CORNELL UNIVERSITY LIBRARY Cornell University Library A short history of science, 3 1924 011 799 198 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924011799198 A SHORT HISTORY OF SCIENCE The whole of modern thought is steeped in science. . . . The greatest intellectual revolution mankind has yet seen is now slowly taking place by her agency. — HUXLET. The history of science familiarizes us with the ideas of evolution and the continuous transformation of hu- man things. . . It shows us that if the accomplish- ments of mankind as a whole are grand the contribu- tion of each is small. Sartow The history of science is the real history of mankind. — Dij Bois Retmond. The history of science . . . presents science as the constant pursuit of truth . . a growth to which each may contribute. . . . Science is international. — LiBBT. SHORT HISTORY OF SCIENCE BY W. T. SEDGWICK and H. W. TYLER Professor of Biology Professor of Mathematics at the Massachusetts Institute of Technology Cambridge The histo^'y z'; ecienoe sHotlc Ve " < - ■- the leading tlteead in the history of civilizatioa . = " .o » T . „ — Sarion. He&j garfe THE MACMILLAN COMPANY 1917 All rights reserved Q 6vy CoPVEldlTT, 1917, By the MACMILLAN COMPANY. Set up and clectrotyped. Published November, 1917. • .- '\ .••••• . .\,' ; J. 8. Cushing Co. — Berwick & Smitli Oo. Norwood, Mass., U.S.A. PREFACE This book is the outgrowth of a lecture course given by the authors for several years* to undergraduate classes of the Massa- chusetts Institute of Technology, the chief aims of the course being to furnish a broad general perspective of the evolution of science, to broaden and deepen the range of the students' interests and to encourage the practice of discriminating scientific reading. There are of course excellent treatises on the history of partic- ular sciences, but these are as a rule addressed to specialists, and concern themselves but little with the important relations of the sciences one to another or to the general progress of civilization. The present work aims to furnish the student and the general reader with a concise account of the origin of that scientific knowl- edge and that scientific method which, especially within the last century, have come to have so important a share in shaping the conditions and directing the activities of human life. The specialist in any branch of science is finding it more and more diiBcult to keep himself informed, even to the indispensable mini- mum extent, as to current progress in his own field, — and hence his frequent neglect of all other branches than his own. It may reasonably be expected that some attention to the his- tory of science on the part of students will give them a better understanding of the broad tendencies which have determined the general course of scientific progress, will enlarge their apprecia- tion of the work of successive generations, and tend to guard them against falling into those ancient pitfalls which have bordered the paths of progress. In the words of Mach : — There is no grander nor more intellectually elevating spectacle than that of the utterances of the fundamental investigators in their gigantic power. * By the senior author since 1889. V vi PREFACE Possessed as yet of no methods — for these were first created by their labors and are only rendered comprehensible to us by their performances — they grapple with and subjugate the object of their inquiry and imprint upon it the forms of conceptual thought. Those who know the entire course of the development of science will . . . judge more freely and more correctly the sig- nificance of any present scientific movement than those who, limited in their views to the age in which their own lives have been spent, contemplate merely the trend of intellectual events at the present moment. At a time when the forces of science are being diverted from the promotion and conservation of civilization to its destruction, and when attempts are being made to turn the waters now flowing in the stream of science back into ancient and so-called classical channels, it will be well for the general reader no less than the student of science to review its history, and to judge for himself concerning its proper place in contemporary life and education. Many volumes would be required to depict the lives of the workers, — often marked by self-denial and sometimes by persecution, — to trace the full significance of their achievements, or to portray the spirit animating their labors ; — that spirit of science to which, regarding it as a critic rather than a votary, impressive tribute has been paid by one of our modern seers : — A greater gain to the world . . than all the growth of scientific knowl- edge is the growth of the scientific spirit, with its courage and serenity, its disciplined conscience, its intellectual morality, its habitual response to any disclosure of the truth. — F. G. Peabody. It has naturally been foreign to the purpose of the authors to admit matter too technical for the general student or, on the other hand, too slight in its influence on the general progress of science. The division of responsibility between them corresponds roughly to that implied by the title "mathematical" and "natural sciences", and emphasis has been laid on interrelations rather than on distinctions between the various sciences. The mathe- matical group from their relatively greater age and higher de- velopment afford the best examples of maturity; the natural sciences illustrate more clearly recent progress. No attempt PREFACE vii has been made by the authors to follow an encyclopaedic plan, under which all fields should receive proportional space and treatment, each by a competent representative, but some fullness of presenta- tion has been aimed at in the particular branches with which they are themselves familiar, with briefer indication of developments along other lines. The authors gladly acknowledge their indebtedness to many men of science interested in their undertaking, and to the special histories already referred to, on which their own work is largely based. Many brief typical quotations from the more important authorities are given as a basis for wider or more special study, but no systematic attempt has been made to examine original sources. No one can possibly be more aware than are the authors of the shortcomings of their work, and corrections of errors, from which a book of this kind cannot hope to have escaped, will be "welcomed. Massachusetts Institute op TEcm^OLOGT, CAMBEmGE, 1917. TABLE OF CONTENTS CHAPTER I PAQB Early Civilizations 1 The Antiquity and Ancestry of Man — Archaeology — Prehis- toric Man — The Science of Mankind, Anthropology — Primi- tive Interpretations of Nature — Prevalence of Animism in Antiquity -/Sources of Information Concerning Prehistoric and Ancient Times — Some Ancient Lands and Peoples — Babylo- nia and Assjria -/ Egypt — Phoenicia — The Hebrews — The viilmergence of European Civilization — jEgean Civilization in the Bronze Age -;- The Iron Age ; The Greeks or Hellenes. CHAPTER II Eablt Mathematical Science in Babylonia and Egypt . . 20 Primitive Astronomical Notions — The Planets — Astrology and Cosmology — Primitive Counting — Primitive Geometry — - Relation of Greek to Older Civilizations — Babylonian Arith- metic — Babylonian Astronomy — Babylonian Geometry — Mathematical Science in Egypt — The Ahmes Papyrus — Egyp- tian Land Measurement — Egyptian Geometry. CHAPTER III The Beginnings of Science 35 Geographical Boundaries -¥- Indebtedness of Greece to Baby- lonia and Egypt — The Greek Point of View — Sources — The Calendar — Time Measurement — Greek Arithmetic — Greek Geometry — The Ionian Philosophers — Thales — Milesian Cos- mology — Anaximander — Anaximenes — Pythagoras and his School — Pythagorean Arithmetic — Pythagorean Geometry — Pythagorean Physical Science — Terrestrial Motion: Philolaus, Hieetas. CHAPTER IV Science in the Golden Age of Greece 58 Literature and Art — Parmenides — Empedocles — Anaxagoras — The Atomists — Democritus of Abdera — The Beginnings of TABLE OF CONTENTS PAGE Rational Medicine: Hippocrates of Cos — Tlie Sophists — Hip- pias of Elis — The Criticism of Zeno — Circle Measurement : Antiphon and Bryson, Hippocrates of Chios — Plato and the Academy — The Analytic Method — Platonic Cosmology — Arehytas — MensBchmus: Conic Sections — A New Cosmology: EudoxTis — Aristotle — Aristotle's Mechanics — Aristotelian Astronomy — Theophrastus — Epicurus and Epicureanism — Heraclides : Rotation of the Earth. CHAPTER V Greek Science in Alexandria 87 The Museum at Alexandria — Euclid — Euclid's Elements — Influence of Euchd — Criticism of Euclid — Other Works of Euclid — Archimedes — Archimedes and Euclid — Circle Meas- urement — Quadrature of the Parabola — Spirals — Sphere and Cylinder — Mechanics of Archimedes — Archimedes as an En- gineer — Alexandrian Geography ; Earth Measurement — Era- tosthenes — ApoUonius of Perga — Apollonius and Archimedes — ■ Medical Science at Alexandria ; Beginnings of Human Anatomy. CHAPTER VI The Decline of Alexandrian Science 115 Orbital Motion of the Earth : Aristarchus — Excentric Cir- cular Orbits — Epicycles — Hipparchus: Star Catalogue — Pre- cession of the Equinoxes — Other Astronomical Discoveries ; Planetary Theory — Invention of Trigonometry — Inventions : Ctesibus and Hero — Hero's Triangle Formula — Inductive Arithmetic : Nicomachus — Ptolemy and the Ptolemaic Sys- tem — The Almagest — Other Works of Ptolemy — Pappus — Beginnings of Algebra : Diophantus — Conclusion and Retrospect. CHAPTER VII The Roman World. The Dark Ages 141 The Roman World-Empire — The Roman Attitude towards Science — Roman Engineering and Architecture — Slave Labor in Antiquity — JuUus Caesar and the Julian Calendar — Vitru- vius on Architecture — Frontinus on the Waterworks of Rome — Roman Natural Science and Medicine — Lucretius — Strabo — Pliny the Elder — Galen — Late Roman Mathematical Sci- ence — Capella — Boethius — Science and the Early Christian Church — The Eastern Empire ; Edict of Justinian — The Dark Ages — The Establishment of Schools by Charlemagne. TABLE OF CONTENTS xi PAGE CHAPTER VIII Hindu and Ababian Science. The Moobs in Spain . . . 156 Alexandria — Hindu Mathematics — Hindu Astronomy — Mohammed and the Hegira — Arabian Mathematical Science — Arabian Astronomy — Asiatic Observatories — The Moors in Spain. CHAPTER IX Pbogress op Science to 1450 a.d 172 The Crusades — Trivium and Quadrivium ; Scholasticism — Medieval Universities — Transmission of Science through Moorish Spain — Dawn of the Renaissance — Mathematical Science in the Thirteenth Century — Roger Bacon — Dante Alighieri — Computation in the Middle Ages — ■ Mathematics in the Medieval Universities — The Renaissance — Humanism — Alchemy — The Mariner's Compass — Clocks — Wool and Silk ; Textiles in the Middle Ages — The Invention of Printing. CHAPTER X A New Astbonomt and the Beginnings op Modern Natural Science 191 The Age of Discovery — The Reformation — Pioneers of the New Astronomy — Conditions Necessary for Progress — Nico- laus Copernicus — De RevoluHonibus — Influence of Copernicus — Tycho Brahe — Uraniborg — Kepler — GaUleo — Medical and Chemical Sciences — Anatomy : VesaUus — Revival of Interest in Natural History. CHAPTER XI Progress op Mathematics and Mechanics in the Sixteenth Century 230 Aims and Tendencies of Mathematical Progress — Pacioli — Geometry in Art — Robert Recorde — Algebraic Equations of Higher Degree — Tartaglia, Cardan — ■ Symbolic Algebra : Vieta — Development of Trigonometry — ^ Map-making — The Grego- rian Calendar — A New Invention for Computation — "Two New Branches of Science " — A Pioneer in Mechanics ; Stevinus — Giordano Bruno. CHAPTER XII Natural and Physical Science in the Seventeenth Cbntuby 255 The Circulation of the Blood : Harvey — Atmospheric Pres- sure ; TorriceUi's Barometer — ■ Further Studies of the Atmos- xii TABLE OF CONTENTS phere; Gases — ^From Philosophy to Experimentation — From Alchemy to Chemistry — A False Theory of Combustion; Phlogiston — Beginnings of Organic Chemistry — Organization of the First Scientific Academies and Societies — The New Philosophy: Baoon and Descartes — ^ Progress of Natural and Physical Science in the Seventeenth Century. CHAPTER XIII Beginnings op Modern Mathematical Science .... 273 Mathematical Philosophy; Analytic Geometry: Descartes — Indivisibles: Cavalieri — Projective Geometry : Desargues — Theory of Numbers and Probability : Permat, Pascal — Me- chanics and Optics : Huygens — WaUis and Barrow — Isaac Newton — Optics — The Theory of Gravitation ; Principia — Newton's Mathematics; Fluxions — Leibnitz — HaUey : Pre- diction of Comets. CHAPTER XIV Natural and Physical Science in the Eighteenth Century 304 Chemistry ; Decline of the Phlogiston Theory — A New Chemistry : Priestley and Lavoisier — The Synthesis of Water — Beginnings of Modern Ideas of Sound — The Beginnings of Modern Ideas of Heat ; Latent and Specific Heat, Calorimetry — Eighteenth Century Researches on Light — Beginnings of Modern Ideas of Electricity and Magnetism — Beginnings of Modern Ideas of the Earth — Eighteenth Century Progress in Botany, Zoology, etc. — Progress in Comparative Anatomy and Physiology — The Industrial Revolution ; Inventions ; Power — Influence of Science upon the Spirit of the Eighteenth Century. CHAPTER XV Modern Tendencies in Mathematical Science .... 323 Mathematics and Mechanics in the Eighteenth Century — Progress in Theoretical Mechanics — Celestial Mechanics — The Perturbation Problem — The Nebular Hypothesis — Mod- ern Astronomy; Telescopic Discoveries — Mathematical Prog- ress and Physical Science — Nineteenth Century Mathematics — Non-Euclidean Geometry — Imaginary Numbers — The Dis- covery of Neptune — Cosmic Evolution — ^ Distance of the Stars — Mathematical Physics. TABLE OF CONTENTS xiii CHAPTER XVI PAGE Some Advances in Physical Science in the Nineteenth Cen- TUKY. Energy and the Conservation of Energy . . 348 Modern Physios — Heat, Thermometry: Carnot, Rumford — Light; Wave Theory, Velocity: Young, Fresnel — The Spec- troscope and Spectrum Analysis — Electricity and Magnetism : Faraday, Green, Ampere, Maxwell — Electromagnetic Theory of Light — Kinetic Theory of Gases: Clausius — The Concep- tion of Energy — Dissipation of Energy — Modern Chemistry — Chemical Laboratories : Liebig — Quantitative Relations ; Atoms, Molecules, Valence — Synthesis of Organic Substances — A Periodic Law among the Elements — Chemical Structure — Physical Chemistry; Electrolytic and Thermodynamic De- velopments of Chemistry. CHAPTER XVII Some Advances in Natural Science in the Nineteenth Cen- tury. Cosmogony and Evolution 366 Influence of Eighteenth Century Revolutions — The Scientific Revolution — Effects of the Rapid Increase of Knowledge — Gradual Appreciation of the Permanence and Scope of Natural Law — Natural Theology and an Age of Reason — Natural Phi- losophy and Natural History ; Differentiation and Hybridizing of the Sciences — Progress in Zoology — Progress in Botany — • Progress in Microscopy; the Achromatic Objective — ^ Embry- ology — Progress in Physiology : Johannes MiiUer ; Claude Ber- nard — Pathology before Pasteur — The Germ Theory of Fermentation, Putrefaction and Disease : Pasteur — Antiseptic and Aseptic Surgery : Lister — Rise of Bacteriology and Para- sitology — Biogenesis versus Spontaneous Generation — I*rog- ress of Geological Science — Glaciers and Glacial Theories — Rise of PalffiQntology — Ancient and Modern Theories of Cos- mogony — Relationship of the Heavens and the Earth — The Scale of Life and the Phases of Life — General Resemblance of Man to the Lower Animals — Anatomical and Microscopical Similarity of Animals and Plants ; Organs, Tissues, Cells and Protoplasms — Fundamental Unity of Nature ; Organic versus Inorganic World — Treviranus' Biology and Lamarck's Zoologi- cal Philosophy — Voyages and Explorations of Naturalists — ■ Darwin's Origin of Species — His Descent of Man — Decline of the Theory of Special Creation — Influence of an Age of Invention and Industry — Science in the Dawn of the Twen- tieth Century. XIV TABLE OF CONTENTS APPENDICES PAGE A. The Oath of Hippocrates (about 400 b.c.) . . .399 B. The Opus Majus of Roger Bacon (1267 a.d.). An Anal- ysis of the Sixth Part by J. H. Bridges . . . .400 C. Dedication of The Revolutions of the Heavenly Bodies by Nicolas Copernicus (1543) 407 D. WiUiam Harvey's Dedication of his Work on the Circu- lation of the Blood (1628) 412 E. Galileo before the Inquisition (1633) . . . 414 F. Preface to the Philosophies Naturalis Prindpia Mathe- matica, by Isaac Newton (1686) ...... 420 G. An Inquiry into the Causes and Effects of the Variolce Vaccina:, by Edward Jenner (1798) . ... 422 H. Principles of Geology, by Charles LyeU (1830) . . 429 I. Some Inventions op the Eighteenth and Nineteenth Centuries . ...... 438 Power ; Its Sources and Significance — Gunpowder, Nitro- glycerine, Dynamite — The Steam-Engine — The Spinning Jenny, the Water Frame, and the Mule — The Cotton Gin — Steam Transportation — The Achromatic Compound Micro- scope — Illuminating Gas — Friction Matches — The Sewing- Machine — Photography — Anaesthesia; The Ophthalmoscope — Indian Rubber — Electrical Apparatus ; Telegraph, Telephone, Electric Lighting, Electric Machinery — Food Preserving by Canning and Refrigeration — The Internal-Combustion Engine — Aniline — The Manufacture of Steel : Bessemer — Agricultu- ral Apparatus and Inventions — Applied Science; Engineering. A Table of Important Dates in the History of Science and Civilization . . ...... 449 A Short List of Books op Reference ..... 