Xll
CONTENTS
III. PYTHAGOREAN ARITHMETIC
PAGES 65-117
Numbers and tbe universe .....
Definitions of the unit and of number
Classification of numbers
‘ Perfect ’ and ‘ Friendly ’ numbers ....
Figured numbers
(in) Triangular numbers
(0) Square numbers and gnomons....
(7) History of the term 1 gnomon ’ . . .
(à) Gnomons of the polygonal numbers
(f) Right-angled triangles with sides in rational
number’s .......
(C) Oblong numbers
The theory of proportion and means
(«) Arithmetic, geometric and harmonic means .
(0) Seven other means distinguished
(7) Plato on geometric means between two squares 01
two cubes .......
O A theorem of Archytas .....
The ‘ irrational ’.......
Algebraic equations
(a) ‘Side-’ and ‘diameter-’ numbers, giving successive
approximations to */2 (solutions of 2x 1 - if— + 11
(0) The èircivôryia (‘ bloom ’) of Thymaridas .
(7) Area of rectangles in relation to perimeter (equation
xy=2x + y) .......
Systematic treatises on arithmetic (theory of numbers) .
Nicomachus, Introductio Arithmetica .
Sum of series of cube numbers . . . .
Theon of Smyrna .......
Iamblichus, Commentary on Nicomachus .
The pythmen and the rule of nine or seven
67-69
69- 70
70- 74
74-76
76-77
77
78- 79
79
79- 82
82-84
84- 90
85- 86
86- 89
89- 90
90
90- 91
91-98
94-96
96-97
97-115
97-112
108-110
112- 113
113- 115
115-117
IV. THE EARLIEST GREEK GEOMETRY. THALES. .118-140
The ‘ Summary ’ of Proclus ...... 118-121
Tradition as to the origin of geometry .... 121-122
Egyptian geometry, i.e. mensuration .... 122-128
The beginnings of Greek geometry, Thales ? . . 128-189
(a) Measurement of height of pyramid . . . . 129-180
(0) Geometrical theorems attributed to Thales . . 130-137
’ (7) Thales as astronomer . ..... 137-139
From Thales to Pythagoras ...... 139-140
V. PYTHAGOREAN GEOMETRY 141 169
Pythagoras 141-142
Discoveries attributed to the Pythagoreans
(а) Equality of sum of angles of any triangle to two
right angles ....... 148-144
(0) The ‘ Theorem of Pythagoras 1 . .... 144-149
(7) Application of areas and geometrical algebra (solu
tion of quadratic equations) .... 150-154
(б) The irrational ........ 154-157
(«) The five regular solids ...... 158-162
(C) Pythagorean astronomy 162-165
Recapitulation . . . . . . . . . 165-169
