Two monuments in Paris are so prominent that they're hard to miss. One is the Eiffel Tower, of course, the all-iron tour de force of engineering, standing by the Seine amid the city's spacious and supremely elegant West End. Then to the north, atop Montmartre, there is the Sacré-Cœur: a tall, immaculately white Catholic basilica that looks like a digitized pre-Raphaelite set from "Lord of the Rings." What most visitors—and in fact most Parisians— don't realize is that both monuments were designed and their construction begun at about the same time, in the 1870s and 1880s. Even more surprising: The tower and the church...