My father was a civil rights lawyer, but among his many interests, he enjoyed recreational math, and he was never afraid of trig tables. Our family visited his mother's farm every summer, nestled in the Catskill mountains of New York, near the Ashokan reservoir. A beautiful bridged road called the "Reservoir Road" cuts across the Ashokan on its way to the reservoir ponds and the spellbinding aeration fountains. From that bridge, there's a prodigious view of the Catskills. Looking west you see a high, long ridge, and what looks like a "duckbill," the tip of a mountain peeking over that ridge. Everybody said that the duckbill was the top of Slide, the tallest mountain in the Catskills. We always enjoyed stopping to look at that ridge, along with the whole panoply of mountains surrounding the reservoir.
One day when I was about eleven, my father asked, "Is that really the tip of Slide Mountain?" Out came the topological maps and the trig tables, and my father began to plot triangles to calculate our line of sight. From the middle of the reservoir, the view west is toward the Friday Mountain/Cornell/Wittenberg Ridge (running north/south), about 3800 feet above sea level. Slide Mountain is set well back from the ridge, at a height of 4400'. If you look at a topo map like this one, it's hard to imagine what else the duckbill could be, but still, you have to do the math to see if Slide is visible over that ridge. We did the math, and sure enough it was. I can't convey to you how satisfying it was to know that the math proved that that duckbill was Slide.