The works of Apollonius of Perga were partly unknown or lost

During the 16th and 17th centuries interest in ancient works on mathematics increased and several mathematicians tried to restore and reconstruct works that had been lost, drawing upon the quotations and references in other works of ancient mathematicians.

• During the 16th and 17th centuries interest in ancient works on mathematics increased and several mathematicians tried to restore and reconstruct works that had been lost, drawing upon the quotations and references in other works of ancient mathematicians.

The works of Apollonius of Perga were partly unknown or lost.

• The works of Apollonius of Perga were partly unknown or lost.

• The preface to the seventh book of Mathematical Collection by Pappus outlined the contents of Apollonius’s treaties On Tangencies and On Inclinations.

Apollonius Problem

• Apollonius Problem

• Given 3 things, each of which may be either a point, a straight line or a circle, to draw a circle which shall pass through each of the given points (so far as it is points that are given) and touch the straight lines or circles.

There are 10 possible different combinations of elements, and Apollonius dealt with all 8 that had not already been treated by Euclid.

• There are 10 possible different combinations of elements, and Apollonius dealt with all 8 that had not already been treated by Euclid.

• The particular case of drawing a circle to touch three given circles attracted the interest of Viete and Newton.

Adriaan van Romanus (Belgian mathematician, 1561-1615) gave a solution by means of hyperbola.

• Adriaan van Romanus (Belgian mathematician, 1561-1615) gave a solution by means of hyperbola.

• Vieta thereupon proposed a simpler construction (by means of only compass and ruler), and restored the whole treatise of Apollonius in a small work, which he entitled Apollonius Gallus (Paris, 1600).

If EZ is a tangent to the circle ABC

• If EZ is a tangent to the circle ABC

• ECA  EBC

•  EC/EA = EB/EC

•  EAEB = EC2, which is a constant

• Corollary: if another secant is drawn from E and intersect at A’, B’, then

• EAEB = EA’EB’

• Viete’s lemma: proof by contradiction

• If EAEB = EC2, then EC is tangent to the circle ABC.

Given two points A, B and a line 

• Given two points A, B and a line 

• (Case 2) If AB is not parallel to 

• Produce AB to  at E

• Construct the point C on  by

• EAEB = EC2

• Construct circle ABC

• It remains to prove that the circle touches 

• By the lemma previously proved, the circle ABC touches  at C

Given a point A, two lines BC, DE

• Given a point A, two lines BC, DE

• Assume there is a circle touching BC, DE at M, O respectively

• The center lies on the angle bisector by symmetry

• Construct angle bisector 

• Construct H, L, K, I ()

• Construct M by HM2 = HLHA

• Construct the circle ALM

Claim: The circle touches DE at O

• Claim: The circle touches DE at O

• i.e. to prove NM = NO

• NK=NK, NKI = 90, HK=KI

•  NKH  NKI

•  NHK = NIK, NH = NI

•  NHM = KHM – NHK

• = KIO – NIK

• = NIO

• Also, NMH = NOI = 90

• NMH  NOI  MN = NO

Given a circle A and two points B, D

• Given a circle A and two points B, D

• Assume there is a circle touching circle A at G

• Join GB, GD; draw the tangent HF

• GEF  GDB

• EGF = EFH = FHB

• D, H, F, G concyclic

• BDBH = BFBG , which is constant

• H can be found

• Construct H (BDBH = AB2 – AK2)

• Construct F (HF is tangent to A)

• Construct G (produce BF)

• Construct circle DBG

Claim: GDB  GEF

• Claim: GDB  GEF

• i.e. to prove GEF = GDB

• BHBD = BK2 = BFBG

•  D, H, F, G concyclic

•  GDB = HFB

• HF is a tangent  EGF = EFH

• GEF = 180 – EGF – GFE

• = 180 – EFH – GFE

• = HFB = GDB

• GDB  GEF with parallel bases and having the same vertex G

• Their circumscribed circles are mutually tangent to each other

Analysis – opposed to synthesis

• Analysis – opposed to synthesis

• (Greek) the reversed solution

• Viete – The Analytic Art (1591)

• Zetetics (problem translated to symbols/equations)

• Poristics (examination of theorems)

• Exegetics (solution by derivations)

BC 260-BC 190 Apollonius On Tangencies

• BC 260-BC 190 Apollonius On Tangencies

• 290-350 Pappus Mathematical Collection

• 1600 Viete Apollonius Gallus

• 1811 Poncelet Solutions de plusieurs problemes de geometrie et de mecanique

• 1816 Gergonne Recherche du cercle qui en touche trois autres sur une sphere

• 1879 Petersen Methods and Theories for the Solution of Problems of Geometrical Constructions

• 2001 Eppstein Tangent Spheres and Tangent Centers

Francois Viete, Apollonius Gallus, Paris, 1600

• Francois Viete, Apollonius Gallus, Paris, 1600

• Thomas Heath, A History of Greek Mathematics Vol.II, Oxford, 1921

• Ronald Calinger, Vita Mathematica: Historical Research and Integration with Teaching, The Mathematical Association of America, 1997

• Euclid Elements

• Google Books

• Wikipedia