# Van Schooten's Ruler Constructions - The Other Eight Problems

Author(s):
C. Edward Sandifer

We now give the remaining eight of van Schooten’s ruler construction problems, with links to his solutions.  The Note to Teachers gives ideas and advice about how to use any or all of van Schooten's ten problems, along with a set of trigonometry exercises based on a trig table from van Schooten's time, in class.

Problem III:  Through a given point C draw a straight line parallel to a given straight line AB.  Solution to Problem III

Problem IV:  Above a given indefinitely long straight line, to construct a perpendicular.  Solution to Problem IV

Note that we are not asked to construct this perpendicular at any particular place.  All van Schooten asks is that the resulting line be perpendicular to the given line.  Compare this problem with Problem V.

Problem V:  Given an indefinitely long straight line AB and a point C on it, to draw a line CF which is perpendicular to the given straight line.  Solution to Problem V

Problem VI:  To a given straight line AB and at a given point C in that line, to construct an angle given ACI equal to a given rectilineal angle E.  Solution to Problem VI

Problem VII:  Given an indefinitely long line AB and a point C away from it, to draw CF which makes an angle with the given line AB which is equal to a given angle E.  Solution to Problem VII

Problem VIII:  Above a given straight line AB, to construct an equilateral triangle.  Solution to Problem VIII

This particular problem is fairly important to van Schooten, since it is Proposition 1 of Book I of Euclid’s Elements.

Problem IX:  Given a straight line AB, to extend it to G so that the total AG to the extreme GB has a given ratio C to D.  Solution to Problem IX

Problem X:  Given three straight lines AB, BC and AD, to find a fourth proportional DE, that is so that AB is to BC as AD is to DE.  Solution to Problem X

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Solution to Problem III

Note to teachers

Trigonometry exercises

Conclusions

C. Edward Sandifer, "Van Schooten's Ruler Constructions - The Other Eight Problems," Convergence (August 2010)