![]() In Oliver Byrne's translation, which I think is a bit more poetic on this point than Heath's, the proof starts, "Let the two triangles be conceived, to be so placed, that the vertex of the one of the equal angles shall fall upon that of the other…" In other words, Euclid seems to describe physically placing one triangle on top of the other one. Proposition 4 is the theorem that side-angle-side is a way to prove that two triangles are congruent. Basically, superposition says that if two objects (angles, line segments, polygons, etc.) can be lined up so that all their corresponding parts are exactly on top of each other, then the objects are congruent.įor example, in Book 1, Proposition 4, Euclid uses superposition to prove that sides and angles are congruent. On its face, Axiom 4 seems to say that a thing is equal to itself, but it looks like Euclid also used it justify the use of a technique called superposition to prove that things are congruent. Things which coincide with one another are equal to one another. If equals be subtracted from equals, the remainders are equal.Ĥ. ![]() If equals be added to equals, the wholes are equal.ģ. Things which are equal to the same thing are also equal to one another.Ģ. Without a way to measure angles, what might Euclid have meant by angles being equal? Contemporary Greek astronomers and mathematicians used degrees, and Euclid was probably aware of them, but he doesn't use them in the Elements. He never discusses degrees, radians, or how to measure an angle using a protractor. It's less if it's "more closed." We know it when we see it.īut Euclid never tells us exactly how to compare two angles. Intuitively, we can all imagine what greater and less mean for angles: angle A is greater than angle B if it's "more open" than angle B. Definition 8 states, "A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line." Definition 10 says, "When a straight line set up on a straight line makes the 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." Definitions 11 and 12 are for obtuse and acute angles, which are defined as being greater than or less than a right angle, respectively. In the beginning of the book, he includes a few definitions relating to angles. To understand what it would have meant to Euclid, we need to go back and look at Euclid's treatment of angles. But Euclid knew what he was doing, so there must be a reason for this postulate. Why not a postulate that says that all 45 degree angles are equal to one another? Or all 12 degree angles? The fourth postulate seems a bit bizarre. It's just part of the way we define angles. We don't need a whole postulate that says this. Fair enough.īut why the heck do we need a postulate that says that all right angles are equal to one another? You probably remember learning in a middle or high school geometry class that right angles are 90 degree angles, and two angles are congruent if they have the same degree measure. Those postulates say that if we want to, we can connect two points by a line, draw lines that continue indefinitely, and draw circles wherever we want and of whatever size we want. The first three postulates have a similar feel to them: we're defining a few things we can do when constructing figures to use in proofs. I included the text of the five postulates, from Thomas Heath's translation of Euclid's Elements:ġ) To draw a straight line from any point to any point.Ģ) To produce a finite straight line continuously in a straight line.ģ) To describe a circle with any centre and distance.Ĥ) That all right angles are equal to one another.ĥ) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." ![]() ![]() In February, I wrote about Euclid's parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry. Image: Public domain, via Wikimedia Commons. Euclid's fourth postulate states that all the right angles in this diagram are congruent. An illustration from Oliver Byrne's 1847 edition of Euclid's Elements. ![]()
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