459 Index 469 ILLUSTRATIONS PAGE HeoatsBUs' Map of the World, 517 b.c. 34 Herodotus' Map of the Worl 1 57 Behaim's Globe, 1492 a.d 190 The Copernican System 198 Tycho Brahe's Quadrant opposite 204 Uraniborg opposite 206 Kepler opposite 210 Gahleo opposite 217 Galileo's Dialogue opposite 224 Stevinus' Triangle 252 Huygens opposite 286 Huygens' Clock opposite 288 Newton's Telescope and Newton's Theory of the Rainbow opposite 292 Sketch Map of Places Important in Ancient and Medieval Science opposite 448 A SHORT HISTORY OF SCIEJN^CE CHAPTER I EARLY CIVILIZATIONS 'The night of time far surpasseth the day' said Sir Thomas Browne ; and it is the task of Archaeology to hght up some parts of this long night. — Charles Eliot Norton. The Antiquity and Ancestry of Man. — It is now gen- erally agreed that men of some sort have been living upon this earth for many thousand years. It is also, though perhaps less generally, agreed that mankind has descended from the lower animals, precisely as the men of to-day have descended from men that lived and died ages ago. The history of science, however, is not so much concerned with the ancestry or origin of mankind as with its antiquity ; for while science is a comparatively recent achievement of the human race, its roots may be traced far back in practices and processes of pre- historic and primitive times. Mankind is very old, but science so far as we know had no existence before the beginning of history, i.e. about 6000 years ago, and until 2500 years ago it occurred if at all only in rudimentary form. The best opinion of to-day holds that man has been on this earth at least 250,000 years, and in spite of wide variations is of one zoological " kind " or " species " and three principal types or " races," viz., white or Caucasian, yellow or Mongolian, and black or Ethiopian (Negroid). These great races are believed to have had a common ancestry in a more primitive race, and this in turn to have descended from the lower animals. It is furthermore held that there was prob- 2 A SHORT fflSTORY OF SCIENCE ably one principal place of origin, or " cradle," of the human race from which have spread all known varieties of mankind, alive or extinct, and that this was probably in "Indo-Malaysia" in that remarkable valley which lies between the rivers Tigris and Euphrates and in its upper part is known as Mesopotamia (be- tween the rivers). Mesopotamia, or the broad valley of the Tigris and Euphrates, was the cradle of civihzation in the remotest antiquity. There can be little doubt that man evolved somewhere in southern Asia, possibly during the Pleiocene or Miocene times .... [And] as paleolithic man was certainly interglacial in Europe, we may assume that he was preglacial in Asia. . . . The earliest known civilization in the world arose north of the Persian GuK among the Sumerians .... but the Babylonians of history were a mixed people, for Semitic influences according to Winckler began to flow up the Euphrates Valley from Arabia during the fourth millennium B.C. This influence was more strongly felt, however, in Akkad than in Sumer, and it was in the north that the first Semitic Empire, that of Sargon the Elder (about 2500 B.C. according to E. Meyer) had its seat. . . . The supremacy of Babylon was first established by the Dynasty of Hamurabi (about 1950 B.C., earlier according to Winckler) which was overthrown by the Hittites about 1760 b.c. Then followed the Kassite dominion, which lasted from about 1760 to 1100 B.C. ... It was probably due to them that the horse, first introduced by the Aryans, became common in south- west Asia ; it was introduced into Babylon about 1900 B.C. but was unknown in Hamurabi's reign. — Haddon. Archeology. — The study of antiquity, and especially of prehistoric antiquity, is known as archaeology (the science of antiquities or beginnings), and is based upon finds of ruins, tools, weapons, caves, skeletons, carvings, ornaments, and similar remains or evidences of human life and action in pre- historic times. It has been well described as " unwritten history." Remains of all kinds have long been roughly but conveniently classified into three groups corresponding to three periods of development, viz. : a Stone Age, a Bronze Age, and an EARLY CIVILIZATIONS 3 Iron Age, according to the use of stone, bronze and iron implements. Prehistoric Man. — If therefore we would begin the history of science at the very beginning, we must turn far backward in imagination to a time when the human race was barely superior to the beasts that perish. Absorbed in a fierce struggle for exist- ence, the passing generations had little history and left behind them no permanent records. In one respect nevertheless mankind stood far above the beasts; namely, in possessing the power of language, by which they could not only communicate more readily one with another, but also convey to their descendants through oral tradition something of whatever they might possess of accu- mulated knowledge. Eventually, though slowly, the generations began to leave behind them more enduring records, — at first crude and fragmentary, in the form of tools, cairns, and other monuments, or in drawings, paintings, or carvings, on ivory or rocks or trees, or on the walls of caverns, — which should serve to inform or instruct other men. Finally, but still slowly, and especially out of this so-called "picture-writing," grew the art of writing, which furnished a means of keeping permanent records of the past and a new and more perfect way of communication between living men and races of men. We who have oiu-selves witnessed some of the consequences of improvements in the arts of communication between men and nations, such as have recently been effected by steam transportation and telegraphy and teleph- ony, can to some extent realize how much the introduction of the rudiments of the art of writing may have meant in the progress of prehistoric and primitive mankind. The Science of Mankind. Anthropology. — The various steps in the evolution of mankind and in the earliest development of civilization and the arts form the subject matter of one of the youngest of the sciences, anthropology, to works upon which the reader is referred who would pursue these matters further. One of the earliest and still one of the most interesting of these, Man's Place in Nature, by Huxley, is now a classic. Another, also somewhat out of date but still very valuable, entitled 4 A SHORT HISTORY OF SCIENCE " Anthropology," is of special interest because its author, E. B. Tylor, was the founder of the science and is still living (in 1916). ^ The Childhood of the Race. — There is reason to believe that the human race, in its long and slow development, has passed through periods of essential childhood and youth, very much as the individual human being passes slowly through infancy onwards ; and that, precisely as the individual begins his intellectual life in wonder, questioning, and curiosity, so the race has advanced from a condition of childish wonder, questionings, and interpretations of mankind and the external world, — sun, moon, and stars, thunder and lightning, wind, rain, and snow, — which have gradually developed into more mature and more scientific explanations. This principle of an essential parallelism between individual development and racial, named by Haeckel " the biogenetic law," will be found especially pertinent at many stages in the history of science. Primitive Interpretations of Nature. — As the child thinks he sees in almost everything some living agency, — because most of the things that happen about him are obviously connected with himself, or his parents, or his nurses, or other children, or with his pets, — so man in the childhood of the race and in its earlier development sees in the wind some hidden being or personality bending the tree, or shaking the leaves, or moaning or sighing in the forest, or roaring angrily in thunder. Only a slightly different imagination is required to see in the sun, moon, and planets super- natural beings or gods travelling across the heavens, and by asso- ciation, since they seem to visit his heavens daily or monthly or at other regular intervals, to believe that they are somehow con- cerned with himself and his welfare or destiny. From this primi- tive interpretation to the modern astronomical knowledge of the immensity, the movements and the paths, the temperatures, and 1 The latest edition of Sir John Lubbock's [Lord Avebury's] " Prehistoric Times " should also be consulted. Other easily accessible volumes are A. C. Haddon's " The Wanderings of Peoples " (Cambridge Manuals of Science and Literature) and J. L. Myres' "The Dawn of History" (Home University Library Series). The chapters on " Modern Savages " in Lord Avebury's " Prehistoric Times " are especially instructive. Most important of all is Professor H. F. Osborn's recent work, " Men of the Old Stone Age." EARLY CIVILIZATIONS 5 even the chemical composition, of those enormous lifeless masses which we call sun, moon, and stars, has been a long and laborious journey, — how long no one can tell. It is still almost always possible to find tribes or peoples somewhere on the earth living under one or more of the various conditions which the more highly developed peoples have apparently passed through, and there is no great difiBculty in finding primitive tribes to-day holding such childish interpretations of nature as we have just described. This circumstance enables anthropologists, ethnologists, and his- torians to draw with considerable confidence the broader outlines of the probable history of the more highly developed nations, such as those of western Europe and North America, — nations in the progress of which, since the beginning of the nineteenth century, science has played a notable part. The first stepping-stones towards scientific knowledge are wonder and curiosity, and peoples are still to be found so low in intelligence as to be almost destitute of curiosity. As a rule, however, most human beings, no matter how primitive, have some curiosity concerning, and some sort of explanation for, the commonest events, such as day and night, life, death, sickness, health, sun, moon, stars, winds, seasons, and the like. And one of the commonest, simplest, and probably most natural, is that already referred to as the childish or personal interpretation of nature; viz., that which assumes everything to be in a sense alive and possessed of some sort of being, animation, or personality, kindred to man's own. This primitive interpretation has been called animism. At present, however, the term animalism finds more favor among certain anthropologists, apparently for the reason that the notion of mere diffuse vitality, or general "animation," is even more primitive, as observed in certain peoples of low development, than is the idea of a specific "soul" (anima) differentiated from the body and possessing a separate existence. For example, a tree blown by the wind may seem to a man of very low development to be merely quivering with life, and bending before some more powerful but invisible influence, diffused, hazy, unembodied, and without personality or name 6 A SHORT HISTORY OF SCIENCE (animatism). Or it may seem to be an individual tree, bent by an invisible but powerful being like a man and perhaps having a name such as "Boreas" (the Greeks' name for the north wind). In this latter case we have the assumption of personality and, by analogy with man, of the presence and influence of a spirit or soul (animism). Prevalence of Animism in Antiquity. — Judging by the opinions and beliefs of races which still exist in very low stages of development, prehistoric man when he pondered at all, reasoned largely in the direction of animism. He in- terpreted himself and his actions by his own ideas, will, feelings, and desires, and reasoned that other things were actuated like- wise. If, for example, he killed an ox or a man by a blow, and later an ox or a man were killed by lightning, it was reasonable to assume that some invisible and manlike being had given the ox or man an invisible blow. The oldest records of the human race confirm this idea. The ancient Assyrians, Babylonians, and Egyptians "animated" much of what we today call inanimate, i.e. inorganic, nature ; and Greek and Hebrew poetry are full of survivals of this view of man and nature, which on the higher levels passes into personification and anthropomorphism. The establishment of a hierarchy of the gods of Greece, such as was supposed to dwell upon Mt. Olympus, is merely a further differ- entiation of the same kind. "The Hellenic gods and goddesses are glorified men and women." Sources of Information concerning Prehistoric and Ancient Times. — ^'These are of three kinds, tradition, monuments (including tools, implements, pottery, and other objects which have survived to the present time, more or less in their original form), and inscriptions. Of these tradition, because readily sub- ject to perversion, is the least reliable and need not be further considered. It is monuments, such as ruins, tombs, weapons, pottery, implements, ornaments, furniture, and the like, upon which we must chiefly depend for our knowledge of prehistoric times, and the evidence which has been gradually accumulated from finds of this sort is extensive and trustworthy and corre- EARLY CIVILIZATIONS 7 spondingly valuable. With the introduction of inscriptions of all sorts, including drawings, pictures, hieroglyphics, and writings of every kind, upon tablets, monuments, walls, caves, clay cylin- ders, papyri, parchments, and the like, from about the eighth or tenth century B.C., we enter upon the historical period. From that time forward we have more or less of the raw material from which we may reconstruct the beginnings, not only of civilization and art, but also of literature and science. Some Ancient Lands and Peoplps. — From the standpoint of European history, and especially the history of science, the most important peoples of antiquity were the Babylonians, Assyrians, Egyptians, and Phoenicians. The Babylonians and Assyrians occupied the fertile valley of the Tigris and Euphrates ; the Egyp- tians, that of the Nile ; and the Phoenicians the eastern slopes of the Mediterranean basin (modern Syria). The first three peoples were chiefly agricultural ; the last, chiefly seafaring, mercantile, and industrial. Babylonia and Assyria. — These, lying almost side by side, may be considered together, although Babylonia furnishes the older and the more important civilization. Babylon and Nineveh were the chief cities of the two countries, the former in Mesopo- tamia on the Euphrates, the latter above and to the northeast, and much nearer the mountains, on the Tigris. In that part of Asia which borders upon Africa, to the north of Arabia and the Persian Gulf, in an almost tropical region at the foot of the Armenian highlands, defended by mountains on the east and bounded by desert on the west, opens the broad valley of the Tigris and the Euphrates rivers which, flowing from the same mountains and in the same direction and maintaining for a long distance a parallel but independent com-se, join at last and fall together into the Persian Gulf. In the month of April these two rivers, swollen by the melted snows in the moimtains of Armenia, overflow, sinking again to the level of their beds in June. The country around them therefore was very similar to the Nile valley. A large number of canals joined the Tigris to the Euphrates, and distributed the water rendered by the tropical climate necessary for agriculture. 8 A SHORT HISTORY OF SCIENCE The upper part of the country inclosed between the two rivers was properly called Mesopotamia, a term used also roughly to desig- nate the whole. The valley of the Upper Tigris, or Upper Mesopo- tamia, was Assyria, and the lower part of both valleys Babylonia. . . . In these two fertile regions flourished two empires, the Chaldean- Babylonian and the Assyrian. The Chaldeans, says a trustworthy authority, appear to have been a branch of the great Hamite race of Akkad, which inhabited Baby- lonia from the earliest times. With this race originated the art of writing, the building of cities, the institution of a religious system, and the cultivation of all science, and of astronomy in particular. In the primitive Akkadian tongue were preserved all the scientific treatises known to the Babylonians. It was in fact the language of science in the East, as the Latin was in Europe during the Middle Ages. When Semitic tribes established an empire in Assyria in the thirteenth cen- tury B.C., they adopted the alphabet of the Akkad, and with certain modifications applied it to their own language. . . . The mythological, astronomical, and other scientific tablets found at Nineveh, are ex- clusively in the Akkadian language, and are thus shown to belong to a priestly class, exactly answering to the Chaldeans of profane history and of the Book of Daniel. . . . From about 747 B.C., the accession of Nabonassar, the line of kings at Babylon is supplied by the well-known work of Ptolemy, the geographer. . . . Babylon, according to ancient historians, was surrounded by walls over three hundred feet in height and eighty in thickness, and was divided into two parts by the river Euphrates, which flowed through it. Narrow streets led to the river, on which they opened by gates. Quays enclosed the water, and towards the centre a bridge crossed it, but the bridge was movable and was only used during the day. At night the two sides of the river were completely separated. . . . When, at the present time, we visit these formerly prosperous countries, we can scarcely believe in the universal fertility that so many witnesses have described. The carelessness of the Turkish administration has allowed the irri- gation canals to be silted up, and the inundations now form unhealthy swamps in the delta of the Tigris and Euphrates. Mesopotamia was wonderfully productive in wheat and barley, the enormous returns obtained by Bablyonian farmers from their corn-lands being un- exampled in modern times ; but it possessed neither olives, figs, nor EARLY CIVILIZATIONS 9 vines ; millet and sesame, however, grew luxuriantly. Date-palms abounded, and furnished a large part of the food of the inhabitants. The people of Assyria and Chaldea were as skilled in manual handicrafts as in the cultivation of the earth. They wove cloths of brilliant colors ; they also ornamented their garments with a pro- fusion of embroideries, and wore magnificent tiaras. Babylonian embroidery was celebrated even in the days of the Roman empire. The manufacture of carpets, one of the chief luxuries in the East, attained wonderful perfection at Babylon, as well as the manufacture of personal attire. Their furniture, by its richness and shape, dif- fered completely from anything we find in present use amongst Ori- entals ; the Assyrians used arm-chairs or sat on stools, and dined as we do from tables. The tables and chairs were handsomely decorated and in good taste, and it is curious to note that the same designs for ornamentation were in use then as we have now — lions' claws, animals' heads, etc. ; and even at the present time the ancient models might be studied with profit and copied with advantage. They were skilful in working hard as well as soft materials. The cylinders of jasper and crystal and the bas-reliefs of Khorsabad sculptured in gypsum or in basalt equally denote their proficiency. They were acquainted with glass and with various kinds of enamel, and they knew how to bake clay for the manufacture of bricks or of porcelain vases. Moreover, the art of varnishing earthenware and of cover- ing it with paintings by means of coloured enamel was well known at Nineveh. The cuneiform writing — so called because it is formed by pres- sure of the stylus on the soft surface of the clay tablets, producing a mark like a wedge or arrow-head — is a development of hieratic, itself an improvement on the primitive hieroglyphic. The hieratic characters had been scratched with the point of the stylus on the clay that served the Mesopotamian peoples for paper. The use of the stylus in cuneiform, gave a single element, by the employment of which in various combinations, all the letters of the alphabet were formed. When the Persians conquered Mesopotamia they published their decrees, etc., in the three chief dialects of their subjects — the Persian, Median, and Assyrian. Hence the trilingual inscriptions which have supplied the key to cuneiform interpretation. The dis- covery of the interpretation of the famous inscription at Behistun, on the Persian frontier, in three languages, Persian, Median, and 10 A SHORT HISTORY OF SCIENCE Assyrian, enabled Sir Henry Rawlinson to find the key to the Assyr- ian characters. . . It is very difficult, in spite of the numerous texts deciphered by modern savants, to form any idea of Assyrian literature; yet the literature must have been considerable, for Layard found a complete library founded by King Asshurbanipal in two of the rooms of his palace at Nineveh. This library consisted of square tablets of baked earth, with flat or slightly convex surface, on which the cuneiform writing had been impressed while the clay was soft, before baking. The characters were very clearly and sharply defined, but many of them so minute as to be read only with the help of a magnifying glass. These tablets, which are preserved at the British Museum, contain a kind of grammatical encyclopedia of the Assyrio-Babylonian lan- guage, divided into treatises ; and also fragments of laws, mythology, natural history, geography, etc. Treatises on arithmetic were also found in the library, proving that mathematical sciences were known, with catalogues of observations of the stars and planets. We have already mentioned that astronomy was greatly honored amongst the Chaldean priesthood, who had studied the course of the moon with so much precision that they were able to predict its eclipse. Science and literature developed, in spite of a primitive writing engraved upon clay tablets ; the art of sculpture was already highly refined ; monuments, which without being majestic like the Egyptian were imposing in their size and splendid in their colours ; rare ele- gance in clothing and furniture, denoting great wealth, the result of active commerce ; a cruel, even ferocious character, revealed by their treatment of prisoners, and indeed by all their history ; a learned caste, devoting themselves to the sciences and also to the unscientific methods of astrology ; a religion elevated by the primitive idea of a supreme god, yet degraded by polytheism