Euclid's second proof in Book VI depends on the theory of similar triangles . Let AHB be a line segment of length * a* + * b* with AH = * a* and HB = * b* . Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as * h* . Then, using Thales' theorem and, as in the proof of Pythagoras' theorem by similar triangles , triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, .
a
/
h
=
h
/
b
,
{\displaystyle \ a/h=h/b,}
from which we conclude by cross-multiplication that
h
2
=
a
b
,
{\displaystyle \ h^{2}=ab,}
and finally that
h
=
a
b
{\displaystyle h={\sqrt {ab}}}
. Note further that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length
(
a
+
b
)
/
2
{\displaystyle (a+b)/2}
then clearly OC > CH, .
(
a
+
b
)
/
2
≥
a
b
{\displaystyle (a+b)/2\geq {\sqrt {ab}}}
(with equality if and only if * a* = * b* ), which is the arithmetic–geometric mean inequality for two variables and, as noted above , is the basis of the Ancient Greek understanding of "Heron's method".

How? “Through this love of money some have wandered away from the faith .” The love of money works its destruction by luring the soul to forsake faith . Faith is the contented trust in Christ that Paul referred to in verse 6: “Godliness with contentment is great gain.” Faith says, “I have learned in whatever situation I am to be content ” (Philippians 4:11). Faith has contentment in all circumstances because it has Christ, and Christ makes up for every loss: “I count everything as loss because of the surpassing worth of knowing Christ Jesus my Lord” (Philippians 3:8).

As technology and globalization generate ever-greater surpluses while disrupting traditional livelihoods, the challenge of reallocating money for moral purposes will become more acute. If profits from self-driving trucks are shared with unemployed truck drivers, everyone wins. Negative income taxes, a guaranteed income and charities that give directly to beneficiaries are probably less corrupting than people think , and offer a pathway to enjoying the windfall of technological advances in the face of Luddites and isolationists — but only if we overcome the intuition that money is the root of all evil.