## 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 ## EAEB = EC2, which is a constant ## Corollary: if another secant is drawn from E and intersect at A’, B’, then ## EAEB = EA’EB’ ## Viete’s lemma: proof by contradiction ## If EAEB = 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 ## EAEB = 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 ## Construct angle bisector ## Construct H, L, K, I () ## Construct M by HM2 = HLHA ## 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 ## GEF GDB ## EGF = EFH = FHB ## D, H, F, G concyclic ## BDBH = BFBG , which is constant ## H can be found ## Construct H (BDBH = 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 ## BHBD = BK2 = BFBG ## 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
## 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
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