and often by gross de- bauchery ; kings sufficiently intelligent to construct splendid palaces and immense cities, and yet inflated with pride and glorying in the most stupid cruelty — such is the picture opened to us by the records of Assyrian and Babylonian history. When we observe on the Assyrian bas-reliefs all the industries and all the arts, we are inclined to acknowledge that they were superior to the nations that surrounded them, and we understand how the Greeks drew inspiration from Assyrian work as well as from Egyptian. — Verschoyle. History of Civilization. EARLY CIVILIZATIONS 11 If, in a final summing up, the question be asked, What was the legacy which Babylonia and Assyria left to the world after an exist- ence of more than three millenniums, the answer would be, that through the spread of dominion the culture of the Euphrates Valley made its way throughout the greater part of the ancient world, leaving its im- press in military organization, in the government of people, in com- mercial usages, in the spread of certain popular rites such as the various forms of divination, in medical practices and in observation of the movements of heavenly bodies — albeit that medicine continued to be dependent upon the belief in demons as the source of physical ills, and astronomy remained in the service of astrology — and lastly in a certain attitude towards life which it is difEcult to define in words, but of which it may be said that, while it lays an undue em- phasis on might, is yet not without an appreciation of the deeper yearnings of humanity for the ultimate triumph of what is right. — Morris Jastrow, Jr. The Civilization of Babylonia and Assyria. Egypt. — Another highly important ancient civilization whose beginnings are lost for us in the darkness of prehistoric times is that which flourished in the valley of the Nile. Near the point where Africa approaches Asia lies a narrow valley, walled in by two ranges of mountains, enclosed on the farther side by two deserts, and fertilized by the periodical inundations of a mighty river. This long and narrow strip of verdure, surrounded by moun- tains and menaced by the desert sands, is Egypt. ... A .few years ago, the beginnings of Egyptian history, and even the source of the great river that fertilizes the land of Egypt, were hidden in mystery. The sources of the Nile have been at last discovered, and archaeo- logists have now retraced the commencement of a history which is practically the commencement of all authentic history. To Speke and Grant in 1862, and to Baker in 1864, we owe the knowledge of the lakes Victoria Nyanza and Albert Nyanza, whence come the abundant waters, that swollen by the equatorial rains, at fixed in- tervals overflow and fertilize with their mud the soil that borders their bed, and refresh a land which lies beneath a sky where a rain-cloud is seldom seen. We know, too, how the abundant harvests that regu- larly result from the inundations of the Nile, returning ample food to moderate labour, promoted the development of the Egyptian nation ; 12 A SHORT fflSTORY OF SCIENCE how the Nile itself supplied to them a highway for communication, rendered doubly useful by the north winds that blow up stream more than eight months of the year, carrying the traffic up into the interior while the ciu-rent carries it down; how the Arabian desert on the east and the Libyan on the west secured to them comparative immunity from invasion and opportunity for internal progress. . . . One of the prizes of Napoleon's expedition, a black basalt stone, disinterred at Rosetta in 1798, and now in the British Museum, bore three parallel and horizontal inscriptions, all quite distinct. One was in hieroglyphics, the second in the characters called demotic or popu- lar, the third in Greek. Although a great many scientific men ex- hausted their skill upon this trilingual inscription, which was a triple inscription of the same text, no one could make the Greek characters exactly apply to the hieroglyphic signs. Champollion was the first to obtain any success, as early as 1812, but further progress was largely aided by the labors of Thomas Young [a name associated in the history of science with those of Fresnel and Helmholtz]. . . . Protected from invasion by the same deserts that isolated them, the people who came from Asia and settled in the Nile valley applied themselves to the regulation of the periodical inundations, and to the distribution of the water. They built towns on the hillocks, in order that the water should not reach them ; and afterwards, with the stones that the two mountain ranges of Libya and Arabia contain in abundance, and by the means of transit afforded by the Nile, they erected monuments that have defied the course of the centuries. . . . The paintings in the tombs also show us men at work upon all the arts and all the handicrafts. 'We see there the workers in stone and in wood, the painters of sculpture and of architectm-e, of furni- ture and carpenters' work ; the quarrymen hewing blocks of stone ; all the operations of the potter's art; workmen kneading the earth with their feet, or with their hands ; men at work making stocks, oars and sculls; curriers, leather-dyers, and shoe-makers, spinners, cloth- weavers with various shaped looms, glass-makers; goldsmiths, jew- ellers, and blacksmiths.' Among the antiquities still in a state of good preservation there is much pottery, including vessels of simple earthenware and enamelled faience, enamelled and sculptured terra- cotta, glass, often resembling Venetian, metal work and jewellerN', and linen cloth as fine as Indian muslin. — Verschoyle. EARLY CIVILIZATIONS 13 Ph(enicia. — Two other civilizations of importance, the Phoenician and the Hebrew, existed in antiquity between the Mediterranean Sea and the great Arabian desert, in what are to-day called Syria and Palestine. By the side of the Hebrew nation, which owed its grandeur to its moral and religious development, dwelt the Phoenicians, a people who owed their fame to their maritime and commercial enterprise. They occupied a narrow strip of land between Lebanon and the Mediterranean, Phcenicia proper being but 28 miles long by one to five mUes broad, and the territory of the Phoenicians being, at the utmost, no more than 120 miles long by 20 wide. . . . The forests which clothed the chain of Lebanon supplied the Phcenicians with timber for their ships, and they soon made the Mediterranean a high road for their navy. Enclosed by mountains in a country that prevented their acquiring any inland empire, they became a maritime power, the first in the ancient world in order of importance as in order of time. Egyptian documents mention the Phoenician towns of Gebal, Beryta, Sidon, Sarepta, etc., as early as sixteen or seven- teen centuries before the Christian era. The Phoenicians served as middlemen to the great civilizations of the Nile and the Euphrates, their vessels easily coasting along to the mouth of the Nile, and their caravans having but a short journey to reach the point where the mid- dle Euphrates almost touches Upper Syria, whence the current would carry them down to the quays of Babylon. ... To the westward the Phoenicians sailed beyond the Mediterranean and ventured upon the Atlantic Ocean. They coasted the western side of Africa, and early accounts record their discoveries of wonderful islands of mar- vellous fertility and charming climate, the ' Fortunate Isles,' — probably Madeira and the Canaries. They also sailed along the coasts of Spain and Western France and reached Northern Europe. Gades (Cadiz) was the starting point for these long and dangerous voyages, which extended as far as Great Britain, where a considerable trade in tin was carried on. . . . The Phoenicians were the great mining people of the ancient world. Gold, silver, iron, tin, lead, cop- per, and cinnabar were obtained from Spain, still the chief metallif- erous country of Southern Europe. The details given by Diodorus concerning the Spanish mines are very circumstantial. 'The cop- per, gold, and silver mines are wonderfully productive,' and 'those 14 A SHORT HISTORY OF SCIENCE who work the copper mines draw from the rough ore one quarter of the weight in pure metal.' . . . The Phoenicians not only brought the mineral wealth of Spain to the Eastern world, but they had also a great trade in wheat, wine, oil, fruits of all kinds, and fine wool. They provided Asia with the products of Spain and Gaul, Sicily and Africa with the products of Asia. But this maritime commerce could only be supplied by an inland trade, which served to connect the countries that were a long distance from the sea. Phoenicia found itself one of the ports of Asia, the merchandise of distant countries was brought to it, and from it was exported all the produce of the Asian continent. The caravans supplemented the fleets, and the fleets distributed the burdens of the caravans. The land trade was chiefly in three directions — to the south it followed the route to Arabia and India ; to the east, that to Assyria and Babylon ; to the north, that to Armenia and the Caucasus. The Phoenicians were not only the great maritime, the great commercial, and the great mining power of antiquity, they were also one of the chief manufacturing powers. Like the Egyptians and Assyrians, they were skilful potters, and they discovered the art of making glass. 'It is said,' writes Pliny the elder, 'that some Phoeni- cian merchants, having landed on the shores of the river Belus, were preparing their meal, and not finding suitable stones for raising their saucepans, they used lumps of natron, contained in their cargo, for the purpose. When the natron was exposed to the action of the fire, it melted into the sand lying on the banks of the river, and they saw transparent streams of some unknown liquid trickling over the ground ; this was the origin of glass.' No matter how it may have originated, there is no doubt that the Phoenicians manufactured glass on a large scale, and their glass-work became celebrated all over the world. Dyeing works, however, take the first rank among Phoenician indus- tries, and Tyrian purple was one of the chief objects of luxury among the ancients. The word ' purple ' was not used only for a single colour, but for a particular kind of dye, for which animal coloiu-s obtained from the juice of certain shellfish were used. The dyeing works could not be carried on without cloths, for the Phoenicians dyed woollen materials chiefly with their famous purple. The wool came from Damascus, and the greater part of their export of woollen stuffs was doubtless of their own manufacture. Sidon was the first town that became noted for these fabrics. Homer often mentions tunics from EARLY CIVILIZATIONS 15 that town, but afterwards they were manufactured all over Phcenicia, and particularly at Tyre. Among the products of Phoenician in- dustry we must also mention the numerous ornaments and the articles whose value depends largely on their workmanship. The trade of barter which they had so long maintained with barbaric races, amongst whom these objects always find an appreciative market, had incited the Phoenicians to apply themselves to these industries. Chains of artistically worked gold were worn by Phoenician navigators in Homer's time, and Ezekiel mentions their curious work in ivory, which they procured through Assyria from India, and from Ethiopia. Accident has preserved the names of only a small number of the articles pro- duced by the Phoenicians, but the existence of these among a rich and luxurious people implies the existence of others. The Phoenician religion was a worship of personified forces of nature, especially of the male and female principles of reproduction. It was in a popular and simple form a worship of the sun, the moon, and the five planets, regarded as intelligent powers actively affecting human life. . . . And the Phoenician religion not only consecrated licentiousness, it also sanctioned cruelty. Living children were offered as burnt sacrifices to Baal as well as to Moloch. One can scarcely understand how human sacrifices could have been endured by an intelligent people ; but this abominable ritual was in force in all the colonies, and especially at Carthage, where during the siege of the city by Agathocles, about 307 B.C., two hundred boys of the best families were offered as burnt sacrifices to the planet Saturn. Though we have but few fragments of Phoenician antiquities and literature, we at least know their system of writing. It is now proved that the Phoenicians did not invent writing ; they merely communi- cated letters to the Greeks. . . . The Greeks adopted the Phoenician characters with only a few modifications; the Latin races used the same letters designed more simply; they had received them at a very remote date, for the Latin tongue was a sister not a daughter of the Greek. The French, Spanish, and Italian languages are all derived from the Latin and use the same characters, while even the Teutonic languages, like English and German, have adopted this alphabet. The Phoenicians must, on this groimd alone, take high rank in the history of civilization. . . . The Phoenicians were not only the pioneers of industry, but by theu- commerce they brought together the peoples of the three con- 16 A SHOET HISTORY OF SCIENCE tinents of the Old World. The first carriers by sea, acting as inter- mediate agents between the different nations, they exchanged ideas as well as merchandise; their exploration of different countries led to the discovery of new riches; they endowed the West with the products of the East, and the East with the products of the West. They proved to the world that cities can attain a high degree of pros- perity by labour, activity, and economy, and they remain examples of the highest development of purely commercial qualities. — Verschoyle. The Hebrews. — South of Phoenicia and lying between the Arabian Desert and the Mediterranean was Palestine : — The whole literature of the Hebrews is included in the collection of prose and poetry which we call the Bible, or, more accurately, the Old Testament. The simplicity of its narratives, the enthusiasm of its hymns, the joyful or plaintive melody of the psalms, the fiery eloquence of the prophets, place the Bible, independently of its re- ligious and historical importance, high among the great literary monuments of antiquity. Their literature is a proof that the poetic imagination was fully developed amongst the Hebrews, and that the people were deeply thoughtful as well as passionately religious. . . . The Hebrews were not an artistic or an industrial people ; but they possessed an indisputable superiority to all other nations of antiquity in their purely spiritual religion, and in their appreciation of the supreme importance of morals as the proper expression of religion. Religion was their rule of life, the maker of their laws, the pervading spirit of the whole community, as in no other nation before or since. — Verschoyle. The Emergence of Extropean Civilization. — Until very recently little was known of European events before the writings of Herodotus (484-425 B.C.). Within the last half century, how- ever, and largely as a result of the labors of the archseologists Schliemann and Evans, the existence of a wonderfully rich and complete prehistoric civilization has been revealed on the shores and islands of and near the Greek Peninsula. The recent discoveries in Crete have added a new horizon to European civilization. A new standpoint has been at the same time EARLY CIVILIZATIONS 17 obtained for surveying not only the ancient classical world of Greece and Rome, but also the modern world in which we live. — Sir Arthur Evans. iEoEAN Civilization in the Bronze Age. — The newer European civilization had also its prehistoric times, and the investigations and especially the excavations of the last half century have revealed such treasures as the site of Homeric Troy, the palace and tomb of Agamemnon, and the cities of Minos and others of the sea kings of Crete. It is now known that the Trojan War was fought about Hissarlik on the eastern shore of the Dar- danelles; that Agamenmon's palace was at Mycenae in Greek Argolis; and that Minos had his home and his naval base of Mediterranean sea power on the island of Crete. In Mycenae and in Crete the arts were highly developed. Painting, sculpture, and pottery, tools, weapons, implements of various kinds, with systems of water supply and drainage, testify to the remarkable degree of civilization attained in the later Bronze Age, although this has left behind it no written records and was formerly known to us only through the poems of Homer. Even to classical students twenty, nay, ten years ago, Crete was scarcely more than a land of legendary heroes and rationalized myths. It is true that the first reported aeronautical display was made by a youth of Cretan parentage, but in the absence of authenticated records of the time and circumstances of his flight, scholars were sceptical of his performance. And yet within less than ten short years we are faced by a revolution hardly more credible than this story; we are asked by archaeologists to carry ourselves back from a.d. 1910 to 1910 B.C., and witness a highly artistic people with palaces and treas- ures and letters, of whose existence we had not dreamed. . . . The theme is a fresh one, because nothing was known of the subject before 1900 ; it is important, because the Golden Age of Crete was the forerunner of the Golden Age of Greece, and hence of all our western cultiu-e. The connection between Minoan [Cretan] and Hellenic civilization is vital, and not one of locality alone, as is the tie between the prehistoric and the historic of America, but one of relationship. Egypt may have been foster-mother to classical Greece, but the mother, never forgotten by her child, was Crete. . . . c 18 A SHORT HISTORY OF SCIENCE Members of three foreign nations have worked in friendly rivalry to learn the buried history of Crete ... and Cretan soil may be said to have been found to teem with pre-Hellenic antiquities. The hopes of archeeologists have been abundantly justified. We have followed them and arrived at the home of the first European civili- zation. — Hawes. Crete, the Forerunner of Greece. Even at this exceedingly early stage of human progress, the va- rious branches of industry had become fairly separated and specialized, more so, perhaps, than in the Homeric period, and a considerable variety of tools was employed in the various crafts. The carpenter was evidently a highly skilled craftsman, and the tools which have survived show the variety of work which he undertook. At Knossos a carefully hewn tomb held, along with the body of the dead artificer, specimens of the tools of his trade — a bronze saw, adze, and chisel. ' A whole carpenter's kit lay concealed in a cranny of a Gournia house left behind in the owner's hurried flight when the town was attacked and burned. He used saws long and short, heavy chisels for stone and light for wood, awls, nails, files, and axes much battered by use ; and what is very important to note, they resemble in shape the tools of to-day so closely that they furnish one of the strongest links between the first great civilization of Europe and our own.' Such tools were, of course, of bronze. Probably the chief industry of the island was the manufacture and export of olive oil. The palace at Knossos has its Room of the Olive Press, and its conduit for convey- ing the product of the press to the place where it was to be stored for use ; and probably many of the great jars now in the magazines were used for the storage of this indispensable article. — Baikie. Sea Kings of Crete. The Iron Age. The Greeks or Hellenes. — Soon after the arrival of the Iron Age, and probably not far from 1200-1000 B.C., a new people became prominent on the shores of the iiEgean. These were the Greeks or, as they called themselves, Hellenes, — inhabitants of Greece or Hellas. Their precise origin is unknown, but they were undoubtedly of Indo-European stock and probably came, in part at least, from the north. It has