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. 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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Van Schooten's Ruler Constructions - Solution to Problem III
Problem III: Through a given point C draw a straight line parallel to a given straight line AB.
Construction: Draw from C through A an indefinitely long straight line and put on it AD equal to AC and from D through B make DBE. In that, place BE equal to BD and join CE. I say that it is parallel to AB.
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Van Schooten's Ruler Constructions - Solution to Problem IV
Problem IV: Above a given indefinitely long straight line, to construct a perpendicular.
Construction: Conceive the given straight line as going through points A and B, and a perpendicular is to be constructed above it; make BC equal to AB [along the same line] and from B draw BD, making with AB any angle whatsoever, and locate D on that line so that it equals BA or BC, and draw the line from point D through point C. If in that line CF is made equal to CA, and in the line ABC, CE is made equal to CD, I say joining EF makes it be perpendicular to AB.
This may be van Schooten’s trickiest construction. The key to the proof of correctness is to note that the points A, C and D are all the same distance from B, so they lie on a circle centered at B and with AC as a diameter. This makes angle ADC a right angle. Now you only have to figure out why angle CEF is also a right angle.
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Van Schooten's Ruler Constructions - Solution to 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.
Construction: Draw, as in the previous problem, any perpendicular DE above AB, and then, from C, by the third, problem, draw a line CF parallel to that. It will be the one sought.
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Van Schooten's Ruler Constructions - Solution to Problem VI
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.
Construction: Construct above DE by the 4th problem, a perpendicular DF, meeting EF at F, and put at the point C in AB a line CG equal to ED, and from G construct over AB, by the preceding problem a perpendicular GH: I say that GI equals DF, and joining CI, the angle ACI will equal the angle E.
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Van Schooten's Ruler Constructions - Solution to Problem VII
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.
Construction: To the point A in AB, construct, following the previous problem, and angle DAB, equal to E, and draw C, from the 3rd problem, line CF parallel to the line AD: I say the angle CFB is equal to the angle E.
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Van Schooten's Ruler Constructions - Solution to Problem VIII
Problem VIII: Above a given straight line AB, to construct an equilateral triangle.
Construction: Cut AB by the second problem, into two equal parts at C, and from C, above AB, by the 5th problem, erect a perpendicular CF and locate on it DC equal to AC or CB. Draw DB and above that construct, as in the 5th problem, a perpendicular to DE at D, and equal to DC, and join EB. I say that if CF is made equal to BE and the triangle AF, FB is made, then the triangle AFB is equilateral.
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Van Schooten's Ruler Constructions - Solution to Problem IX
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.
Construction: Draw from A a line AE making any angle with AB, and put on it AF equal to C and EF equal to D. Join FB. I say that if a parallel EG is drawn to that, intersecting AB extended at G, then AG is to GB as AE is to EF, that is, C to D.
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Van Schooten's Ruler Constructions - Solution to Problem X
Problem X: Given three straight lines AG, BC and AD, to find a fourth proportional DE, that is so that AB is to BC as AD is to DE.
This problem can be done in the manner of Euclid, putting the first two lines AB, BC in a straight line AC, and the third on another line AE, which forms an angle CAE with AC. If BD is drawn parallel to CE, then DE is the fourth proportional being sought.
