Table of Contents

XVI. Aristarchus Of Samos—


(a) General


(b) Distances of the sun and moon


(c) Continued fractions (?)


XVII. Archimedes—


(a) General


(b) Surface and volume of the cylinder and sphere


(c) Solution of a cubic equation


(d) Conoids and spheroids—


(i) Preface


(ii) Two lemmas


(iii) Volume of a segment of a paraboloid of revolution


(e) The spiral of Archimedes—


(i) Definitions


(ii) Fundamental property


(iii) A verging


(iv) Property of the subtangent


(f) Semi-regular solids


(g) System of expressing large numbers


(h) Indeterminate analysis : the Cattle Problem


(i) Mechanics : centres of gravity—


(i) Postulates


(ii) Principle of the lever


(iii) Centre of gravity of a parallelogram


(j) Mechanical method in geometry


(£) Hydrostatics—


(i) Postulates


(ii) Surface of fluid at rest


(iii) Solid immersed in a fluid


(iv) Stability of a paraboloid of revolution


XVIII. Eratosthenes—


(a) General


(b) On means


(c) The Platonicus


(d) Measurement of the earth


XIX. Apollonius of Perga—


(a) The conic sections—


(i) Relation to previous works


(ii) Scope of the work


(iii) Definitions


(iv) Construction of the sections


(v)Fundamental properties


(vi) Transition to new diameter


(b) Other works—


(i) General


(ii) On the Cutting-off of a Ratio.


(iii) On the Cutting-off of an Area


(iv) On Determinate Section.


(v) On Tangencies


(vi) On Plane Loci


(vii) On Vergings


(viii) On the dodecahedron and the icosahedron


(ix) Principles of mathematics


(x) On the Cochlias


(xi) On unordered irrationals


(xii) Measurement of a circle


(xiii) Continued multiplications


(xiv) On the Burning Mirror


XX. Later Developments In Geometry—•


(a) Classification of curves


(b) Attempts to prove the parallel postulate—


(i) General


(ii) Posidonius and Geminus


(iii) Ptolemy


(iv) Proclus


(c) Isoperimetric figures


(d) Division of zodiac circle into 360 parts: Hypsicles


(e) Handbooks—


(i) Cleomedes


(ii) Theon of Smyrna


XXI. Trigonometry—


1. Hipparchus and Menelaus


2. Ptolemy—


(a) General


(b) Table of sines—


(i) Introduction


(ii) sin 18\degree\and sin 36\degree\


(iii) sin\sup\2\theta\+ cos\sup\2\theta\=1


(iv) Ptolemy's theorem


(v) sin (\theta\-\phi\) = sin\theta\cos\phi\-cos\theta\sin\phi\


(vi) Sin\sup\2 1/2\theta\= 1/2(l -COS\theta\).


(vii) cos (\theta\+\phi\) = cos\theta\cos\phi\-sin\theta\sin\phi\


(viii) Method of interpolation


(ix) The table


(c) Menelaus's theorem—


(i) Lemmas


(ii) The theorem


XXII. Mensuration : Heron Op Alexandria—


(a) Definitions


(b) Measurement of areas and volumes—


(i) Area of a triangle given the sides


(ii) Volume of a spire


(iii) Division of a circle


(iv) Measurement of an irregular area