been conjectured that their conquest of the existing inhabitants was facilitated by, if not due to, their possession of weapons of iron. Of the earlier EARLY CIVILIZATIONS 19 part of this new period (1200-800 B.C.) we have only the legendary accounts of the Homeric and Hesiodic poems, which are now gen- erally believed to be based upon, if not actually descriptive of, episodes of this age. The Hellenes soon supplanted the Phoenicians as traders in the southern Mgean ; and " if we now leave the monuments of the Egyptian temple or the Assyrian palace and turn to the pages of the Iliad and the Odyssey ... at once we are in the open air, and in the sunshine of a natural life. The human faculties have free play in word and deed. . . From the first the Greek is re- solved to confront the facts of life." — Jebb. Refeeences for Reading OsBORN, H. F. Men of the Old Stone Age. Lubbock, Sir John (Lord AvEBtrar). Prehistoric Times (7"" Edition). Mtres, J. L. The Dawn of History. Ttlor, E. B. Anthropology and Primitive Culture. Haddon, a. C. History of Anthropology. Jastrow, Morris, Jr. The Civilization of Babylonia and Assyria. Hawes, C. H. and H. Crete, the Forerunner of Greece. Spearing, H. G. The Childhood of Art. CHAPTER II EARLY MATHEMATICAL SCIENCE IN BABYLONIA AND EGYPT In most sciences one generation tears down what another has built and what one has estabhshed another destroys. In Mathematics alone each generation builds a new story to the old structure. — Hankel. A HISTORY of science may be based on some more or less definite logical system of definitions and classifications. As a matter of historical evolution, however, such systems and such points of view belong to relatively recent and mature periods. Science has grown without very much self-consciousness as to how it is itself defined, or any great concern as to the distinction between pure and applied science, or as to the boundaries between the different sciences. Mathematics, for example, has had its roots in the human need of exact statement as to both number and form in all sorts of affairs, and on the other hand in the analytical faculties of the human mind, which have shaped the development of the pure science and given it in course of time its deductive stamp. The origin of a science can seldom be precisely determined, and the more ancient the science the more difficult is the attainment of such precision. The periods at which primitive men of different races began to have conscious appreciation of the phenomena of nature, of number, magnitude, and geometric form, can never be known, nor the time at which their elementary notions began to be so classified and associated as to deserve the name of science. Very early in any civilization, however, mathematics must ob- viously have taken its rise in simple processes of counting and adding, of time measurement in primitive astronomy, of the geom- etry and arithmetic involved in land measurement and in archi- tectural design and construction. We can safely sketch certain rough outlines of the prehistoric picture, and we can to some extent 20 BABYLONIA AND EGYPT 21 verify these, on the one hand, by archaeological evidence, on the other, by present-day observations of backward races — still in their prehistoric stage. Primitive Astronomical Notions. — On the astronomical side the most obvious fact is the division of time into periods of light and darkness by the apparent revolution of the sun about the earth. With closer attention it must soon have been observed that the rela- tive length of day and night gradually changes, and that this change is attended by a wide range of remarkable phenomena. At the time of shortest days, vegetable and animal life (in the north temperate zone) is checked by severe cold. With the gradually lengthening days, however, snow and ice sooner or later disap- pear, vegetation is revived, birds return from the warmer south, all nature is quickened. In the symbolism of the beautiful old myth, the sleeping princess, our earth. Is aroused by the kiss of the sun-prince. The longest days and those which succeed them are a period of excessive heat and of luxuriant vegetation, followed by harvests as the days shorten, towards the completion of the great annual cycle. In time, closer observers, noting the stars, dis- covered that corresponding with this great periodic change are gradual variations in the starry hemisphere visible at night, that in other words the sun's place among the stars is progressively changing, that it is in fact describing a path completed in a large number of days, which after repeated counting is found to be 365. It is also found that the midday height of the sun above the southern horizon shares in the annual cycle. The determination of the number of days in the year is a matter of very gradual ap- proximation, possible only to men who have already attained some command of numbers and the habit of preserving records extending over a long series of years. For there is no well-marked beginning of the year as of the day. An erroneous determination of the number of days becomes apparent only after a number of years, increasing with the accuracy of the original approximation. If, for example, the year is assumed to be exactly 365 days, that is, about six hours too short, the festivals and other dates will slip back about 24 days in a century, and thus lose their original cor- 22 A SHORT HISTORY OF SCIENCE respondence with climatic conditions. A revision of the calendar will become necessary. Still another natural period is introduced by the motion of the moon, which seems like the sun to have a daily motion about the earth, and also to describe a closed path among the stars in a period of about 29 days. Unlike the sun, however, the moon has during this period a remarkable change of apparent shape and luminosity from " new " to " full " and back again. The study of the day, the year, the month, thus naturally determined by the great heavenly bodies has led to the development of the calendar with greater and greater accuracy, the most recent rectification of the length of the year dating only (in England) from 1752. The difficulty of expressing the precise length of the month and the year in days, causing the imperfection of early calendars, has, on the other hand, reacted to the advantage of mathematical as- tronomy by demanding the greatest possible precision both of observation and of the computation based upon it. The Planets. — Another celestial phenomenon, though less ob- vious than the foregoing, must have found wide recognition in pre- historic times. The stars vary widely in grouping and individual brilliancy, but in general their relative positions are sensibly constant. To this constancy, however, five exceptions are easily discovered in the wandering motion of the planets Mercury, Venus, Mars, Jupiter, and Saturn, which like sun and moon have their several paths among the stars but with seemingly irregular mo- tions. Corresponding to these seven bodies there was set up by prehistoric people an arbitrary division of time into weeks of seven days, "the most ancient monument of astronomical knowledge." The correspondence with the planets is still preserved in the names of the days of the week in several modern languages.^ The French Italian 1 Sunday dimanohe domenica (Sun) Monday- lundi lunedi (Moon) Tuesday mardi martedi (Mars) Wednesday mercredi mercoledi (Mercury) Thursday jeudi giovedi (Jupiter) Friday vendredi venerdi (Venus) Saturday samedi sabato (Saturn) BABYLONIA AND EGYPT 23 further division of time into hours, minutes, and seconds has followed more arbitrarily, and in connection with the develop- ment of progressively improved methods of time measurement. Astrology and Cosmology. — Side by side with the develop- ment of elementary astronomy on its observational and mathemati- cal sides were evolved in intimate connection with it, but sometimes in extraordinary imaginative forms, astrology and cosmology, dealing respectively with the supposed influence of the heavenly bodies on human afl^airs, and with the structure and organization of the world. Both these pseudo-sciences were inextricably blended, under priestly and literary influences, with a bewildering mass of superstition and mythology, legend and invention. In their earlier stages, both doubtless contributed powerfully to interest and progress in real science. Ultimately both have had to be torn away, as the scaffolding from a cathedral, in the never ending process of releasing truth from error. Primitive Counting. — On the arithmetical side the present counting processes of primitive peoples have particular interest. The distinction between one and two similar objects, and that between two and three or more, belong to a relatively early stage of development, but tribes are known to-day in which the entire number scale is one, two, many (i.e. more than two). The pro- cess of counting is naturally facilitated by the use of fingers and toes as counters, their number 10 being the well-known anatomi- cal basis for our denary or decimal number system. This may be illustrated by the following passages from E. B. Tylor's Primitive Culture : — Father Gilij, describing the arithmetic of the Tamanacs on the Orinoco, gives their numerals up to 4; when they come to 5, they express it by the word amgnaitone, which being translated means ' a whole hand'; 6 is expressed by a term which translates the proper gesture into words itacono amgnapona tevinitpe, 'one of the other hand,' and so on up to 9. Coming to 10, they give it in words as amgna aceponare, 'both hands.' To denote 11 they stretch out both the hands, and adding the foot they say puittorpona tevinitpe, 'one to the foot,' and so on up to 15, which is iptaitone, 'a whole 24 A SHORT HISTORY OF SCIENCE foot.' Next follows 16, 'one to the other foot,' and so on to 20, temn itoto, 'one Indian;' 21, itacono itoto jamgnar bona tevinitpe, 'one to the hands of the other Indian'; 40, acciache itoto, 'two In- dians,' and so on for 60, 80, 100, 'three, four, five Indians,' and be- yond if needful. South America is remarkably rich m such evidence of an early condition of finger-counting recorded in spoken language. The Zulu counting on his fingers begins in general with the little finger of his left hand. When he comes to 5, this he may call edesanta ' finish hand ; ' then he goes on to the thumb of the right hand, and so the word tatisitupa 'taking the thumb' becomes a numeral for 6. Then the verb hornha 'to point,' indicating the forefinger, or ' pointer,' makes the next numeral, 7. Thus, answering the question ' How much did your master give you ? ' a Zulu would say ' U kom- bile' 'He pointed with his forefinger' i.e. 'He gave me seven,' and this curious way of using the numeral verb is shown in such an ex- ample as ' amahashi akombile' 'the horses have pointed' i.e. 'there were seven of them.' In like manner, kijangalobili 'keep back two fingers,' i.e. 8, and kijangalolunje 'keep back one finger' i.e. 9, lead on to kumi, 10 ; at the completion of each ten the two hands with open fimgers are clapped together. The most instructive evidence I have found bearing on the forma- tion of numerals, other than digit-numerals, among the lower races, appears in the use on both sides of the globe of what may be called numeral-names for children. In Australia a well-marked case occurs. With all the poverty of the aboriginal languages in numerals, 3 being commonly used as meaning ' several or many,' the natives in the Ade- laide district have for a particular purpose gone far beyond this narrow limit, and possess what is to all intents a special numeral system, extend- ing perhaps to 9. They give fixed names to their children in order of age, which are set down as follows by Mr. Eyre : 1, Kertameru; 2, Warritya; 3, K-udnutya; 4, Monaitya; 5, Milaitya; 6, Marru- tya; 7, Wangutya; 8, Ngarlaitya; 9, Pouarna. These are the male names, from which the female differ in termination. They are given at birth, more distinctive appellations being soon afterwards chosen. The mathematical advantage of 12 as a base conveniently divisible has often been pointed out, but the choice unfortunately had to be made long before its real significance could possibly be apprehended, and the difiBculty of subsequent change would be BABYLONIA AND EGYPT 25 prohibitive. Vestiges of the use of 5 and of 20 are famlHar ; the former, for example, in the Roman numerals IV, VI, etc., the latter in such expressions as "three score and ten" and in the French qvMre-mngt. Increasing matrnty of a tribe or race, as of an in- dividual, is accompanied by gain in the command of larger and larger numbers, the rate of progress being very dependent, however, on a fortunate choice of notation. However great the capacity for inventing number-words, it soon becomes necessary to employ some system which shall lead to a regular development of higher from lower names. The selection of a point at which dependent names, and later, symbols shall begin, is one of the most important steps in the history of mathematics. It is difficult for us to realize the extent of our indebtedness to the comparatively recent so-called Arabic — or more properly, Hindu — notation, in which numbers of whatever magnitude may be expressed by means of only ten symbols. In any case, however, the appreciation of large numbers soon becomes vague. To most of us the word million is nearly equivalent to an innumerable multitude. Primitive Geometry. — On the geometrical side data are naturally more meagre. The notions of a primitive society in regard to areas and perimeters and the ratio of a circumference to its diameter may quite escape discovery. On the other hand skill in making and reading maps is well known — as among the Esquimaux. Relation of Greek to Older Civilizations. — Mathematical science seems to have first assumed- definite form in Greece, and it is of particular interest to study the indebtedness of the Greeks to the older civilizations referred to in the preceding chapter. Some degree of civilization doubtless existed further back than any records run, in China, in India, in Babylonia, and in Egypt. But of these only the latter two exerted a determining influence on the general evolution of European science, India making minor though funda- mental contributions at a much later stage. Babylonia and Egypt exchanged ideas with each other, and, after unnumbered centuries, furnished Greece with a certain nucleus of scientific knowledge of which the Greeks made enormous use. In practical engineering 26 A SHORT HISTORY OF SCIENCE the achievements of the older civihzations were marvellous, but for the creation of real science as systematized, organized knowl- edge, containing within itself the seeds of infinite growth, they were quite unequal. Babylonian Arithmetic. — In Babylonian arithmetic whole numbers were expressed in general by only three of the so-called cuneiform or wedge-shaped characters employed on the tablets, 1 = Y ; 10 = ~^» 100 = Y ^) but the numbers known to have been used run into the hundred thousands, this naturally im- plying a higldy developed command of the fundamental operations by means of which large numbers are made to depend upon smaller ones. The use of the words for thousand and ten thousand in characterizing an indefinite multitude is illustrated in many scriptural passages, for example : " Saul hath slain his thousands, and David his ten thousands" (1 Sam. xviii. 7); "a thousand thousands ministered unto him, and ten thousand times ten thou- sand stood before him " (Dan. vii. 10). With such expressions may be compared : " I will make thy seed as the dust of the earth " ; "He telleth the number of the stars; he calleth them all by their names" (Gen. xiii. 16; Ps. cxlvii. 4). The number 40 also plays a special role in such expressions as the "forty years in the wilderness," the "forty days and forty nights" of rain which caused the flood. Of remarkable interest in the Babylonian inscriptions is the oc- currence, side by side with a decimal system, of a number system based on 60, employed for mathematical and astronomical pur- poses. A table of squares of the natural numbers presents, for example, nothing novel for the first seven numbers, after which follow, however, the equivalent of 1 4 is the square of 8 1 21 is the square of 9 1 40 is the square of 10 2 1 is the square of 11 Just as in our notation, for example, 325 means three times the square of ten plus twice ten, plus five, so this table must mean : — BABYLONIA AND EGYPT 27 once sixty plus four once sixty plus twenty-one once sixty plus forty twice sixty plus one, etc., necessarily implying the representation of 60 by 1 in the second place. In a table of cubes, the perfect cube 4096 is represented similarly by 1 8 16, that is 1 x 60^ + 8 X 60 + 16 = 4096. The origin of this sexagesimal system has been ingeniously attributed to the blending of two civilizations, one possessing a system based on 10, the other a system based on 6, — a combination suggested by the command of the Persian king that the Ionian troops wait 60 days at the bridge over the Ister; by the splitting of the river by Cyrus into 360 rivulets, etc. Fractions were employed to a limited extent with denominators 60, and 3600 (= 60 X 60). The great step of completing the number system by a character for zero seems not to have been successfully made, though there are indications of an approach to it in later Babylonian times. There is evidence of a mystical or magical use of numbers. Each god, for example, was designated by a number from 1 to 60 ac- cording to his rank. A rational system of weights and measures was introduced, the unit of weight depending on that of length, as in the modern metric system. Babylonian Astronomy. — In connection with astronomical observations the Babylonians invented a method of measuring time by means of the water clock or clepsydra. From a vessel kept full, water was allowed to escape very slowly into a second vessel in which it could be weighed. To equal weights of water corresponded equal intervals of time. Starting the flow at the moment the upper edge of the sun first appeared in the east and stopping as soon as the whole sun was visible, the amount of water collected was compared with that escaping from sunrise to sunrise, and the sun's diameter thus determined as ri^ oi its whole path in the sky. The time 28 A SHOET HISTORY OF SCIENCE required for traversing the whole path — i.e. the day — was then divided into 12 double hours, in one of which the sun's disk advanced by its own diameter multiplied by 60. Their use of the number 60 as a base led also to the fin-ther subdivision of the hour into 60 minutes of 60 seconds each. The year was reckoned as 365 days, and even the unequal rate of the sun's motion at different periods was recognized. . Particularly noteworthy in connection with Chaldean astronomy is the discovery of a period of 6585 days, — a little more than 18 years, — for the recurrence of eclipses. This would appear to have been based on a long series of observations, but to have taken no account of the region of visibility of eclipses of the Sun. The periods of the planets in their orbits were approximately deter- mined, but there is no evidence of a systematic geometrical theory of celestial motions. As to accuracy of direct observation it is said that in later Baby- lonian times angles were measured to within 6 minutes and time to less than a minute. Quantities obtained indirectly by observa- tions extended over long periods, as the length of the lunar month, were naturally determined with correspondingly greater precision. A list of eclipses of the moon from 747 B.C. was known to Ptolemy, while an astrological work prepared about 3700 B.C. con- tains evidence of a long series of pre-existing observations. To the Romans the Chaldeans were known as star-gazers, and the art of augury or divination was much cultivated, making some of the earliest known use of geometrical forms. Herodotus ascribes the origin of the sun-dial to Babylonia. Babylonian Geometry. — In geometry the elementary use of the circle quickly leads to the discovery that a chord