http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI4.html
In the standard reference for Euclid’s Elements, David Joyce writes, “It is not clear what is meant by ‘superposing a triangle on a triangle…’ Whatever the intended meaning of superposition may be, there are no postulates that allow any conclusions based on superposition. One possibility is to add postulates based on a group of transformations of space, or if restricted to plane geometry, on a group of transformations of the plane… Yet another alternative is to simply take this proposition [side-angle-side congruence] as a postulate, or part of it as a postulate. Hilbert, in his Foundations of Geometry takes as given that under the hypothesis of this proposition the remaining angles equal the remaining angles. Then Hilbert proves that the base equals the base.”
Clearly, superposition is controversial and it always has been; Paolo Mancosu details the several positions taken four hundred years ago in his Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, pp. 28-33. What Common Core has done is to take the easy way out, adding postulates based on a whole group of transformations of space but, since any mention of postulates is taboo, this really amounts to waving their hands in the air and simply announcing – one of the many factoids to be memorized – that geometric figures can be reflected, translated, rotated and dilated ad libitum. They are assuming their conclusions!
This is not geometry in the traditional sense. It is actually what is known as coordinate geometry, which is normally taught at the end of high school in preparation for calculus and is sometimes called pre-calculus. But teaching it immediately after Algebra I does not mean that the students are more advanced than they would have been otherwise. At this level it is just a review of Algebra I with the triangles serving only as a foil to set up the algebra problems being reviewed. (I discuss this in more detail at the beginning of the red-belt chapter of my book.) It just wastes the students’ time by inserting a review of material they have already mastered between Algebra I and Algebra II. They learn nothing about geometry and deductive logic.
I have written a short geometry textbook, Geometry-Do, which avoids the pitfall of superposition while erecting geometry from the ground up, starting with a modern formulation of Euclid’s five postulates and with the axioms generally accepted in modern abstract algebra, which was unavailable to Euclid. Have I successfully avoided superposition in geometry?
https://www.researchgate.net/publication/291333791_Volume_One_Geometry_without_Multiplication
Article Volume One: Geometry without Multiplication
Robin Hartshorne (Geometry: Euclid and Beyond, p. 2) writes:
“The method of superposition used [by Euclid] in the proof of [Book I, Proposition 4, SAS Congruence], which allows one to move the triangle ABC so that it lies on top of the triangle DEF, cannot be justified from the axioms.”
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