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If you ask mathematicians what they do, you always get the same answer. They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers.

Mathematics Magazine, v. 65 no. 5, December 1992.

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# Van Schooten's Ruler Constructions

## The other eight problems

We now give the remaining eight of van Schooten’s ruler construction problems, with links to his solutions.

Branch point:          Note to teachers.

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

Branch point:         Solution to Problem III

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

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.

Branch point:         Solution to Problem IV.

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.

Branch point:         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.

Branch point:         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.

Branch point:         Solution to Problem VII.

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

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

Branch point:         Solution to Problem VIII.

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.

Branch point:         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.

Branch point:         Solution to Problem X.