equal to the radius subtends one-sixth of the four right angles at the centre, and is thus one side of a regular inscribed hexagon, a figure found on Babylonian monuments. A failure to distinguish between the length of the arc and that of its chord led to the first approximation to the ratio of a circumference to its diameter, tt = 3, which occurs in the Old Testament where King Solomon's molten sea is said to be "ten cubits from the one brim to the other: it was round all BABYLONIA AND EGYPT 29 about, . . . and a line of thirty cubits did compass it round about." There is some evidence of a knowledge of the fact that a triangle of sides 3, 4, and 5 has a right angle, and the trisection of the right angle was accomplished. The circle was divided into 360 degrees. The sun-dial and its division into degrees are very clearly men- tioned in the books of Kings and Isaiah. Parallels, triangles, and quadrilaterals were used. We may summarize what we know as to the main features of Babylonian mathematical science as follows : — In astronomy, records of observations extending over many cen- turies, the determination of an 18-year eclipse period, the approximate determination of the year as 365 days, a good system of measuring time, the identification of Mercury, Venus, Mars, Jupiter, and Saturn as planets; In arithmetic, a well-developed sexagesimal system, tables of squares and cubes, arithmetic and geometric progressions, the use of large numbers; In geometry, the identification of the right triangle of sides 3, 4, and 5, the inscribed hexagon, the division of the circum- ference into 360 degrees, the crude approximation for the ratio of circumference to diameter, ir = 3. Mathematical Science in Egypt. — Josephus asserts that the Egyptians learned arithmetic from Abraham, who brought it with astronomy from Chaldea, and that the Egyptians in their turn taught the Greeks. The indebtedness of Greek science to Egyptians must at any rate have been very considerable. The pyramids are monumental evidence of appreciation of geometric form and of a relatively high development of engineering con- struction nearly 4000 years before the Christian era. Their builders must have had precise geometrical and astronomical notions. In nearly all of the pyramids, for example, the slope of the lateral faces is 52°, and the direction of their base- edges is nearly uniform. The regular inscribed hexagon was known. After an earlier year of 12 months of 30 days each, the 30 A SHORT HISTORY OF SCIENCE Egyptians added 5 days at the end of each year. According to their legend the god Thot won these days at play from the moon goddess. An edict of 238 B.C. introduced the leap-year, but the innovation was afterwards forgotten. The Egyptian records number more than 350 solar, and more than 800 lunar eclipses before the Alexandrian period. The Ahmes Papyrus. — Our most important source of in- formation in regard to early Egyptian mathematics is the so-called Ahmes manuscript, dating from some time between 1700 and 2000 B.C. " Direction for attaining knowledge of all dark things" are the opening words of this oldest known mathematical treatise. Rules follow for computing the capacity of barns and the area of fields. The text consists, however, rather of actual examples than of rules, the inferring of these being left to the reader. Reference is made to writings some 500 years older, presumably based in their turn on centuries of tradition. In the computations fractions are used as well as whole numbers, but fractions other than f are expressed in terms of fractions with unit numerators. The problem of decomposing other frac- tions into a limited number of such reciprocals is interestingly treated, examples occurring of considerable complexity. It would appear that such decompositions, effected by special de- vices or hit upon accidentally, were gradually tabulated as rec- ords of mathematical experiment. The problems discussed by Ahmes include a class equivalent to our algebraic equations of the first degree with one unknown quantity, — the first known appearance of this important idea. Thus, for example : — "Heap (or quantity) its |, its |, its }, its whole makes 33." In 2 a; fl- our notation -x -\ 1 |-a; = 33. The solution requires the number to be found which multiplying 1 + I + 2 + T shall produce 33. The result appears in the sufii- ciently intricate form BABYLONIA AND EGYPT 31 Again : " Rule for dividing 700 loaves among four persons, | for one, I for the second, | for the third, | for the fourth, . . . Add f , i, i, and I that gives 1 + | + |. Divide 1 by 1 + i + | that gives I + T^t. Make i + ^ of 700 that is 400." Thus, to modernize this solution, the four persons A, B, C, and D receive on one round 1 + | + | = | loaves ; the number of rounds is :; — -^ — —[ or - — — x 700 = 400, from which the respective ■■■+2+4 1+2+4 shares are readily obtained. Certain problems show an acquaintance with arithmetic and geometric progressions. Thus, for example, a series is given of the numbers 7, 49, 343, 2401, 16807, the successive powers of 7, accompanied by the words person, cat, mouse, barley, measure. Almost 4000 years later this was interpreted to mean : 7 persons have each 7 cats, each cat catches 7 mice, each mouse eats 7 stalks of barley, each stalk can yield 7 measures of grain ; what are the numbers and what is their siun ? Special symbols are used for addition, subtraction, and equality. The Egyptian seems never to have had a multiplication table. Multiplication by 13, for example, was accomplished by repeated doubling, and then by adding to the number itself, its products by 4 and by 8. Herodotus reports from the fifth century B.C. that the Egyptians reckoned with stones, a practise independently developed in many lands, notably in the form of the abacus. This little comput- ing machine of beads on wires was invented independently in different parts of the ancient world. In China and other parts of the Orient it is still widely and very skilfully employed. The handbook of Ahmes is also rich on the geometrical side. It contains information in regard to weights and measures, and treats of the conversion from one denomination into another. As in case of the progressions, geometrical problems are given, de- pending on the use of formulas not derived in the text itself. They include computation of areas of fields bounded either by straight lines or circular arcs, including in the former case only isosceles tri- angles, rectangles, and trapezoids. An isosceles triangle of base 4 32 A SHORT HISTORY OF SCIENCE and side 10 is said to have as its area f X 10 = 20, the actual area being of course f X VlOO- 4(=19.6 approximately). It is interesting that this and similar crude methods continued in use by surveyors for many centuries, even after EucHd had given geo- metrical science its modern form. Another problem amoimts to finding two squares having a given total area and their sides in a given ratio, being thus equivalent to solving the equations a;2 + 2/2 = 100 a: : 2/ = 1 : 1 By trial a; = 1, y = f , give x^ -V y^ = (f)^- Since 100 = {W X 8^, the trial values must be multiplied by 8, so that X = 8 and 2/ = 6. The classical problem of " squaring the circle" is attempted, the result being equivalent to the approximation tt = ^^ = 3.16, as against the actual 3.14 — an excellent result for the time. Other computations deal with the capacity of storehouses — of unknown shape — for grain. A remarkable group of problems deals with a certain geometrical ratio in pyramids equivalent to a modern cosine or cotangent, and of interest in connection with the uniform slope of the great pyramids. Egyptian Land Measurement. — Greek writers emphasize the methods of land measurement of the Egyptians consequent on the obliteration of boundaries by floods of the Nile. Herodotus relates that Sesostris had so divided the land among all Egyptians that each received a rectangle of the same size, and was taxed ac- cordingly. Whoever lost any of his land by the action of the river must report to the king, who would then send an overseer to meas- ure the loss, and make a proportionate abatement of the tax. Thus arose geometry {geometria = earth measurement). Dio- dorus, for example, says: "The Egyptians claim to have intro- duced alphabetical writing and the observation of the stars, like- wise the theorems of geometry, and most of the arts and sciences." The priests "occupy themselves busily with geometry and arith- metic, for as the river annually changes the land, it causes many controversies as to boundaries between neighbors. These cannot be easily adjusted unless a geometer ascertains the real facts BABYLONIA AND EGYPT 33 by direct measurement. Arithmetic serves them in domestic affairs and in connection with the theorems of geometry ; it is also of no shght advantage to those who occupy themselves with the stars. For if the position and motions of the stars have been care- fully observed by any people it is by the Egyptians ; they preserve records of particular observations for an incredibly long series of years. . . . The motions and times of revolution and stationary points of the planets, also the influence of each on the development of living things and all their good and evil influences have been very carefully observed by them." Egyptian Geometry. — In a passage written about 420 B.C., the Greek mathematician, Democritus, boasts that " In construct- ing lines according to given conditions no one has ever surpassed me, not even the so-called rope-stretchers of the Egyptians." The exact orientation of the Egyptian temples required the deter- mination of the meridian and of a right angle. Both processes were naturally an important part of the mathematical lore of the priesthood. The first step was accomplished by observation of the stars. It is believed that the second step was the function of the " rope-stretchers," the name being due to their dependence on a rope of length 12, divided by two knots into sections of 3, 4, and 5. When the two ends of the rope are joined and the three sections drawn taut by the knots, the angle opposite the section 5 is a right angle. The geometrical knowledge thus attributed to the Egyp- tians of a special case of the Pythagorean proposition does not, of course, imply knowledge of the proposition itself, or even the ability to prove the particular case, which was probably known only em- pirically. Egyptian architecture made use of geometrical figures as wall decoration and even employed the principle of propor- tionality, by dividing a blank wall-space into squares before apply- ing the design. The idea of perspective drawing seems, however, not to have been attained. The existence of such a problem book as that of Ahmes may be considered as fairly implying also the existence of comparable treatises of a more theoretical character, but other evidence of this is lacking. 34 A SHORT HISTORY OF SCIENCE The main features of Egyptian mathematical science are thee as follows : about 2000 B.C. a well-developed use of whole numbers and fractions ; a method of solving equations of the first degree with one unknown quantity; an approximate method for find- ing the circumference of a circle of given radius ; approximate methods for finding areas of isosceles triangles and trapezoids; the rudiments of a theory of similar figures. References for Reading Ball. A Short History of Mathematics, Chapter I. Cajori. A History of Mathematics, pages 1-15. Berry. A History of Astronomy, Chapter I. Dreyer. Planetary Systems, Introduction. Gow. History of Greek Mathe- matics, Chapters I, II. the ^ -- ■ .1. -f-t^ "v=^ K ^^ . u ■v=... V O ^v ^^ ,5....J .„. f ^^ f^^ ^"^- •■" - ;^-.."^^ / 1 f\ 64 ^i*N. / \ ^% ^ J >:fe4^=\ [ _1 f^^r^^^^ THRACE^ ARMENIA ..0^ 1 zp,nrr^l^t-fj^''i^j':=Sr-}r |^^^|■^-l-jBL JF p^ Carthafj^^^^M'-' V \ <^^^=it2r^^^ ^A\, /^ \-- \ r^ ^ x=f ^-^ '^ ^m ^-W/ / / %, ym ^^^ V — "N >■ V r ^-'^ ^ *i .^^"^ .■' / N rj — *>^ r ^■^^ .,..* y -w- ^ - ° p t „ « ^..■- ,v y Map of the World by Hecat^us (517 B.C.) (From Breasted's Anciml Times. Courtesy of Messrs. Ginn & Co.) wrora*g^ir°'ap'SvlT?hrworld^''l™tV*'r'^l!''"^ '?'^"=^ ""'"-^'"^ => i°"™«y "P ^e Nile, aDd The centre a?d?heknds\bout it 4ere all tlose known''? ^^h" ' Vi,*''" MediterrLean Sea was Historian of the Hebrews ,abrur8Io"B%1S^as^ZXst^rsto^;fe'a^°^,He.ofttfa?l°f^^^^^^^ — Breosied, CHAPTER III THE BEGINNINGS OF SCIENCE IN GREECE Except the blind forces of Nature nothing moves in this world which is not Greek in its origin. — Sir Henry Sumner Maine. A spirit breathed of old on Greece and gave birth to poets and thinkers. There remains in our classical education I know not what of the old Greek soul — something that makes us look ever upward. And this is more precious for the making of a man of science than the reading of many volumes of geometry. — Poincare. Number, the inducer of philosophies, The synthesis of letters. — J^!schylus. Mathematics, considered as a science, owes its origin to the idealistic needs of the Greek philosophers, and not as fable has it, to the practical demands of Egyptian economics. . . . Adam was no zoologist when he gave names to the beasts of the field, nor were the Egyptian sur- veyors mathematicians. — Hankel. Geogeaphical Boundaries. — From the twilight of civili- zation and the first faint suggestions of science in Chaldea and Egypt, we pass to the more brilliant dawn of science and civili- zation in Greece. Geographically we shall be concerned not merely with Greece itself, but, as time passes, with other Hellenic countries, especially the Ionian shores and islands of western Asia Minor, and the Greek colonies in southern Italy, Sicily, and, after its conquest by Alexander the Great, northern Egypt. Greece and its civilization seem immeasurably closer to us both in time and in spirit than do ancient Babylonia and Egypt. In these more remote civilizations science had been cultivated chiefly as a tool, either for immediate practical applications or as a part of the professional lore of a conservative priesthood. In Greece, on the other hand, for the first time in the history of our race, 35 36 A SHORT HISTORY OF SCIENCE human thought achieved freedom, and real science became pos- sible. Mathematics as a science commenced when first some one, prob- ably a Greek, proved propositions about any things or about some things, without specification of definite particular things. — White- head. Indebtedness of Greece to Babylonia and Egypt. — It is plain, nevertheless, that Greek civilization and Greek science owed much to Egypt and Chaldea. Herodotus has been quoted already, and Theon of Smyrna (second century a.d.) says : — In the study of the planetary movements the Egyptians had employed constructive methods and drawing, while the Chaldeans preferred to compute, and to these two nations the Greek astrono- mers owed the beginnings of their knowledge of the subject. Again in the third century a.d. Porphyry observes : — From antiquity the Egyptians have occupied themselves with geometry, the Phcenicians with numbers and reckoning, the Chal- deans with theorems. The Greek Point of View. — It is not, however, so much the achievements of the Greeks in positive science which compel our attention and admiration as it is the remarkable spirit which they displayed toward man and the universe. Here for the first time we meet with a new point of view, and while Shelley's well-known dictum, " We are all Greeks, our laws, our literature, our religion, our art have their roots in Greece," must be dismissed as in- correct as well as extravagant, and even Sir Henry Maine's maxim, which stands at the head of this chapter, is undoubtedly an exaggeration, these famous sayings serve well to illustrate the fact that with the Greeks came into the world a new spirit and a new interpretation of Nature. In a striking essay entitled "What we owe to Greece," Butcher has portrayed with extraordinary clearness those characteristics of the Greeks which lifted them above all of their predecessors and above most if not all of those that have come after them : BEGINNINGS IN GREECE 37 The Greeks before any other people of antiquity possessed the love of knowledge for its own sake. To see things as they really are, to discern their meaning and adjust their relations, was with them an instinct and a passion. Their method in science and philosophy might be very faulty and their conclusions often absurd, but they had that fearlessness of intellect which is the first condition of seeing truly. . . . Greece, first smitten with the passion for truth, had the courage to put faith in reason and in following its guidance to take no account of consequences. 'Those,' says Aristotle, 'who would rightly judge the truth must be arbitrators and not litigants.' 'Let us follow the argument wheresoever it leads' may be taken not only as the motto of the Platonic philosophy but as expressing one side of the Greek genius. . . . At the moment when Greece has come into the main current of the world's history, we find a quickened and stirring sense of per- sonality and a free people of intellectual imagination. The oppres- sive silence with which Nature and her unexplained forces had brooded over man is broken. Not that the Greek temper is irreverent or strips the xmiverse of mystery. The mystery is still there and felt . . . but the sense of mystery has not yet become mysticism. . . . Greek thinkers are not afraid lest they should be guilty of prying into hidden things of the gods. They hold frank companionship with thoughts that had paralyzed Eastern nations into dumbness or inactivity, and in their clear gaze there is no ignoble terror. . . . Know thyself, is the answer which the Greek ofl'ers to the sphinx's riddle. . . . But to the Greeks, 'know thyself meant not only to know man but the less pleasing task to know foreigners. . . . The people of ancient India did not care to venture beyond their mountain barriers and to know their neighbors. The Egyptians, though in certain branches of science they had made progress, — in medicine, in geometry, in as- tronomy, — had acquired no scientific distinction for they kept to themselves, but the Greeks were travellers. . . . Aristotle thought it worth his while to analyze and describe the constitutions of 58 states, including in his survey not only Greek states but those of the barbarian world. . It was the privilege of the Greeks to discover the sovereign efficacy of reason. . . . And it was Ionia that gave birth to the idea which was foreign to the East but has become the starting-point of modern science, the idea that Nature works by fixed laws. . . . Again, in 38 A SHORT HISTORY OF SCIENCE history the Greeks were the first who combined science and art, reason and imagination. . . The appHcation of a clear and fearless intel- lect to every domain of life was one of the services rendered by Greece to the world. It was connected with an awakening of the lay spirit. In the East the priests had generally held the keys of knowledge. . . To Greece then we owe the love of science, the love of art, the love of freedom. . . . And in this union we recognize the distinctive features of the West. The Greek genius is the European genius in its first and brightest bloom. Sources. — The sources of our information as to the details of the scientific ideas of the Greeks are exceedingly meagre, some of the most important historical and scientific treatises being known to us only by title or by detached quotations, or indirectly through Arabic translations. Among specific ancient sources of infor- mation in regard to Greek mathematical science the following may be mentioned : — About 330 B.C., Eudemus, a disciple of Aristotle, wrote a his- tory of geometry of which a summary by Proclus has been preserved. About 70 B.C., Geminus of Rhodes wrote an Arrangement of Mathematics with historical data. This has also been lost, but quotations are preserved in some of the later authors. About 140 A.D., Theon of Smyrna wrote Mathematical Rules necessary for the Study of Plato. About 300 A.D., Pappus' Collections contain much information in regard to the previous development of geometry. In the fifth century a.d., Proclus published a commentary on Euclid's Elements with valuable historical data. The Calendar. — The Greek calendar was based at an early period on the lunar month, the year consisting of 12 months of 30 days each. About 600 B.C. a correction was made by Solon, making every two years contain 13 months of 30 days and 12 of 29 days each, giving thus 369 days per year. In the following century a much closer approximation — 365 j days — was at- tained by confining the thirteenth month to three years out of eight. This arrangement naturally failed, however, to meet the BEGINNINGS IN GREECE 39 Greek desire that the months begin regularly at or near new moon, and Aristophanes makes the Moon complain: Chorus of Clouds " The Moon by us to you her greeting sends, But bids us say that she's an ill-used moon. And takes it much amiss that you should still Shuffle her days, and tm-n them topsy-turvy ; And that the gods (who know their feast-days well,) By yoiu false count are sent home supperless. And scold and storm at her for your neglect." About 400 B.C., Meton the Athenian observed that 19 years consist of almost exactly 235 lunar months, and accordingly pro- posed a new calendar with 125 months of 30 days and 110 of 29 days, corresponding to an average year of 365 days, 6 hours and 19 minutes — only about 30 minutes too long. Of this Meton's cycle the traditional rule for determining the date of Easter still preserves traces. On account of so much confusion in the official calendar the almanacs of the time even designated the dates for agricultural operations by means of the constellations visible at the corresponding time. Time Measurement. — While sun and moon suffice for large- scale measurement of time, the approximate determination of its subdivisions early became important, and this problem has been solved with continually increasing precision to our own day. Early time measurement depended either on some form of sun- dial as a natural means, or on an apparatus analogous to the hour-glass as an artificial method. In Isaiah xxxviii. 