Van Schooten's Ruler Constructions - Problem II

Author(s): 
C. Edward Sandifer

Here is van Schooten's second problem and his first solution to it.

Problem II: Given a straight line with endpoints A, B, to bisect it.

Construction:  Take any point C at random away from the line AB, and from A through C draw the indefinitely long line ACD, and locate on that line CD equal to the double of AC, and from D through B make a straight line.  In that line, put BE equal to BD, and, joining C, E, I say it bisects AB at F.

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Second solution to Problem II

The other eight problems

Van Schooten's Ruler Constructions - Second Solution to Problem II

Author(s): 
C. Edward Sandifer

Van Schooten then presented a second solution to Problem II.

Problem II: Given a straight line with endpoints A, B, to bisect it.

Another way:  Find, as before, away from AB, any point C, and draw from it through the points A and B straight lines CAD, CBE, and make AD equal to AC and BE equal to BC.  Then join DB, AE, and let them intersect at F.  I say that if FC is made, it will bisect AB at G.

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Third solution to Problem II

The other eight problems

Van Schooten's Ruler Constructions - Third Solution to Problem II

Author(s): 
C. Edward Sandifer

Finally, van Schooten presented a third solution to Problem II.

Problem II: Given a straight line with endpoints A, B, to bisect it.

Another way:  Assume, as before, C is a point away from AB, and from B through C is drawn an indefinitely long line, and find on it CD equal to CB, and join it to A to form AD.  Then in DA, assume DE equals DC and EF equals CB.  [Here, van Schooten gives five illustrations, showing each of the following five cases:  1) DF = DA, 2) DF < DA, 3) DF > DA but DE < DA, 4) DE = DA, and 5) DE > DA.  We illustrate case #2 only.)]  Make BE, FC intersecting at G, then DGH cutting FB in H.

If now F falls on the point A, then the lines FB and AB coincide, and then H bisects it.

But if the point F falls beyond or within A, I say that if the line is drawn through the points C and H, it will bisect the line AB at I.

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         The other eight problems