A Renaissance math puzzle

In 1471, Johannes Müller asked where you should stand so that a vertical bar appears longest.

To be more precise, suppose a vertical bar is hanging so that the top of the bar is a distance a above your eye level and the bottom is a distance b above your eye level. Let x be the horizontal distance to the bar. For what value of x does the bar appear longest?

Note that the apparent length of the bar is determined by the size of the angle between your lines of sight to the top and bottom of the bar.

Please don’t give solutions in the comments. I’ll post my solution tomorrow, and you can give your solutions in the comments to that post if you’d like.


Update: See this post for more historical background.

8 thoughts on “A Renaissance math puzzle

  1. I’ve seen this problem in calculus textbooks. It’s interesting to me that it predates calculus.

  2. This puzzle has a sporting application. Imagine a sport where you have to aim at a goal of width |b – a| on the y axis. You can take your shot from a point whose y coordinate is chosen for you, but you can choose x, i.e. where you shoot from on this line. Assume that achieving the distance with your shot is not an issue, you just want the goal to “look” as big as possible from where you stand to have the best chance of hitting the goal.

    This is in fact a pretty good model of the problem of taking a conversion kick in rugby: you have to take the place kick from a point in line with where the try (touchdown) was scored, which might be over by the side line, but you can choose how far back from the end line to take it.

    Or at least, that’s how this puzzle was presented to me in high school math class growing up in New Zealand (a rugby-mad country).

  3. I slogged it out with calculus, but a solution that simple should have a more elegant derivation. Curious to see what it is.

  4. Aaaah. Begs the question: What is the curve traveling along which a given segment appears to be of constant angular size?

  5. The solution is kinda cool and there is really no need for calculus here, as you can change the maximization problem to finding the maximum of a parabola with a change of variables.

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