8, in connection with a promise of prolonged life to Hezekiah, it is said And this shall be a sign unto thee from the Lord, that the Lord will do this thing that he hath spoken ; behold, I will bring again the shadow of the degrees, which is gone down in the sun-dial of Ahaz, ten degrees backward. So the sun returned ten degrees, by which degrees it was gone down. 40 A SHORT HISTORY OF SCIENCE The first sun-dial of which a description is preserved belongs to the time of Alexander the Great, and consisted of a hollow hemi- sphere with its rim horizontal and a bead at the centre to cast the shadow. Curves drawn on the concave interior divided the period from sunrise to sunset into twelve parts, these lengths being thus proportionate to the lengths of the daylight period. The use of the clepsydra, or water clock, in Greece dates from the fifth century B.C. It consisted there of a spherical bottle with a minute outlet for the gradual escape of water. Its use in regulat- ing public speaking is illustrated by Demosthenes' demand when interrupted, "You there : stop the water." For the sake of conformity with the sun-dial division of each day and each night into twelve equal parts, the rate of flow in the clepsydra required continual adjustment. Ingenious improvements were made in the mechanism in course of time, but in considering the work of the Greek astronomers, the impossibility of what we should consider accurate time measiu'ement must not be for- gotten. Greek Arithmetic. — In Greek arithmetic the earliest known numerals are merely the initials of the respective number words. Two other systems came into use later. In one of these the numbers from 1 to 24 are represented by the 24 letters of the Ionian alphabet ; in the other the letters represent numbers, but no longer in consecutive order. This use of letters for numbers was not confined to Greece, but appears to have originated there. The Greeks had no zero, and never discovered the immense ad- vantage of a position-system, such as that by which we are able to express all numbers by only ten symbols. Fractions occur not infrequently. The change from the earlier notation to that with 24 characters was a disastrous one. There were not only more characters to memorize, but computation became materially more complicated. These disadvantages far more than offset the su- perior compactness, the sole merit of the new system. The special importance of such compactness for coins has led to the suggestion that they were the medium through which this nota- tion was introduced. BEGINNINGS IN GREECE 41 A simple numerical computation of late date in the Greek alphabetic numerals and its modern equivalent are I e 265 ^ e 265 S a 40 000, 12 000, 1000 M M ,0 ,a M 10 ,7X"^ 12 000, 3 600, 300 la T Jce 1 000, 300, 25 i 70 225 M a K e ■Gov). Division was an exceedingly laborious process of repeated sub- traction. Probably nothing in the modern world would have more aston- ished a Greek mathematician than to learn that, imder the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. — Whitehead. Approximate square roots were found by the later Greeks. Theon in the fourth centiu-y a.d. for example gives the following rule : — When we seek a square-root, we take first the root of the nearest square-number. We then double this and divide with it the re- mainder reduced to minutes and subtract the square of the quotient, then we reduce the remainder to seconds and divide by twice the degrees and minutes (of the whole quotient). We thus obtain nearly the root of the quadratic. The reckoning board, or abacus, — known in so many different forms throughout the world, — came into very early use, but actual evidence in regard to its form is meagre. A sharp dis- tinction was made between the art of calculation (logistica), and the science of numbers (arithmetica). The former was deemed unworthy the attention of philosophers, and to their attitude may be fairly attributed the fact that Greek mathematics was always 42 A SHORT HISTORY OF SCIENCE weak on the analytical side, and seemed in a few centuries to reach the limit of its possible development. Greek Geometry. — It was in geometry that Greek mathe- matics chiefly developed, and for several fundamental reasons. The Greek mind had a strong predilection for formal logic, a keen aesthetic appreciation of beauty of form, and, on the other hand, with no adequate symbolism for arithmetic or algebra, a distinct disdain, at any rate among the educated, for the commercialized mathematics of computation. The history of Greek mathematics is therefore to a great extent the history of geometry. Formal geometry as distinguished from the solving of particular geo- metrical problems, had, indeed, no previous existence, and we have to do with the beginnings of elementary geometry as we now know it. The Ionian Philosophers. — The sense of curiosity, the feel- ing of wonder, the spirit of inquiry, — these are the common ele- ments of philosophy and science. It is thus not strange that the earliest names in science are likewise the earliest in philosophy. In the childhood and youth of the race specialization has not begun, all knowledge lies invitingly open to the expanding mind. We have seen how much had been accumulated in Egj^t and Babylonia of knowledge and skill in observing and recording the phenomena of the heavens, in irrigation and in measiu-ement of land. Much of the same general character was doubtless true of the Phoenicians, the Trojans, the Cretans, and other precursors of the Greeks. But nothing deserving the name of science has come down to us from the Mgean or Greek civilization before the time of Thales of Miletus, chief of the Ionian philosophers, and one of the " seven wise men of Greece." Thales. — • The ancient and fragmentary register of Greek mathematicians, or history of Greek geometry before Euclid, attributed to Eudemus, begins : As it is now necessary to consider also the beginnings of the arts and sciences in the present period, we report that, according to the evidence of most, geometry was invented by the Egyptians, taking its origin from the measurement of land. This last was necessary BEGINNINGS IN GREECE 43 for them on account of the inundation of the Nile, which obliterated every man's boundaries. It is however, nothing wonderful that the invention of this as of the other sciences has grown out of necessity, as everything in its beginnings proceeds from the incomplete to the complete. A regular transition takes place from perception to thought- ful consideration, from this to rational knowledge. Just as now with the Phoenicians an exact knowledge of numbers took its rise in the needs of trade and commerce, so geometry began with the Egyptians for the reason mentioned. Thales, who went to Egypt, first brought this science into Greece. Much he discovered himself, of much however he transmitted the beginnings to his successors. Some things he made more general, some more comprehensible. The significance packed into this terse quotation may well be emphasized. The mathematics of the Chaldeans, the Egyptians, the Phoenicians, was merely a tool, crudely shaped to meet vital concrete needs; it had little possibility of development. The Greek intellect, seizing upon the fragmentary knowledge of these practical races, refined from it the germs of a new pure science, making the knowledge "more general" and "more comprehen- sible," and at the same time discovering much that was new. On the other hand, inclining in its zeal for pure science to the op- posite extreme of disregard for the concrete applications, Greek science eventually reached its own limit of possible growth. In the long run scientific progress must depend on due appreciation of the complementary importance of both pure and applied science. Thales was of Phoenician descent, and was born about 624 B.C. in Miletus, a city of Ionia, at that time a flourishing Greek colony in what is now Asia Minor. As an engineer he was employed to construct an embankment for the river Halys. As a merchant he dealt in salt and oil, and, visiting Egypt, learned there something of the wisdom of the Egyptian priesthood. He oc- cupied himself with the study of the stars as well as of geometry, and in particular, announced to the inhabitants of Miletus that night would enter upon the day, the sun hide himself, the moon place herself in front, so that his fight and radiance would be intercepted. 44 A SHORT HISTORY OF SCIENCE Herodotus says that there was a war between the Lydians and the Medes, and after various turns of fortune in the sixth year a conflict took place, and on the battle being joined, it happened that the day suddenly became night. And this change, Thales of Miletus had predicted to them, definitely naming this year, in which the event really took place. The Lydians and the Medes, when they saw the day turned into night, ceased from fighting, and both sides were desirous of peace. This eclipse is supposed to have taken place in 585 B.C. The prediction of the year of an eclipse gained Thales a great repu- tation with his contemporaries, though his designation with six others as "wise men of Greece" appears to have had a primarily political significance. None of the other six at any rate had any scientific standing. He taught that the year has 365 days ; that the equinoxes divide the year imequally; that the moon is illu- minated by the sun. The mathematical attainments attributed to Thales include the following theorems of elementary geometry ; the angles at the base of an isosceles triangle are equal ; when two straight lines cut each other the opposite angles are equal ; the first proof that the circle is bisected by its diameter; the in- scription of the right triangle in the semicircle ; the measurement of height by shadow, involving the principle of similar triangles. Plutarch relates that Niloxenus, conversing with Thales con- cerning King Amasis, says : — Although he also admires you on account of other things, he prizes above everything the measurement of the pyramids, in that you have without any trouble and without needing an instrument, merely placed your staff at the end of the shadow cast by the pyramid, showing from the two triangles formed by the contact of the solar rays that one shadow has the same relation to the other as the pyra- mid to the staff. Some writers even attribute to Thales a knowledge that the sum of the angles of a triangle is two right angles, also of the idea of a circle as a locus of a point having a certain property. BEGINNINGS IN GREECE 45 plied knowledge of similar triangles is doubtful. In connection with his shadow measiu-ements it is interesting that his scholar Anaximander, born 611 B.C., introduced the sun-dial into Greece. While our knowledge of Thales and his work is extremely meagre, the mathematical results above mentioned have consid- erable significance in connection with the comparison between Greece and Egypt. The Egyptian standpoint was fundamentally practical, specific, inductive; the Greek shows already its char- acteristic tendencies to abstract generalization, to logical proof, and to the methods of deductive science. Most of the facts as- cribed to Thales may well have been known to the Egyptians. For them these facts would have remained unrelated; for the Greeks they were the beginnings of an extraordinary develop- ment of the science of geometry. Milesian Cosmology. — The cosmological ideas of the Milesian philosophers were sufficiently ingenious and picturesque. To Thales the earth is a circular disk floating in an ocean of water. This water is the fundamental element of the whole. Ice, snow, and frost turn readily into water, even rocks wear away and disappear in it. Man himself seems capable of turning into it, while the waters of sea and land shrink into solid residues. By evaporation of the water air is formed, its agitation causes earth- quakes. The stars between their setting and rising pass behind the earth. The following passages (Fairbanks' translation) indicate the estimation in which Thales was held by later Greek philosophers. As to the quantity and form of this first principle or element, there is a difference of opinion ; but Thales, the founder of this sort of philosophy, says that it is water (accordingly he declares that the earth rests on water), getting the idea I suppose because he saw that the nourishment of all beings is moist, and that warmth itself is generated from moisture and persists in it (for that from which all things spring is the first principle of them) ; and getting the idea also from the fact that the germs of all beings are of a moist nature, while water is the first principle of the nature of what is moist. . . . 'Some say that the earth rests on water. I have ascertained that 46 A SHORT HISTORY OF SCIENCE the oldest statement of this character is the one credited to Thales, the Milesian, to the effect that it rests on water, floating like a piece of wood or something else of that sort.' . . . And Thales, according to what is related of him, seems to have regarded the soul as some- thing endowed with the power of motion, if indeed he said that the loadstone has a soul because it moves iron. . . Some say that soul is diffused throughout the whole universe ; and it may have been this which led Thales to think that all things are full of gods. — Aristotle. Of those who say that the first principle is one and movable, to whom Aristotle applies the distinctive name of physicists, some say that it is limited ; as for instance Thales of Miletos . . . who seems also to have lost belief in the gods. These say that the first principle is water, and they are led to this result by things that appear to the senses ; for warmth lives in moisture and dead things wither up and all germs are moist and all nutriment is moist .... Thales is the first to have set on foot the investigation of niitiu-e by the Greeks ; although so many others preceded him, he so fai surpassed them as to cause them to be forgotten. It is said that he left nothing in writing except a book entitled Nautical Astronomy. — Theo- phrastus. It is said that Thales of Miletos, one of the seven wise men, was the first to undertake the study of Physical Philosophy. He said that the beginning (the first principle) and the end of all things is water. All things acquire firmness as this solidifies, and again, as it melts, their existence is threatened ; to this are due earthquakes and whirlwinds and movements of the stars . . Thales was the first of the Greeks to devote himself to the study and investigation of the stars and was the originator of this branch of science ; on one occa- sion he was looking up at the heavens and was just saying he was in- tent on studying what was overhead, when he fell into a well ; where- upon a maid-servant named Thratta laughed at him and said : ' In his zeal for things in the sky he does not see what is at his feet.' And he lived in the time of Krcesos. — Hippolytus. Thales of Miletos regards the first principle and the element as the same thing. . . So we call earth, water, air, fire, elements. . Thales declared that the first principle of things is water. The Physicists, followers of Thales, all recognize that the void is really a void. The earth is one and spherical in form. It is in the midst of the imiverse. Thales and Democritus find in water the cause BEGINNINGS IN GREECE 47 of earthquakes. . . . Thales thinks that the Etesian winds blowing against Egypt raise the mass of the Nile, because its outflow is beaten back by the swelling of the sea which lies over its mouth. — JStius. Anaximander. — A second native of Miletus, Anaximander (about 611-545 B.C.) had a different interpretation of nature, holding that the fundamental stuff, out of which all things are made, is something between air and water. He believed the earth to be balanced in the centre of the world, because being in the centre and having the same relation to all parts of the circum- ference, it ought not to tend to fall in one direction rather than in any other. This point of view, not easily taken by the layman, illustrates the natural tendency of the Greek philosopher to em- phasize geometrical symmetry. Among those who say that the first principle is one and movable and infinite is Anaximander of MUetos, son of Praxiades, pupil and successor of Thales. He said that the first principle and element of all things is infinite, and he was the first to apply this word to the first principle ; and he says that it is neither water nor any other one of the things called elements, but the infinite is something of a dif- ferent nature from which came all the heavens and the worlds in them ; and from what source things arise, but that they return of necessity when they are destroyed. . . . Evidently when he sees the four elements changing into one another, he does not deem it right to make any one of these the underlying substance, but something else besides them. — Theophrastus. The earth is a heavenly body, controlled by no other power and keeping its position because it is the same distance from all things. The form of it is curved, cylindrical, like a stone column. It has two faces. One of these is the ground beneath our feet and the other is opposite to it. The stars are the circle of fire, separated from the fire about the world, and surrounded by air. There are certain breathing-holes like the holes of a flute through which we see the stars ; so that when the holes are stopped up there are eclipses. The moon is sometimes full and sometimes in other phases, as these holes are stopped up or open. The circle of the sun is 27 times that of the moon. . Man came into being from another animal, namely the fish, for at first he was like a fish. — Hippolytus (on Anaximander). 48 A SHORT HISTORY OF SCIENCE Anaximander, collecting data from the Ionian sailors frequent- ing Miletus, constructed a map of the earth, and speculated on the relative distances of the heavenly bodies. Herodotus relates that during the reign of Cleomenes, Aristagoras, prince of Miletus, ar- rived at Sparta; the Lacedsemonians affirm, that desiring to have a conference with their sovereign, he appeared before him with a tablet of brass in his hand, on which was inscribed every known part of the habitable world, the seas, and the rivers. Anaximenes. — A third Ionian Greek, often associated with those just mentioned, is Anaximenes (sixth century B.C.), like them a native of Miletus. For him the stars are fixed upon the celes- tial vault, and pass behind the northern (highest) part of the earth on setting. Air, not water, is the first cause of all things, the others being formed by its compression or rarefaction. The heat of the sun is due to its rapid motion, but the stars are too remote to give out heat. Anaximenes arrived at the conclusion that air is the one movable, infinite, first principle of all things. For he speaks as follows : ' Air is the nearest to an immaterial thing ; for since we are generated in the flow of air, it is necessary that it should be infinite and abundant, because it is never exhausted. ' (A fragment accredited to Anax- imenes.) Most of the earlier students of the heavenly bodies believed that the sun did not go underneath the earth but rather around the earth and this region, and that it disappeared from the view and produced night because the earth was so high toward the north. . . . Anaxim- enes and Anaxagoras and Democritus say that the breadth of the earth is the reason why it remains where it is. . . . Anaximenes says that the earth was wet, and when it dried it broke apart, and that earthquakes are due to the breaking and falling of hills. — Aristotle. The school of Thales and his successors in this Ionian outpost of Greek civilization was soon succeeded by developments of still greater importance in the more remote Italian colonies. BEGINNINGS IN GREECE 49 Pythagoras and his School. — The register of mathemati- ians proceeds : — " After these Pythagoras transformed the occu- ation with this branch into a true science, by considering the aundation of it from a higher standpoint, and investigated its heorems in a more abstract and intellectual way. It is he also ^ho invented the theory of the irrational and the construction f the cosmical bodies." These few words like those quoted of ?hales are full of meaning. The Egyptian priests knew eometrical facts, the raw material of mathematical science; ^'hales adapted this material to building purposes, Pythagoras legan the systematic foundations of the structure. Both in lame and in substance mathematics as a science begins with 'ythagoras. Pythagoras founded in the Greek cities of southern Italy a chool which had much of the character of a fraternity or secret ociety, this with political tendencies ultimately arousing hos- ility which proved destructive to it. Beyond these undisputed acts his life and work are obscured by a great mass of tradition ,nd myth, even the date of his birth being doubtful. A native if the island of Samos not far from Miletus, he appears _to h ave leen m uch affected by Egyptian influences during a residence in hat cou ntry. A visit to Babylon even is alleged, but with doubt- ul authority. The etiquette of the Pythagorean school required hat all discoveries should be attributed to the "Master" and LOt revealed to outsiders. To Pythagoras himself must probably le ascribed the so-called Pythagorean theorem, this forming the lecessary basis for the theory of the irrational mentioned in the egister. A similar inference may be drawn in regard to the reg- ilar polyhedra. On the other hand, Pythagoras appears to have Qterested himself in the theory of numbers, particularly in con- lection with music and geometry. He is said to have first in- roduced weights and measures among the Greeks. The attribution of particular results or beliefs to individuals of his period is however very doubtful on account of the fact that 'ythagoras left no writings whatever, that his school was es- entially a secret society, and that in later centuries it became 50 A SHORT HISTORY OF SCIENCE the custom to credit its founder with all sorts of knowledge which he could not possibly have possessed. Pythagoras makes the classification, arithmetic (numbers absolute), music (numbers applied), geometry (magnitudes at rest), astronomy (magnitudes in motion), this fourfold division or " quadrivium " continuing in vogue for some two thousand years. The distinction between abstract and concrete arithmetic had been emphasized among the Greeks in comparatively early times. Arithmetic and geometry were distinguished on one side from mechanics, astronomy, optics, surveying, music, and computation on the other. The aim of Greek arithmetic "was entirely differ- ent from that of the ordinary calculator, and it was natural that the philosopher who sought in numbers to find the plan on which the Creator worked, should begin to regard with contempt the merchant who wanted only to know how many sardines, at 10 for an obol, he could buy for a talent." The limited mathematics of the practical Egyptians had con- sisted of numerical cases. It was an easy step for Pythagoras to make number in a somewhat mystical sense the central element in his philosophy. Pythagorean Arithmetic. — In pure arithmetic or number theory as we should call it, the Pythagoreans enunciated such dicta as, for example, "Unity is the origin and beginning of all numbers but not itself a number." Prime and composite num- bers were also distinguished, and theorems of considerable alge- braic complexity discovered. There is naturally no algebraic symbolism, but "unknown" and "given" quantities are employed in the modern sense. Odd and even numbers received special names, and besides the series of squares and cubes and the arith- metic and geometric progressions previously known, • • other series were derived from these, for example, • • • the triangular numbers: 1, 3, 6, 10, 15, etc., by successive addition of the natural numbers. The reason for the name triangular will be clear if one counts the dots in the triangle formed by taking one, two, three or more rows beginning at the top of the figure. BEGINNINGS IN GREECE 51 11 9 7 6 3 1 The series of squares is formed by adding the odd numbers jccessively ; 1+3 = 4, 1+3+5 = 9, etc. The series 2, 6, 2, 20, 30, etc. is formed by adding the even umbers, or again by multiplying adjacent atural numbers. If we construct a series of juares or parallelograms with a common angle nd sides of length 1, 2, 3, 4, 5, etc. the figure 'hich must be added to any one to produce le next larger was called by the Greeks a nomon, the area of which would be repre- ;nted by one of the series of odd numbers, — an interesting and ^^pical example of the Greek habit of combining geometry with umber-theory. As products of two numbers were associated with reas — "square" or "oblong" — so products of three factors ere interpreted as volumes. A later Pythagorean calls the cube le " geometrical harmony " — an expression embodying the as- Dciation of mathematics with music. The cube has indeed 6 ices, 8 vertices, 12 edges ; 6, 8, and 12 are in harmonic progres- on, that is, 8 is the harmonic mean between 6 and 12. .Pythagorean Geometry. — In geometry the Pythagoreans )rmulated definitions of the fundamental elements, line, sur- ice, angle, etc. They are credited with a number of theorems epending on the application of one surface to another,^ and im- lying a knowledge of methods of determining area and of the roperties of parallel lines. They developed a fairly complete leory of the triangle, including the fundamental proof that the im of the angles of a triangle is two right angles, by a method ot very different from our own. The theory of the "cosmical bodies" mentioned in the register is of special in- terest. Any solid angle must have at least three faces. If three equal equilateral triangles have a common vertex they will when cut or folded so that their edges are brought together, form a solid igle, and a fourth equal triangle will complete a regular tet- ihedron. Similarly, if we start with four triangles, we may 1 Some of these are equivalent to the solution of the quadratic equation. 52 A SHORT HISTORY OF SCIENCE build up with four others a regular octahedron, or starting with five, an icosahedron with 20 faces. Six triangles, however, will fill the angular space about a point, and thus not permit the formation of a regular polyhedron. Using squares instead of triangles, we obtain only the cube; using pentagons (angle 108°), the regular dodecahedron — 12 faces, 3 at each vertex. The Egyptians must have been familiar with the cube, the regular tetrahedron, and the octahedron. To these, with the icosahedron, the Pythagoreans as- sociated the four cosmical elements — earth, air, fire, and water. Their discovery of an additional body, the regular dodecahedron, formed by 12 pentagons, made a break in the correspondence, and the need was met by the addition of the universe, or, according to others, the ether, as a fifth term in the cosmical series. This correspondence was not merely symbolical, but physical, the earth being supposed to consist of cubical particles, etc. We cannot infer that the impossibility of a sixth regular polyhedron was known. That only these five regular polyhedra are pos- sible was in fact first proved by Euclid. There is a tradition that the Pythagorean discoverer of the dodecahedron was drowned at sea on account of the sacrilege of announcing his discovery publicly. A later commentator records a similar tradition that the discoverer of the irrational perished by shipwreck, since the inexpressible should remain forever con- cealed, and that he who touched and opened up this picture of life was transported to the place of creation and there washed in eternal floods. The regular polygons were naturally studied, and in particu- lar the decomposition of them into right triangles of 45° and 30° -60°. With the pentagon the attempt naturally failed, but the five-pointed star formed by drawing diagonals was a special emblem of the Pythagoreans. With the inscribed pentagon connects itself naturally the division of a line in extreme and mean ratio, or, as it was later characterized, the "golden sec- tion." This division, by which the square on the greater segment of a line is equivalent to the rectangle whose sides are the other segment and the whole line, occurs repeatedly in Greek archi- BEGINNINGS IN GREECE 53 ecture of the fifth century, with fine effect, and must have been ystematically employed. As to the celebrated theorem which bears the name of Pythag- )ras, he may well have learned from the Egyptian rope-stretchers ;hat a right angle is formed by taking sides of lengths 3 and 4 and leparating the other ends a distance 5, while his study of numbers vould easily have led to the discovery that in the series of squares ;he adjacent 9 and 16 make 25. It would naturally be investi- gated whether a similar relation could be verified for other right ;riangles. In the most familiar case of the isosceles right tri- mgle it soon appears that the length of the equal sides being taken IS 1, the length of the hypotenuse could be only approximately expressed. It cannot indeed be exactly expressed by any whole lumber, or fraction; it is irrational. If it is true as Whewell says, that the essence of the triumphs of science and its progress consists in that it enables us to consider evident ind necessary, views which our ancestors held to be unintelligible and ifeve unable to comprehend, then the extension of the number concept :o include the irrational, and we will at once add, the imaginary, s the greatest forward step which pure mathematics has ever taken. — Hankel. In this case the proof of the Pythagorean theorem is easily effected by a simple graphical construction, involving merely the drawing of diagonals of squares. The smaller triangles in the figure are evidently all equal. The larger square contains fom- of them, the smaller squares, two each. It seems possible that this was the Pythagorean method, but as to how the proof was accomplished in other cases we have no information, the simpler proof of Euclid having com- pletely superseded the earlier. On the other hand, for the cor- responding arithmetical problem of finding three whole numbers which can be the sides of a right triangle, Pythagoras is said to liave given a correct solution, equivalent in our notation to (2a + 1)2 + (2a2 + 2a)2 = i2a'' +2a + l)^ 54 A SHORT HISTORY OF SCIENCE a denoting any positive integer. How this method was discovered remains a matter of conjecture. We may recognize here the characteristic elements of the in- ductive method, first, observation of the particular fact that in a certain right triangle, with sides 3, 4, and 5, the sum of the squares on the two sides is equal to that on the hypotenuse ; second, the formation of the hypothesis that this may be true also for right triangles in general; third, the verification of the hypothesis in other particular cases. Then follows the deductive confirmation of the hypothesis as a law for all right triangles. Pythagorean Physical Science. — It has been already noted that one of the most fundamental principles of the Pythagorean school was the significance attached to number in connection with all sorts of phenomena, the regular motions of the heavenly bodies, the musical tones, etc. There is a tradition that Pythag- oras, walking one day, meditating on the means of measuring musical notes, happened to pass near a blacksmith's shop, and had his attention arrested by hearing the hammers as they struck the anvil produce sounds which had a musical relation to each other. It was found that vibrating cords emitted tones de- pendent in a simple way on their length ; for example, cords of lengths 2, 3, and 4 giving a tone, its fifth and its octave re- spectively. The monochord used in studying these numerical relations is said to have been the first apparatus of experimental physics. It was even supposed that each of the various heavenly bodies and the sphere of the fixed stars had a char- acteristic tone, these all uniting to produce the so-called " music of the spheres." Terrestrial Motion ; Philolaus, Hicetas. — The universe was believed to consist of the four elements, — earth, air, fire, water, — to be a sphere with a spherical earth at its centre, and to have life. Pythagoras identified the morning and evening stars, and attributed the moon's light to reflection. It is of peculiar interest that later Pythagoreans, in particular Philolaus, about 400 B.C., attributed the apparent daily motion of the heavenly bodies from east to west not to their own actual motion but to a motion of BEGINNINGS IN GREECE 55 the earth in the opposite direction. This latter motion, however, was thought of, not as a rotation, but as an orbital motion about a so-called "central fire." Just as the moon revolved about the earth, always turning the same face towards the latter, so the earth might revolve about the central fire which would be forever invisible to the inhabitants of the other side of the earth. While we say that the moon rotates about its axis in the same time in which it revolves about the earth, to the ancients such a motion was not considered to include rotation at all. A further essen- tially arbitrary assumption introduced between the earth and the central fire a counter-earth (antichthon) , which was required to make up the supposed number of the heavenly bodies, and which would hide the central fire from dwellers in the antipodes. Aristotle, criticising this theory, says of the Pythagoreans : — They do not with regard to the phenomena seek for their reasons and causes, but forcibly make the phenomena fit their opinions and preconceived notions. . . . When they anywhere find a gap in the numerical ratios of things, they fill it up in order to complete the sys- tem. As ten is a perfect number and is supposed to comprise the whole nature of numbers, they maintain that there must be ten bodies moving in the universe, and as only nine are visible, they make the antichthon the tenth. All the other heavenly bodies describe orbits, each in its own hollow sphere about the central fire, the generally adopted order, based on the apparent rate of motion among the stars, being Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn. Pythagorean speculations as to relative distances of the different planets were naturally mystical notions merely. The sun was said to move around the central fire in an " oblique circle," i.e. the ecliptic. The moon was believed to be inhabited by plants and animals. The moon might be eclipsed either by the earth or by the counter-earth. This remarkable system, admitting the earth to move and not to be the centre of the universe, was not generally or long accepted, but had a share in securing the acceptance of the theories of Copernicus nearly 2000 years later. One at least 56 A SHORT HISTORY OF SCIENCE of the Pythagoreans made the great further step, somewhat loosely described by Cicero in the words : — Hicetas of Syracuse, according to Theophrastus, believes that the heavens, the sun, moon, stars, and all heavenly bodies are standing still, and that nothing in the universe is moving except the earth, which, while it turns and twists itself with the greatest velocity round its axis, produces all the same phenomena as if the heavens were moved and the earth were standing still. The activity of the Pythagorean school continued to be im- portant until about 400 B.C., that is, until the rise of the Athenian school under Plato and his successors. It had not only created the science of mathematics ; it had developed, however vaguely and imperfectly, the idea of a world of physical phenomena governed by mathematical laws. Dr. AUman says of Pythagoras : — In establishing the existence of the regular solids he showed his deductive power; in investigating the elementary laws of sound he proved his capacity for induction; and in combining arithmetic with geometry ... he gave an instance of his philosophic power. These services, though great, do not form, however, the chief title of the Sage to the gratitude of mankind. He resolved that the knowl- edge which he had acquired with so great labom-, and the doctrine which he had taken such pains to elaborate, should not be lost; and . . . devoted himself to the formation of a society d'elite, which would be fit for the reception and transmission of his science and philosophy ; and thus became one of the chief benefactors of human- ity, and earned the gratitude of countless generations. In medicine, we meet before the fifth century only with the anatomist Alcmaeon (508 B.C.) of the early medical school at Crotona, in Italy, and in natural philosophy (besides Thales and others already mentioned) with Xenophanes, who, like Pythagoras, held that fossils are in fact what they appear to be, and not mere " freaks of nature," as was generally beUeved. BEGINNINGS IN GREECE 57 References for Reading Allman. Greek Geometry. Chapters I, II. Ball. A Short History of Mathematics. Chapter II. Berry. A History of Astronomy. Chapter II to page 26. Cajori. a History of Mathematics. Pages 16-23. Dreyer. Planetary Systems. Chapters I, II. GoMPERZ. Greek Thinkers. Vol. I, pp. 1-164. Gow. History of Greek Mathematics. Chapters III, IV, VI. Heath. Aristarchus of Samos. Butcher. Aspects of the Greek Genius. Map of the World according to Herodotus (From Breasted'a Ancient Times. Courtesy of Measra. Ginn & Co.) From long journeys in Egypt and other Eastern Countries Herodotus returned with much information regarding these lands. His map showed that the Red Sea connected with the Indian Ocean, a fact unknown to his predecessor, Hecatseus. [See p. 34,] — Breaated. CHAPTER IV SCIENCE IN THE GOLDEN AGE OF GREECE Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in engineer- ing. She not only lends a hand to the progressive natural sciences, but participates at the same time in the abstract investigations of logicians and philosophers. — Klein. There still remain three studies suitable for freemen. Calcu- lation in arithmetic is one of them ; the measurement of length, surface, and depth is the second ; and the third has to do with the revolutions of the stars in reference to one another . . there is in them something that is necessary and cannot be set aside ... if I am not mistaken, [something of] divine necessity. — Plato. Literature and Art. — The fifth century B.C. witnessed that astonishing flowering of the Greek genius in Uterature and mili- tary glory which has made it ever since famous. The battles of Marathon and Salamis had flung back the Asiatic hosts which threatened to overrun and enslave Europe, and had transformed the Greeks from a group of jealous and parochial city states into a great democratic nation. Trade prospered, wealth increased, and for about a century letters, art, and science flourished as never before and never since. History began to be written by Herodotus and Thucydides. The drama was developed by iEschylus, Sophocles, and Euripides to such a pitch that even to-day, after the lapse of nearly 2500 years, crowds listen with eager interest to the Oedipus of Sophocles and the Iphigenia of Euripides, while the poetry of Pindar and the wit of Aristophanes have never lost their charm. In architecture and the plastic arts the Parthenon and its sculptures still testify to Greek supremacy. 