Mercator's World Map and the Calculus 2nd Edition (UMAP)

This Module explains the need for angle-preserving nautical charts, celebrates Mercator’s angle-preserving world map of 1569, and explores the mathematical foundations of the Mercator projection. Those foundations include evaluating numerically the integral ϕ 0 sec ϕ dϕ in order to place the meridian lines on the map, a feat that was achieved long before the discovery of the form of the antiderivative of sec ϕ.

EDITOR’S NOTE: This is the 2nd edition of a Module originally published as a typed fascicule in 1978 and reprinted in 1980 in UMAP Modules 1977–79: Tools for Teaching, 677–727 (Boston, MA: Birkhauser). It has been typeset with an abstract added, material from the Intermodular Description Sheet put into Notes for Instructors, references updated, and the tables edited for better clarity.

Table of Contents:

1. Mercator’s Achievement

1.1 A Strategy for Navigation with Map and Compass 
1.2 Rhumb Lines
1.3 The Need for a Map on Which Rhumb Lines Are Straight
1.4 Mercator’s Successful Map
1.5 Modern Navigators Use Mercator Charts
1.6 The Integral R sec ϕ dϕ Is Involved

2. Calculus and the Mercator Map

2.1 The Framework of the Mercator Map 
2.2 Horizontal Distances at Latitude ϕ Get Stretched
2.3 Mercator’s Insight: Vertical Distances Too Must Be Stretched
2.4 The Vertical and the Horizontal Stretches Must Be Equal
2.5 Summary: How We Get Straight Rhumb Lines
2.6 How To Place the Parallels of Latitude

3. More History

3.1 Mercator’s Map: Cartography in His Time 
3.2 Edward Wright’s Discovery
3.3 Later Mathematical History

4. Several Calculations of R sec x dx

4.1 The Usual Integration 
4.2 A Partial-Fractions Integration
4.3 Gregory’s Form of the Integral

5. A Series for R sec x dx

5.1 Derivation of Wallis’s Series 
5.2 Numerical Approximation of the Integral

6. What Has Calculus Contributed to the Mercator Projection?

7. Hints for Exercises

8. Special Assistance Supplement

8.1 Latitude, Longitude, and the Compass 
8.2 Great Circles

9. References

10. Answers to the Exercises

11. Notes for Instructors

11.1 Output Skills 
11.2 Suggested Uses
11.3 Prerequisites
11.4 Suggested Support Materials
11.5 Output Skills
11.6 Extensions

References

Acknowledgments

About the Author

Note: The information below was created with the assistance of AI.

Mathematics Level

This module is intended for second-semester undergraduate calculus students. It is also appropriate for advanced high school students in AP Calculus BC or honors calculus if supported by an instructor. The module goes beyond basic differentiation and focuses on integration, trigonometric identities, numerical approximation, and infinite series. Some sections are suitable for standard Calculus II, while the later material involving series and historical integration methods is better for honors calculus or independent study.

Application Areas

The module applies calculus to geography, cartography, navigation, and physics. It explains why the Mercator projection became important for nautical navigation: on a Mercator map, rhumb lines, or paths of constant compass bearing, appear as straight lines. The visual examples in the module compare several map projections and show why a straight-line course on a non-Mercator map can lead a navigator off course, while the Mercator projection preserves angles and makes compass navigation practical.

Prerequisites

Students should know basic latitude and longitude, trigonometric functions and identities, trigonometric derivatives, integration of x−1x^{-1}x−1 to $\ln x$, substitution, partial fractions, and geometric series. The module also assumes comfort with radians, secant and tangent functions, Riemann sums, and numerical approximation. More advanced exercises require double-angle formulas, trigonometric identity manipulation, convergence of series, and term-by-term integration.

Subject Matter

The module explains the mathematical foundations of the Mercator projection. It begins with the navigation problem of following a constant compass direction across the curved Earth. 

Correlation to Mathematics Standards

This module aligns strongly with Calculus II standards, especially integration techniques, applications of integration, numerical approximation, series, and interpretation of integrals as accumulated change. It also connects well with AP Calculus BC, including definite integrals, substitution, integration techniques, improper behavior near vertical asymptotes, infinite series, and mathematical modeling.

For Common Core High School Mathematics, the strongest connections are to modeling, functions, trigonometry, quantities, geometric reasoning, and interpreting mathematical representations. The module is less focused on statistics or algebra standards and more focused on advanced functions and modeling.

For NCTM Process Standards, it strongly supports problem solving, reasoning, representation, connections, and communication. At the undergraduate level, it aligns well with MAA CUPM recommendations because it emphasizes applications of calculus, historical context, interdisciplinary modeling, quantitative reasoning, and mathematical communication. It also fits SIAM/COMAP modeling practices by requiring students to identify assumptions, represent a real-world system mathematically, derive a model, analyze the model, and interpret the result in context.

Overall Classification

Mercator’s World Map and the Calculus is best classified as an undergraduate Calculus II applied mathematics module. Its primary value is showing students that integration is not only a symbolic procedure but also a tool for solving a historically important problem in navigation and mapmaking. The module is especially useful in calculus, mathematical modeling, history of mathematics, geography, cartography, and STEM applications courses.