narrowly focused on the topics which you actually need to cover.

3. In the question, it is noted that the language "may sound archaic to some students". I think that this fails to capture the magnitude of the problem—I think that students are likely to bounce hard off of a topic which is presented with unfamiliar notation (and, by the way, unfamiliar notation which they won't ever see again) written in a form of English which is distinctly old fashioned. They inconsistent typesetting also does the book no favors (but now I'm being catty).

I am struggling for an analogy which is not hyperbolic, but I think that the following works: you don't teach students Russian by asking them to read Война и Мир (War and Peace) or Евгений Онегин (a novel by Pushkin) in the original pre-reform language right out of the gate. Get your students to read something from the 20th or 21st Century, first (maybe something like Один День Ивана Денисовича—the language is modern and pretty accessible, and it is still a classic). Anyone who wants to become a scholar of Russian literature should probably get familiar with pre-reform works eventually, but that isn't the place to start. Start with modern Russian, and work back.

Similarly, students of mathematics should start with a modern presentation and then, if they want to study the history of mathematics, start trying to tackle the classics (presuming that Loney is, indeed, a "classic", and not simply "old").

Basically, my recommendation would be to find another text. I don't think that Loney is appropriate for a modern introductory audience. At best, you might use it supplement the course (for example, one of my critiques of the text, above, is that if over-emphasizes computation, at least by modern standards; however, there appear to be a lot of exercises in the text, which might prove useful). Moreover, there is nothing wrong with reading a book such a Loney's and drawing inspiration from it (personally, I have gotten a lot of milage in my precalc classes out of Gelfand's Method of Coordinates and Klein's Elementary Mathematics from an Higher Standpoint).

Unfortunately, I also don't have a lot of advice about which book you should use. There are a lot of books out there with titles like Precalculus (with Trigonometry!) and Trigonometry for the Precalculus Student and whatnot. Most of these books are 1,000 pages long, weigh 10 lbs., and cost $200+. They make good doorstops, but are otherwise too dilute and broad to be of much use. I'm also not a huge fan of the Schaum's Outlines as course texts. They are useful for exercises, but leave a lot to be desired vis-à-vis exposition (of course, that is kind of the point—they are outlines). Perhaps consider one of the open source texts that are out there? For example, the OpenStax Algebra and Trigonometry or the Open Textbook Library's Trigonometry. I suspect that these books will suffer from many of the same problems as the$200+ tomes, but at least they are free. :\

• "you don't teach students Russian by asking them to read Война и Мир (War and Peace) or Евгений Онегин (a novel by Pushkin) in the original pre-reform language right out of the gate." — the major difference of pre-reform language are two extra letters, one of them is a dead letter, another is pronounced as another, existing letter. As for style and vocabulary, Pushkin in fact is considered one of the first modern Russian writers, that is, using the language which is largely used today. He is no Shakespeare. And Tolstoy's is quite modern too, but he used lots of French in the War and Peace. – Rusty Core Sep 24 '19 at 18:59
• @RustyCore You are attacking an analogy which I prefaced by stating outright was likely flawed (though I would still argue that Solzhenitsyn, or a Soviet Realist like Kataev, is a lot more approachable than Pushkin or Tolstoy---"modernity" and "approachability" are related, but not identical). Would you like to offer a better analogy? If you have no constructive criticism to add, why comment? – Xander Henderson Sep 24 '19 at 21:23

I think the language is too much of a turnoff for class use. If you want a cheap, concise text, I would go with the Schaum's: