The Curves of Life

Overview


Classic, well-thought-out examination of the function of the spiral, or helix, in both nature and art. Demonstrates how spiral is fundamental to structure of shells, leaves, horns, human body, drawings of Leonardo, Leaning Tower of Pisa, more. 1914 edition. 426 illustrations.
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Overview


Classic, well-thought-out examination of the function of the spiral, or helix, in both nature and art. Demonstrates how spiral is fundamental to structure of shells, leaves, horns, human body, drawings of Leonardo, Leaning Tower of Pisa, more. 1914 edition. 426 illustrations.
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Table of Contents

CHAPTER I INTRODUCTORY-THE SPIRAL
  Growth and Beauty and Spiral Formations
  Letters from Sir E. Ray Lankester and Dr. A. R. Wallace
  Measurement of Bones
  Nature not mathematically exact
  Gravity and Perfect Motion: Spirals and Perfect Growth
  "Spirals in Shells, Whirlwinds, Human Organs, Nebulæ, etc."
  "Classification, Utility, and Antiquity of Spirals"
  Need of Theory
CHAPTER II MATHEMATICAL DEFINITIONS
  Spiral Appearances subjective
  Flat Spirals
  Left Hand and Right Hand
  Conical and Cylindrical
  Ionic Volute drawn by means of a Shell
  Ways of making Spirals
  Curious Nomenclature used by Botanists
CHAPTER III UPRIGHT SPIRALS IN SHELLS
  Formation of Spirals in Shells
  Tube coiled round Axis
  Life History of a Series in One Shell
  Acceleration and Retardation
  Natural Selection
  Adjustment to Environment
  Survival and Spiral Variation
  Right-hand and Life-hand Shells
  Ammonite and Nautilus
  External and Internal Spirals
  Supporting the Central Column
  Comparison with Insects and Plants
  Multiple Spirals
CHAPTER IV FLAT SPIRALS IN SHELLS
  Nautilus and Logarithmic Spiral
  Equiangular Spiral a Manifestation of Energy
  Deviation from Cuve of Perfect Growth
  Leonardo da Vinci as Student of Shells
  Work of Professor Goodsir
  Varying Inversely as the Cube and the Square
  Significance of the Position of the Siphuncle
  Verticle and Horizontal Views of Shells and Plants
CHAPTER V BOTANY-THE MEANING OF SPIRAL LEAF ARRANGEMENTS
  Provision for Air and Sunlight
  Overlapping of Old Leaves by Young
  Advantages of Overlapping in Intense Glare
  Spiral Plan for Minimum Overlap
  The Ideal Angle
  Fibonacci Series
  Mr. A. H. Church on Logarithmic Spirals in Phyllotaxis
CHAPTER VI SPECIAL PHENOMENA IN CONNECTION WITH SPIRAL PHYLLOTAXIS
  The Sprial Theory of Schimper
  Growing Systems in place of Adult Construction
  A Logarithmic Spiral on a Plane Surface
  The Fibonacci Series again
  Radial Growth and Spiral Patterns
  A Standard for Comparison
  Examples of Different Systems
CHAPTER VII RIGHT-HAND AND LEFT-HAND SPIRAL GROWTH EFFECTS IN PLANTS
  Twist Effects:
  (i.) Spiral Leaf Arrangements
  (ii.) Overlapping Effects
  (iii.) Unequal Growth in Main Axis
  (iv.) Spiral Movement of Growing Ends
  (v.) Spiral Growth of Twining Plants
  (vi.) Spiral Effects after Death
  Nomenclature of Spirals
  Numerical Proportions of Right and Left Hand
CHAPTER VIII RIGHT-HAND AND LEFT-HAND SPIRAL GROWTH EFFECTS IN PLANTS (continued) DEAD TISSUES AND SPINNING SEEDS
  Spiral Twisting of Dead Tissues
  Coiling when Drying: Straightening when Wet
  Predominace of Right-hand Fibres
  Seed Spinning in Flight
  The Mechaanism of Winged Fruits
CHAPTER IX RIGHT-HAND AND LEFT-HAND SPIRAL GROWTH EFFECTS IN PLANS (continued) SOME SPECIAL CASES
  Anomalous Variation producing Spirals
  "Spiral Staircase" Construction"
  Peculiarities of Spiral Spermatozoids
  Male Cells of Cycads and Chinese Maidenhair Tree
  Spirals and Locomotion
  Prevalance of Right-hand and Left-hand Spirals
CHAPTER X RIGHT-HAND AND LEFT-HAND SPIRALS IN SHELLS
  Contrast of Trees with Shells
  Spiral Fossils in Nebraska
  Determination of Hand in Shells
  Different Hand in Fossils and Survivors of same Species
  Left-hand Spirals of Tusks
  "Sinistral Shell, but Dextral Animal"
  Shells among Primitive Peoples
  Following the Sun
  The Swastika
  Spiral Formation and the Prinicple of Life
CHAPTER XI CLIMBING PLANTS
  The Purpose of Climbing
  With and Without Tendrils
  Hand and Species
  Mr. G. A. B. Dewar on Climbers
  "Feeling" for Supports "
  Inheritance and Memory
  Circumnutating
  "Sense Organs" for Gravity and Light"
  The Statolith Theory
  Influence of Light and Moisture
  Effects of Climate Reversal of Spirals
CHAPTER XII THE SPIRALS OF HORNS
  Pairs of Horns
  Odd-toed and Even-toed Hoofed Animals
  The Angle of the Axis in Horns
  Suggested Geometrical Classification
  Distinctions between Horns of Wild Animals and Tame
  Homonymous Horns
  "Perversion" and Heteronymous Horns"
  "Comparison with other Spiral Growths, as of Plants and Shells"
  Exceptions to Dr. Wherry's Rule
  Tame Animals showing Twists of their Wild Ancestors
  Development or Degeneration?
  The Problem stated
CHAPTER XIII SPIRAL FORMATIONS IN THE HUMAN BODY
  Natural Objects do not consciously produce Spirals
  Deviation from Mechanical Accuracy
  Spiral Formations of Upper End of Thigh Bone
  Growth and Change
  Corresponding Structures in Birds and Mammals
  Conical Spiral of Cochlea
  "Spiral Formations: Umbilical Cord, Skin, Muscular Fibres of Heart, Tendo Achillis, The Humerus (Torsion), Ribs, Joints, Wings and Feathers, Eggs, Animalculæ"
CHAPTER XIV RIGHT AND LEFT-HANDED MEN
  Right and Left-handedness
  Legs and Arms of Babies
  Leonardo da Vinci
  Preference of Orientals for Left-hand Spirals
  Prehistoric Man generally Right-handed
  Skill of Left-handed Men: Examples from the Bible
  The Hand of Torques
  The Rule of the Road
  "Left-handed Sportsmen: Anglers, Archers, etc."
  Left-handed Artists
  More about Leonardo
  Letter from Mr. A. E. Crawley
CHAPTER XV ARTIFICIAL AND CONVENTIONAL SPIRALS
  Spiral Decoration in Prehistoric Times
  The Successive Races of Man
  Artistic Skill of Aurignacians
  Magdalenian Civilisation
  The Spiral as a Link between Aurignacians and Greeks
  The Mycenaean and Minoan Age
  Late Neolithic Ornamentation
  Distribution of Spirals in United Kingdom
  Scandinavia and Ireland
  Egyptian Spiral in Danish Celts
  Neolithic Stones and Etruscan Vases
  The Sacred Lotus
  "The "Unlucky" Swastika"
  Spirals in Greek Art
  Origin of the Volute
  Theory and Experiment
  The Iron Age
  Uncivilised Communities of the Present Day
  Mediæval Gothic
  Violin Heads
  "Cylindrical Spirals: Torques, Armlets, "Collars"
CHAPTER XVI THE DEVELOPMENT OF THE SPIRAL STAIRCASE
  Spiral Columns
  Rarity of Left-hand Spirals
  Right-handed Architects and Workmen
  Accidental Cause of a Twisted Spire
  Efficiency and Beauty
  Practical Origin of Spiral Staircases
  Gradual Evolution
  Central Support
  The Hand Rail
  Defence against Attack
  Double Spiral Staircases
CHAPTER XVII SPIRALS IN NATURE AND ART
  Shells and Spiral Staircases
  Practical Problems and Beauty of Design
  Efficiency and Beauty
  Leaning Companiles Intentionally Designed
  Charm of Irregularity
  The Parthenon
  Architecture and Life
  Quality of Variation in Greek Architecture
  Expression of Emotions
  Artistic Selection from Nature
CHAPTER XVIII THE OPEN STAIRCASE OF BLOIS
  The Staircase designed by Leonardo da Vinci
  Voluta Vespertilo
  The King's Architect
  A Left-handed Man
  Work of Italians in France
  Leonardo's Manuscripts
  His Theories of Art
CHAPTER XIX SOM
  "Dürer and the "Cavallo"
  Dürer's Mathematical Studies
  "Dante, Leonardo, Goethe"
  The Experimental Method
  "Beauty is "Fitness Expressed"
  The Value of Delicate Variations
  "Good Taste"
  Processes of Scientific Thought
CHAPTER XX FINAL RESULTS
  The Logarithmic Spiral as an Abstract Conception of Perfect Growth
  Spiral Nebulæ
  Dr. Johnstone Stone's Spiral of the Elements
  Infinite Series and the Rhythmic Beat
  Phyllotaxis
  The Ratio of Pheidias
  The F Spiral
  Space Proportion
  Art and Anatomy
  The Theory of Exceptions
  "Value of a "Law"
  Complexities of the Higher Organism
  "A Flame to Curiosity"
  The Methods of Science
APPENDIX
I. Nature and Mathematics (illustrated)
II. The F Progression. By William Schooling
III. Infinite Series and the Theory of Grouping
IV. Origins of a Symbol (illustrated)
V. The Spiral in Pavement-toothed Sharks and Rays (illustrated). By R. Lydekker
VI. The Spiral in Bivalve Shells (illustrated). By R. Lydekker
VII. The Shell of Travancore
VIII. The Growth of Shells (illustrated)
IX. The F Progression in Art and Anatomy (illustrated)
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