Archimedes' Book of Lemmas - Index

Geometry, Math Education



Archimedes' Book of Lemmas



Proposition 1.
Tangent circles and parallel diameters 


Proposition 2
Semicircle, diameter, tangents, secant line. 


Proposition 3
Semicircle, perpendicular to diameter, chords.


Proposition 4
Semicircles, perpendicular to diameter, Arbelos.


Proposition 5
Arbelos, Archimedean Twins, Semicircles, Circles


Proposition 6
Arbelos, Semicircles, Circle.


Proposition 7
Square and inscribed and circumscribed Circles


Proposition 8
Angle Trisection. Neusis construction.


Proposition 9.
Perpendicular chords, sum of arcs.


Proposition 10
Tangents to a circle, secant line and chords, Parallel.


Proposition 11
 Perpendicular chords  and radius, square.


Proposition 12
Semicircle, Diameter, Chords, Tangents, Perpendicular.


Proposition 13
Circle, Diameter, Chord, Perpendicular, Congruence.


Proposition 14
Salinon: Semicircles, Perpendicular, Circles, Areas.


Proposition 15
Circle, Diameter, Inscribed Regular Pentagon, Arc, Midpoint, Perpendicular, Radius

Archimedes (287 - 212 B.C.) was born at Syracuse of Sicily as a son of the astronomer Pheidias. He learnt from the disciples of the mathematician Euclid when he was young. Archimedes is considered one of the three greatest mathematicians of all time (together with Newton and Gauss).

According to "The Works of Archimedes" by T. L. Heath, Cambridge 1897, Archimedes' works included "On the Sphere and Cylinder", "On the Measurement of a Circle", "On Conoids and Spheroids", "On Spirals", "On the Equilibriums of Planes", "The Sand-reckoner", "Quadrature of the parabola", "On Floating Bodies", "Book of Lemmas" and "The Method".

In the book "Book of Lemmas", attributed by Thabit ibn-Qurra to Archimedes, there were 15 propositions on circles, with the first proposition referred in the subsequent fifth and sixth propositions. The statements in "Book of Lemmas" do not seem to concur to a central theme.

Lemma is a subsidiary proposition that is assumed to be true in order to prove another proposition. There is no technical distinction between a lemma, a proposition, and a theorem.

Exercise your brain. Archimedes wrote the "Book of Lemmas" more than 2200 years ago. Solve these 15 high school level problems and lift up your geometry skills.