1% common-notions.tex — Euclid's five common notions (axioms of magnitudes).
23\section*{Common Notions}
4\label{sec:common-notions}
56\begin{rrxivremark}[Common Notion~1]
7\label{cn:1}
8Things which are equal to the same thing are also equal to one another.
9\end{rrxivremark}
1011\begin{rrxivremark}[Common Notion~2]
12\label{cn:2}
13If equals be added to equals, the wholes are equal.
14\end{rrxivremark}
1516\begin{rrxivremark}[Common Notion~3]
17\label{cn:3}
18If equals be subtracted from equals, the remainders are equal.
19\end{rrxivremark}
2021\begin{rrxivremark}[Common Notion~4]
22\label{cn:4}
23Things which coincide with one another are equal to one another.
24\end{rrxivremark}
2526\begin{rrxivremark}[Common Notion~5]
27\label{cn:5}
28The whole is greater than the part.
29\end{rrxivremark}
30
to
AB
extended through
A
to
D
(I.12), so
that the foot
D
falls outside segment
AB
on the far side of
A
.
In the right-angled triangle
BCD
, Proposition I.47 gives
\[
BC^2 \;=\; BD^2 + CD^2.
\]
By the binomial-square identity II.4 applied to
BD
cut at
A
(with
BD=BA+AD
as a straight line, since
D
lies on
AB
extended through
A
):
\[
BD^2 \;=\; BA^2 + AD^2 + 2\cdot(BA \cdot AD).
\]
Substitute, and use I.47 in the right-angled triangle
ACD
to
write
AC2=AD2+CD2
; then
AD2+CD2=AC2
, and
substitution gives:
\[
BC^2 \;=\; BA^2 + AC^2 + 2\cdot(BA \cdot AD),
\]
which is the law of cosines as Euclid states it: the square on the
side subtending the obtuse angle exceeds the sum of the squares on
the sides containing it by twice the rectangle on