# ^^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.

## Links

Book1 propositions.