OK, so I did write something. This is a marked improvement from nothing. It needs a concluding paragraph, but the little grey cells are presently not up to the task.
The Elements of Euclid: constructing Right Angles
When reading the definitions again after looking at some of the propositions from book 1 of Euclid’s Elements, I found it striking how some off the definitions I had found puzzling at first glance came to make sense in how they were used. For instance, it seemed unnecessarily awkward to say that a circle is a figure wherein “all the straight lines falling upon it from one point among those lying within the figure are equal to one another” (15). After a few instances of the circle as it is applied to the other figures, however, most often to draw lines of equal length, it made a good deal more sense why Euclid might insist on that attribute of a circle above any others. A circle becomes that which permits one to draw equal lines at any angle from a given point. The definition of right angles and perpendicular lines likewise seems written in such a way as to be as useful as possible for the propositions, but in a different way. Instead of suggesting the ways in which right angles might be used, it apparently answers in advance the question: how might one know when a right angle has been constructed? Does the definition of a right angle already suggest its construction, and if so, how?
When a straight line set up on a straight line makes adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. (Elements, 1, def. 10)
In contrast with most of the other definitions, def. 10 begins not with a part of a figure as such — “a ⊾ is” — but with an instance of the thing being defined, a when. In def. 8 there are plane angles, in 9, when those plane angles are made with straight lines they are rectilinear, and then in 10, when the adjacent rectilinear plane angles are equal, then they are right. More interestingly, the lines do not simply form an angle, but are “set up” one upon another, as though the perpendicular line had been hoisted like a mast. It is as though one were looking at a construction and asking if the angles in it were right, and how that might be known. A line has been set upon another straight line; what angles do they make? Do they equal one another? Unlike the method of counting degrees, in this definition the presence of a right angle necessarily and explicitly implies another, were the base line continued past the perpendicular. Defining a right angle not only as a particular angle, but as a particular equality of angles anticipates not only the construction of right angles as such, but also the ability to prove non-right angles as being equal to two right angles, and therefore of proving the equality of vertical angles and the straightness of intersecting lines (props. 13-15).
In definition 10, then, there is already a suggestion for a possible construction: can a line be set up on another line such that the resulting angles are equal? as well as a relationship between right angles and angles equal to two right angles and the straight line upon which they stand. Then the question comes up: how might it be possible to prove a definite relationship between two adjacent angles without them simply being given as such? Here prop. 8 comes in: it is possible to go from knowing nothing about a set of angles to knowing them to be equal if they are within triangles with equal sides. That conclusion seems intuitively obvious, but once it has been proven with some certainty, its applications are less so. If there were two triangles with equal bases and equal sides set up against each other so that there was no space between them, the shared angle must be right on account of both sides being proven equal. So there’s a line, AB; how might two such triangles be constructed upon it?
At first I had thought that props. 9 and 10 were necessary for prop. 11, but upon closer inspection, although they involve similar constructions, they are quite independent unless a perpendicular needed to be constructed bisecting a finite line, in which case props. 10 and 11 could be used. Otherwise, while both those figures do contain right angles, that fact is not important to what is being proved, and therefore not relied upon in prop 11.
So, while the two triangles needed to construct a set of right angles are a bisected equilateral triangle, the bisection is only incidental. The important part is that the equality of the two triangles be known so that the equality of the two adjacent angles becomes apparent. C is given and D can be chosen arbitrarily, as the length of all the other lines are drawn relative to it. CE is then drawn equal to CD, and DF, FE equal to DE as in prop. 1. It seems the triangle need not necessarily be equilateral if an isosceles could be constructed as easily, but it seems that one cannot be constructed at all, at least at this point. Then FC is connected, which is to be the perpendicular line. Conveniently enough, DCE and FCE must be equal on account of the shared CF, and the equality of their other sides. As an aside, is it important that Euclid calls FD, FE bases and FC, DC, CE sides? Is the hypotenuse in a right triangle always the base? More generally, is the longest side always the base? In any event, definition 10 is so written that very little ends up being needed in order to construct a set of right angles and know that they are such: to cut off lines as equal; to draw a triangle with equal sides; and to know that two triangles with equal sides also have equal angles. In other words, props. 1 and 8.
Prop. 12 shows the other possible case: what if the perpendicular line must go through a point off the original line? That answers the previous question about using an isosceles rather than an equilateral triangle: it does work if the given point is not on line AB. It is also quite attractive, what with the triangle in the circle and the little perpendicular line inside the triangle. Does it further expand what is known about right angles? It is good to know that perpendicular lines can be drawn both up from a given line and down toward one. Also, this case does need prop. 10, in order to establish a point on AB to connect to point C. Is there a reason why in prop. 11 the enunciation asks for right angles, and in 12 it asks for a perpendicular line? It seems important at the end of prop 12 to make explicit the connection between a line that meets another with equal adjacent angles and a line that is perpendicular, so that at the end of prop 12 after the equality of the angles is proven it is added that, when such equality is the case the lines are also perpendicular. That is in keeping with Euclid’s customary thoroughness.