Mathematical Textbooks and Teaching during the 1700s

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Mathematical Textbooks and Teaching during the 1700s

Overview

The eighteenth century was a time of great progress in the field of mathematics. Included in this progress were advances in ways of teaching mathematics, including a number of textbooks at all levels of complexity by some of the world's greatest mathematicians. Because of these textbooks, mathematics education became more standardized and more formalized than had previously been the case, especially at introductory levels. This, in turn, helped in the education of still more mathematicians who were able to continue producing further mathematical research of very high quality.

Background

The great majority of students are not likely to consider mathematics textbooks a great boon and, in fact, are more likely inclined to curse the authors than praise them. However, any society that strives towards technical and scientific mastery needs to be able to teach its citizens the basic mathematical tools necessary for survival; at the very least, basic arithmetic and algebra skills and an appreciation of concepts in geometry and more advanced mathematical fields such as trigonometry should be realized by its citizens. These mathematical skills are necessary not just for future scientists and engineers, but for anyone hoping to work in a skilled job or profession. However, until the eighteenth century, mathematics education was not a highly developed field.

In fact, through most of the history of mathematics, math skills were taught to a society's elite: priests, rulers, and selected others. Much of this education most closely resembled an apprenticeship, in which a priest or private tutor taught a student either individually or in very small groups. Most students learned only basic arithmetic—addition, subtraction, multiplication, and division—because no greater knowledge would be required of them during their lives. A select few, destined for careers in finance, might learn the mysteries of interest calculations, and even fewer would learn sufficient math skills to calculate the answers to problems in physics or astronomy. However, the great majority of people learned virtually no math, and their lives were none the poorer for this because their world was a simple one.

Another factor to consider was that many mathematicians, instead of sharing their methods of solving problems, kept them secret. A very small number of mathematicians helped support themselves through mathematical "tricks," such as extracting square roots or performing rapid mental calculations. Teaching others their methods was akin to a magician giving away the secrets to his tricks; it only gave audiences and the "competition" the ability to do the same thing.

Finally, until relatively recently, there were very few students and even fewer books. The peasant class did not attend school, the aristocracy and royalty had private tutors, and religious orders trained their own students. It made no sense to write and publish textbooks unless students read and used them and, without enough students of mathematics, textbooks were simply not a necessary innovation.

This began to change in the seventeenth century, as university attendance began to increase (albeit still very strongly weighted towards the upper classes of society). In addition, scientific societies began to form, encouraging the teaching of science and the dissemination of knowledge, as opposed to its sequestration in the hands of a few select. Also, science began to be viewed as a field in its own right, capable of supporting full-time work by professors and scientists. And, as society took the first steps towards more financial and technological sophistication, the first steps were also taken in the process of training engineers, scientists, and accountants. In addition, the development of the calculus by Isaac Newton (1642-1727) and Gottfried Liebniz (1646-1716), their feud over priority, and the exploits of other famous mathematicians helped bring mathematics to the attention of the upper and merchant classes, raising its profile somewhat. Adding to this was Newton's stature as the world's first famous scientist, which further helped bring him and his mathematical discoveries to the forefront of public attention in England, while his Continental rivals did the same in Germany, France, and Switzerland. Finally, people began to realize that mathematics could be applied to solving everyday problems, again lending some credence towards teaching an increasing number of students at least the fundamentals of mathematics. All of these factors resulted in an effort to begin formalizing mathematics education, including publishing the first textbooks in this subject.

The first mathematics textbooks (as opposed to arithmetic primers) started coming out in the mid-1700s. Leonhard Euler (1707-1783), perhaps the greatest mathematician who ever lived, published a book, Algebra, in 1770-1772. This book went through six editions, and Euler's other textbook, on number theory, was popular as well. In Britain, Colin Maclaurin (1698-1746) wrote texts on both the elementary and advanced levels, including Treatise of Algebra, which also went through multiple editions in the latter half of the century. Other texts on algebra, geometry, and the mathematical treatments of physics were released during the eighteenth century, many by some of the era's greatest minds in mathematics.

