What is "Steinhaus · Long Meter" that can measure the length of the curve drawn on paper?

It is difficult to measure with an ordinary ruler, the mechanism that can measure the length of the curve written on paper is "Stein house · long meter"is. A movie that describes such "Steinhaus / Long meter" is published on YouTube.

Steinhaus Longimeter Review / HowTo


This paper which draws strange patterns with several lines intersecting is Steinhaus · Long Meter.


Stein house long meter was invented to measure the length of the curve drawn on the plane.


Easy to use. On top of the curves you want to measure, overlay Stein house · long meter printed on watermarked paper ... ...


Just count the number of intersections between the straight line and the curve of Steinhaus · Long meter. The length of the curve can be measured in millimeters by the number of curves and straight lines intersecting.


I invented Steinhaus · Long Meter, a Polish mathematician who was active in the 20th centuryHugo SteinhausMr. It is a person who makes many discoveries in fields such as geometry and trigonometry from World War I to the Second World War and is one of the founders of game theory.


At the time, Polish mathematicians including Mr. Steinhaus gathered at the bar and wrote various problems on the notebook and were searching for solutions.


Among them, Mr. Steinhaus explored the method of measuring the length of the curved line and thought whether it was possible to measure the curve by the number of times it crossed the ruled line of the grid of grid paper.


Let's start with a straight line. Draw a 6 inch (about 15 cm) straight line on paper ...


Pile 0.5 inch (about 1.3 cm) grid paper parallel to the straight line.


When counting the number of intersections between the straight line and the ruled line of the squares in this state ... ...


Since it intersected twelve times to the 12.0.5 inch (. 1.3 cm) grid line, if it is assumed that "the length of the line intersects the ruled line of the grid × times 0.5 inches (about 1.3 centimeters)" , The straight line is roughly 6 inches (about 15 centimeters)? "


Next, let's count the intersection of the straight line and the grid line of the grid with the grid paper diagonally.


Next time is 17. As before, if the number of times the straight line intersects the ruled line of the grid × 0.5 inch (about 1.3 centimeters) is the length of the line, this calculation will be 8.5 inches (21.6 centimeters) this time.


Even a straight line gives an error like this, but if it becomes a curve it can not be measured even better.


Mr. Steinhouse who bothered to correct this error thought, "After counting the intersection of the curve and the grid of the grid, once it turns the grid and counts it again?"


"We count the number of intersections at the beginning once, then rotate the squares by 30 degrees, count the intersection again, and rotate the 30th squares last to measure intersections" Invention


I tried to reduce the error by counting the intersection of one curve with the ruled line of the square at three angles.


In addition, to make it easy to understand the number of intersections counted and the length of curves, Mr. SteinhaustrigonometryUse the size to devise the size of the squares. As a result, it was found that the straight line forming the squares gave an interval of 3.82 millimeters, and the length of the curve became close to 1 mm for the most intersection.


And, it is troublesome to rotate 30 degrees of the graph paper manually, so it is easy to get errors ...


We devised a sheet which printed three layers of graph paper stacked from the beginning. Actually, this sheet is "inch Steinhaus · Long meter" made by inch method which was originally a grid paper at 3.82 mm intervals.


The Steinhaus · Long meter, which is made of grid paper which is truly 3.82 mm intervals, is very fine.


You can count the number of intersections of curves and straight lines with Steinhaus · Long Meter, and you can convert the counted number to millimeter length as it is. For example, if you crossed 126 times, you can guess that the length of the curve is approximately 12.6 centimeters. However, the original Steinhaus / Long meter corresponding to the millimeter length is too fine to count and it is very difficult.


Currently"Kilby meterThe meter used to measure the length of the curved line on the map, which is primarily developed, is developed, and scenes where Steinhaus · Long Meter is active seems to rarely appear.


Here, actually measure the same curves with Steinhaus · Long meter and Kilby meter and see how much accuracy the Steinhaus · Long meter boasts.


As a result of measuring with a kilby meter ......


The length of the curve is 34.5 centimeters.


Since counting is too difficult if it is a millimeter-long Steinhouse Long meter, I will try to measure it with a Steinhaus / Long Meter in terms of inches. In Steinhaus · longmeter in terms of inch, the number obtained by dividing the number of intersections of the curve and the meter by 10 is directly inch length of the curve.


1, 2, 3, 4 ...


114. After fixing it to inches and returning it to a further centimeter, it is 28.9 centimeters. that?


It seems quite difficult to measure curved lines that show different ways of bending for each part with a grid paper.


Unfortunately the invasion of Nazis 'Polish began soon, and Steinhouse, who was a Jewish system, escaped the Nazis' eyes and lived quietly quietly, although there was evidence that Mr. Steinhaus had tried to sell this Steinhouse Long Meter I had to do. After all, the Steinhaus · Long meter became known only to a few mathematician lovers without seeing the sun's eyes.


In addition, Mr. Chris Staecker who created this movie has also created a special page of Steinhaus · Long Meter.

Chris Staecker: Steinhaus Longimeter
http://cstaecker.fairfield.edu/~cstaecker/machines/longimeter.html

The original Steinhaus · Long meter, where the number of times the curve crosses the meter is directly the millimeter length of the curve, can be downloaded from the following site.

longimeter mm.pdf
(PDF file)http://cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=longimeter%20mm.pdf

An inch converted Steinhaus · Long meter whose inch length corresponds to the number of times the curve crosses the meter divided by 10 corresponding to inches can be downloaded from the following site.

longimeter in.pdf
(PDF file)http://cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=longimeter%20in.pdf