# ^^Euclide. Book1 map.

map.png

 48 propositions = 14  + 34 technological constructions.logical inferences deduced from the diagrams we can construct.

Es construction, prop1Book1

• To construct an equilateral triangle on a given finite straight line.

Es inference, prop5Book1

In isosceles triangles:
• the angles at the base equal one another, and,
• if the equal straight lines are produced further,
• then the angles under the base equal one another.

The blanket term “Proposition” does not come from Euclid,
in Greek simply they appear as a numbered list without any headings.

Nevertheless, Euclid distinguish 2 classes of propositions in how their proofs are concluded

• 14 technological constructions conclude with the phrase
ὅπερ ἔδει ποιῆσαι (“what was necessary to do”)
• 34 logical inferences conclude with the phrase
ὅπερ ἔδει δεῖξαι (“what was necessary to show”).

Unfortunately, in its transmission through Latin and then into modern languages, these phrases are usually contracted to the catch-all

• Q.E.D. (quod erat demonstrandum, “that which was to be shown”)

thereby erasing this important clue about the 2 types of propositions. Some English editions of Euclid preserve the distinction by calling one type “problems” and the other type “theorems”; we prefer the terms “technological constructions” and “logical inferences” — or just “technologies” and “inferences” for short — as these terms emphasize the distinction more clearly.

# credits:  A Concept Map for Book 1 of Euclid's Elements>>>

Alexander Boxer and Justace Clutter
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)

### Abstract

Book 1 of Euclid's Elements begins with just a few simple assumptions and culminates in a profound statement about our universe — the Pythagorean Theorem. We have created a concept map of Book 1 designed to illustrate graphically this remarkable logical sequence. We hope that our effort, although preliminary, will be of interest to math teachers, devotees of the history of math, and anyone who deals with the graphical display of relational data.