Impact

The effects of this attention to mathematics education were significant.

First and foremost, the character of any subject is dictated in large part by the textbooks out of which students study because it is in the textbooks that students receive their first exposure to a topic.
In addition, these texts reflected the fact that a market for them existed. The growing number of students of mathematics encouraged the writing of texts, while the growing numbers of texts, many by famous mathematicians, encouraged students to study mathematics. As a result, European and American colleges and universities turned out increasing numbers of mathematically literate (or, at least, mathematically aware) students, who went on to use these skills in their jobs.
Finally, much of today's mathematics texts and teaching methods are direct descendants of these first textbooks; we can trace the sequence in which we learn mathematics and many of the topics taught to these first texts. Each of these ideas will be developed in more detail in the following paragraphs.

To start with, textbooks have a vital role in shaping the manner in which any topic is taught and, in turn, with shaping the opinion of students. This fact is perhaps best appreciated by considering recent debates over teaching evolution in the classroom. Religious fundamentalists fully realize that the manner in which evolution, creationism, and creation science are presented in a textbook will affect a student's perception of these mutually exclusive systems. If evolution is taught without reference to any form of creationism, then students perceive that it is unchallenged as the correct theory surrounding the history of life. On the other hand, if evolution is taught as one of several possible theories, then competing theories gain credence and the case for evolution is weakened. Finally, if it is forbidden to teach evolution, regardless of the great weight of scientific evidence in its favor, it will then be perceived as lacking credibility, and a generation of students will consider it to be somewhat flawed.

Similarly, the content of mathematics texts helps to set the tone for how the educated public viewed mathematics in the eighteenth, and subsequent centuries. By portraying mathematics as a useful tool, these textbooks helped to encourage its acceptance by a wider array of merchants, engineers, and scientists than had previously been the case. When mathematics was taught by Mayan priests only to other Mayan priests, its utility was limited because it was a tool dedicated to calculating astronomical events by only a select few. When any literate person at a university could pick up a mathematics text and learn relatively sophisticated techniques, the subject was demystified and became much more useful for daily activities.

This, in turn, helped to advance the field of mathematics. Going back to the Mayans, mathematics barely progressed from a tool for astronomical calculations because that was its entire reason for being. However, as university students graduated and went on to careers in finance, engineering, mathematics, or science, they took with them the ability to adapt their mathematical tools to the uses at hand. This gave mathematics an infusion of new ideas, new applications, and new techniques, contributed by both mathematicians and nonmathematicians alike. Both mathematics and society benefited from this exchange of ideas.

Finally, the way that mathematics is taught today can, in many ways, be traced directly back to these first textbooks, suggesting that today's students continue to be influenced by the authors of the first such books, over two centuries ago. In spite of the occasional jokes (or furor) over new math, the fact is that students are still taught many topics in a manner their predecessors would recognize. Addition, subtraction, multiplication, and division continue to form the basis of elementary mathematics education, as do fractions and decimals. And with good reason; these are the skills that are easiest to learn and with the most applicability to a wide variety of circumstances and professions. After that, students start to learn basic algebra techniques, they may be taught the fundamentals of geometry, and they may study logarithms, trigonometry, and similar mathematical skills. This is a curriculum that Euler would have recognized, because these are among the most elementary tools in the mathematical repertoire. These tools will carry virtually anyone through most nontechnical careers, and constitute the foundation for further study of mathematics at a more advanced level in college. The primary difference between this education and that of the eighteenth century is that of timing; these topics are considered elementary topics to be taught to middle and high school students today, but were considered advanced topics for university students two centuries ago.

We can see, then, that the appearance of the first textbooks in mathematics was an important milestone in the history of mathematics. By writing these books, the authors helped to establish a standard framework that formed the basis for all mathematics education, not just at the time, but continuing until the present. In addition, by educating a larger number of students, these books also helped spread the utility of mathematics and, by so doing, to cross-fertilize mathematics with ideas from other fields. All of these had a significant impact on mathematics, education, and society.

P. ANDREW KARAM