Short answers, pulled from the story.
A topographic map shows both natural and artificial features with large-scale detail. It presents horizontal and vertical positions in measurable form unlike planimetric maps that do not show elevations.
The first multi-sheet topographic map series of an entire country was completed in France in 1789 under the name Carte géométrique de la France.
The Great Trigonometric Survey of India began in 1802 under the East India Company and later came under British Raj control after 1857.
Topographic maps conventionally show land contours by means of contour lines that connect contiguous points of the same altitude. Every point on a marked line of 100 meters elevation sits exactly 100 meters above mean sea level.
In the United States the national map-making function migrated to the newly created United States Geological Survey in 1879. That agency has retained this responsibility ever since while sharing it previously with both the Army Corps of Engineers and the Department of the Interior.