58 THE GOLDEN AGE OF GREECE 59 In science, also, great names testify to memorable deeds. No such perfection, to be sure, was attained in science as in literature and in sculpture, but vast progress was made in mathematical science beyond anything hitherto accomplished, and the founda- tions were securely laid for a rational interpretation of man and of nature. Literature, architecture, sculpture, and the drama re- quire no special apparatus or reagents. Mathematical science also is not dependent upon such externals, being in this respect like literature and art, and we find geometry and arithmetic at the outset moving forward far more rapidly than natural or physi- cal science. Parmenides. — The recognition of the spherical shape of the earth and its division into zones are attributed not. only to the Pythagoreans, but also to Parmenides of Elis, who lived in the early part of the fifth century. He introduced a system of concen- tric spheres analogous to that soon to be so highly developed by Eudoxus. He identified the evening and the morning stars, and attributed the moon's brightness to reflected light. He regarded the sun as consisting of hot and subtle matter detached from the Milky Way, the moon chiefly of the dark and cold. Empedocles. — Passing over the guesses of Heraclitus and Parmenides at the riddle of existence and of man and nature, we may pause for a moment to examine the speculations of Empedocles (about 455 B.C.). A native of Agrigentum in southern Sicily, Empedocles was regarded as poet, philosopher, seer, and im- mortal god. He appears to have been a close observer of nature, understanding the true cause of solar eclipses and believing the moon to be twice as far from the sun as from the earth. The latter is held in place by the rapidly rotating heavens "as the water remains in a goblet which is swung quickly round in a circle." Aristotle attributes to Empedocles that analysis of the universe into the four " elements," earth, air, fire, and water, which until comparatively recent times was universally accepted as fundamental. It is, nevertheless, not only misleading but absurd to hold with Gomperz ("Greek Thinkers," I, 230) that Em- pedocles' theory of the four elements "takes us at a bound into 60 A SHORT HISTORY OF SCIENCE the heart of modern chemistry." The facts seem rather to be that Empedocles put together and hospitably accepted and clarified the theories of his various predecessors. He is the first sanitarian of whom we have any record, for Empedocles is credited with having cut down a hill of his native city and thus cured a plague by letting in the north wind, and to have done a similar service to the neighboring "parsley" city of Selinus (Selinunte) by simply draining a local marsh. The following is a fragment from the writings of Empedocles : — So all beings breathe in and out; all have bloodless tubes of flesh spread over the outside of the body, and at the openings of these the outer layers of skin are pierced all over with close-set ducts, so that the blood remains within, while a facile opening is cut for the air to pass through. Then whenever the soft blood speeds away from these, the air speeds bubbling in with impetuous wave, and when- ever the blood leaps back the air is breathed out; as when a girl, playing with a clepsydra of shining brass, takes in her fair hand the narrow opening of the tube and dips it in the soft mass of silvery water, the water does not at once flow into the vessel, but the body of air within pressing on the close-set holes checks it till she imcovers the compressed stream ; but then when the air gives way the de- termined amount of water enters. And so in the same way when the water occupies the depths of the bronze vessel, as long as the narrow opening and passage is blocked up by human flesh, the air outside, striving eagerly to enter, holds back the water inside behind the gates of the resounding tube, keeping control of its end, until she lets go with her hand. Then, on the other hand, the very opposite takes place to what happened before ; the determined amount of water runs ofl' as the air enters. Thus in the same way when the soft blood, surging violently through the members, rushes back into the interior, a swift stream of air comes in with hurrying wave, and whenever it [the blood] leaps back, the air is breathed out again in equal quantity. — Fairbanks. Anaxagoras. — (500-428 B.C.) For the student of science Anaxagoras, a native of Clazomene in Asia Minor, is more im- portant than Empedocles. Turning aside from wealth and civic distinction in his enthusiasm for science, he seems to have occupied THE GOLDEN AGE OF GREECE 61 himself with the problem of squaring the circle, a problem at- tacked even by the Egyptians with some degree of success, and destined to exercise great influence on the development of Greek geometry. The beginnings of perspective are also attributed to him, in connection with studies of the stage. He was particu- larly interested in a great meteorite — the appearance of which he was afterwards said to have predicted — supposing it to have fallen from the sun, and inferring that the latter was a "mass of red-hot iron greater than the Peloponnesus," not very distant from the earth. Like the Pythagoreans he assigned as the order of distances : — moon, sun, Venus, Mercury, Mars, Jupiter, Saturn. The earth's axis was inclined, in order that there might be variations of climate and habitability. He explained the moon's phases correctly, also solar and lunar eclipses, but he misinterpreted the Milky Way as due to the shadow cast by the earth. His theory of the nature and origin of the cosmos, viz. that it was material and had come by the combination and differentiation of primitive elementary substances or "seeds" of matter, was repugnant to those holding the polytheistic dogmas of his time and brought him into popular disfavor. Convicted of impiety, he died in exile, 428 B.C. By his insistence upon the importance of minute invisible "seeds" or particles of matter he paved the way for the "atomism" of Leucippus and Democritus. The Atomists. — A very little observation of external nature shows that disintegration is forever going on. Ice turns to water, water to vapor, rocks to sand and sand to dust — in other words, masses to particles. Furthermore, dust vanishes and vapor dis- appears, while clouds and fogs, rain and snow, make their appear- ance without obvious cause, and dust accumulates from invisible sources. What is more reasonable than to suppose that visible things — rocks and ice and water — become gradually resolved into invisible particles, and that these in their turn condense into new visible substances at some later time? For these or similar ideas the material "seeds" of Anaxagoras had, as stated above, paved the way, when later emphasized by Leucippus and his more famous pupil Democritus. Of the life of Leucippus 62 A SHORT HISTORY OF SCIENCE almost nothing is known, but he was probably a contem- porary of Empedocles and Anaxagoras, and possibly a pupil of Zeno. Leucippus assumed the existence of empty space as well as of matter, and held that of atoms all things are consti- tuted. Space is infinite in magnitude, atoms infinite in number and indivisible, with only quantitative differences. Atoms are always in activity, and worlds are produced by atoms variously shaped and weighted, falling in empty space and giving rise to an eddying motion by mutual impact. Democritus of Abdera was a pupil and associate of Leu- cippus, whose theories of empty space and material atoms he de- veloped and made so famous that his own name alone is often associated with them. Of his life, his works, and his death little is certainly known, but he may be regarded as marking the culmina- tion and conclusion of the Ionian school ; and his reputation, both in antiquity and in mediaeval times, was immense. Like contem- porary and preceding philosophers, his writings were in verse, and Cicero is said to have deemed his style worthy of comparison with that of Plato. His somewhat boastful comparison of his own geometrical power with that of the Egyptian rope-stretchers has been quoted. Democritus appears to have agreed closely in his interpretation of nature with Leucippus, and regarded empty space and atoms as cosmic elements. He also held that by the motion of the atoms was produced the world with all that it contains. Soul and fire are of one nature, their atoms small, smooth, and round. By inhaling them life is maintained. Hence the soul perishes with and in the same sense as the body, — a doctrine which made Democritus odious to later generations. Dante, for example, places him far down in hell as "ascribing the world to chance." The atomic theory of perception held that from every object "images" of that object are being given off in all directions, some of which enter the organs of sense and cause "sensations." De- mocritus further held that sensations are the only sources of our knowledge. He was regarded as one of the extreme sceptics of antiquity, as e.g. in this saying, "We know nothing: not even THE GOLDEN AGE OF GREECE 63 if there is anything to know." Galileo, himself of a highly scepti- cal turn of mind, refers with approval to Democritus, and it is probably on this side, i.e. by exemplification of the critical spirit, that Democritus rendered his greatest service. His positive con- tributions to science, even in atomism, were apparently neither novel nor important. Democritus explained the Milky Way as composed of a vast number of small stars, but to his dis- ciple, Metrodorus of Chios, it was a former path of the Sun. The Beginnings of Rational Medicine. Hippocrates of Cos. — Before the middle of the fifth century B.C., science in the healing art had no existence. Excepting among a few of the more enlightened, sacrifices and other appeals to the gods still characterized medicine as a priestly rather than a scientific pro- fession, while the prevailing ignorance of anatomy and physiology made rational treatment of the sick difficult if not impossible. Alcmaeon, in the previous century, had taken some steps in the right direction, proving for example that the sperm does not originate, as was currently believed, in the spinal marrow, and that the brain is the organ of mind, and advancing a naturalistic theory of disease which seems to foreshadow that of his great successor Hippocrates. Two island centres of medical lore (they can hardly be called medical schools), both of the cult of Asclepias, existed in the southeastern ^Egean, viz. Cos and Cnidus, and on the former was born, in 460 B.C., Hippocrates, "the Father of Medicine," in the next century already characterized by Aristotle as "the Great." Of his life, education, practice, and writings comparatively little is certainly known. Many of the writings attributed to him are of doubtful authenticity and are more safely assigned to the Hip- pocratic "school." Enough remain, however, especially when added to the references by later authors to him and to his sayings and to his methods of practice, to make it clear that in every re- spect Hippocrates was worthy of the lofty reputation with which his name has come down to us after five and twenty centuries. And yet it is not for the practical arts of medicine or any of its basic sciences that Hippocrates did his most famous work. It 64 A SHORT HISTORY OF SCIENCE was rather in his attitude toward health and disease that his real greatness lay. For, as far as we know, it was Hippocrates who first insisted on regarding disease as a natural rather than a supernatural process, and Hippocrates who first urged that care- ful observation and study of the patient which entitles him to rank as the original "clinician" of medical science. Again, it was Hippocrates who first insisted on the existence and importance of those processes of self-repair which are to-day recognized as fundamental properties of living matter, — processes summed up in that famous phrase of his which has come down to us through the Latin of the middle ages, — vis medicatrix naturce, — "the healing power of nature," one of the finest and truest of the tenets of scientific medicine to-day. Finally, by advancing his famous theory of the four humors, a theory which with minor modifications was for some two thousand years afterwards the prevailing theory of pathology, or the nature of disease, among the most enlight- ened, Hippocrates still further established his right to be regarded as the "father" of medicine, and the first (and only) medical man ever authoritatively entitled "the Great." This theory — crude enough to-day — held that health consists in the right mixture, and disease in the wrong mixture, of four "humors" (juices) of the body, viz. blood, phlegm, yellow bile, and black bile. Here again the great merit of Hippocrates' idea was that it directed attention to the body itself, and hence to natural rather than supernatural phenomena. The tone of the Hippocratic writings is well illustrated by the titles of those accepted as probably genuine, e.g. On Airs, Waters, and Places; On Epidemics; On Regimen in Acute Diseases; On Fractures; On Injuries of the Head; etc. The so-called Hippocratic Oath is rightly described by Gomperz as "a monument of the highest rank in the history of civilization." That this oath is still administered to graduates about to enter on the practice of medicine, is sufiicient evidence of the high char- acter and far-sighted wisdom of its originator. (See Appendix.) The Sophists. — In the fifth century b.c. political events fol- lowing war with Persia made Athens supreme in Greece — the THE GOLDEN AGE OF GREECE 65 finest and richest city in the world. Its citizens aspired to suc- cess in public life, and sought training to that end from the soph- ists. While science was not generally cultivated as a leading subject in the educational system thus developed/ mathematics could not fail to be esteemed as a means of discipline, and several of the sophists made notable contributions to its development. HiPPiAS OF Elis is the first sophist to be mentioned for impor- tant mathematical work. About 420 B.C. Hippias invented a curve called the quadratrix, serving for the solution of two of the three celebrated problems of Greek geometry ; viz. the quadrature of the circle and the trisection of an angle. By means of straight line and circle constructions, the solution of the quadratic equation had been accomplished, though without algebraic symbolism, or any recognition of negative or imaginary results. The tri- section problem, like that of duplicating the cube, was equivalent to the solution of the cubic equation, and could therefore not be accomplished by line and circle methods. The quadratrix was generated by the inter- section P of two moving straight lines, one MQ always parallel to its initial position OA, the other OR revolving uniformly about a centre 0. By means of this curve the trisection problem is reduced to that of tri- secting a straight line, which is elementary.^ The curve meets the perpendicular lines OA and OB at C and B respectively so that OC : 05 = 2 : tt, where tt is the ratio of the circumference of a circle to its diameter. To this quadrature solution the name of the curve is due. Dinostratus showed that the assumptions 0C:0B>2:w and 0C:0B<2:ir both lead to contradictions, therefore 0C:0B = 2:ir — a good example of the Greek redvxiio ad ahsurdum. The study of a problem not capable of solution by elementary means ' See Freeman, " Schools of Hellas." 2 To trisect any angle as AOR. draw MQ paraUel to OA and divide OM into three equal parts by lines paraUel to OA, meeting the curve in D and E respectively. The radii OS and OT wiU then trisect the angle AOQ, by the deBnition of the curve. 66 A SHORT HISTORY OF SCIENCE thus led to the invention of this new curve, the first of which we have any definite record. The Criticism of Zeno. — The Stoic philosopher Zeno, teach- ing in Athens about this time, though not himself a mathematician, represents an important phase of philosophical criticism of mathe- matics. Every manifold, he says, is a number of units, but a true unit is indivisible. Each of the many must thus be itself an indivisible unit, or consist of such units. That which is in- divisible however can have no magnitude, for everything which has magnitude is divisible to infinity. The separate parts have therefore no magnitude, etc. Again, as to the possibility of mo- tion, he maintains that before the body can reach its destination it must reach the middle point, before it can arrive there it must traverse the quarter, and so on without end. Motion is thus impossible ; so the tortoise, if he have any start, cannot be over- taken by the swift runner Achilles, for while Achilles is covering that distance the tortoise will have attained a second distance, and so on. Such specious criticism was naturally, and in a measure justly, evoked by misguided efforts of certain mathematicians to show that a line consists of a multitude of points, etc. These or similar controversies as to the interpretation of the infinite and the infinitesimal have persisted till our own day, resembling in that respect the classical problems of circle squaring and angle tri- section to which reference has been made above. The more or less mystical statements about the new discoveries of the Pythagoreans also invited sceptical epigrams. Zeno was concerned with three problems. . . These are the problem of the infinitesimal, the infinite, and continuity. . . . From him to our own day, the finest intellects of each generation in turn attacked these problems, but achieved, broadly speaking, nothing. . . . — B. Russell. Aristotle accordingly solves the problem of Zeno the Eleatic, which he propounded to Protagoras the Sophist. Tell me, Protagoras, said he, does one grain of millet make a noise when it falls, or does the ten-thousandth part of a grain? On receiving the answer that it does not, he went on : Does a measure of millet grains make a noise THE GOLDEN AGE OF GREECE 67 when it falls, or not? He answered, it does make a noise. Well, said Zeno, does not the statement about the measure of millet apply to the one grain and the ten-thousandth part of a grain? He as- sented, and Zeno continued. Are not the statements as to the noise the same in regard to each ? For as are the things that make a noise, so are the noises. Since this is the case, if the measure of millet makes a noise, the one grain and the ten-thousandth part of a grain make a noise. Circle Measurement : Antiphon and Brtson ; Hippocrates OF Chios. — Two of the sophists, Antiphon and Bryson, made an interesting contribution to the problem of squaring the circle, by means of the inscribed and circumscribed regular polygons. Antiphon started with a regular polygon inscribed in a circle, and constructed by known elementary methods an equivalent square. By doubling the number of sides repeatedly he obtained polygons which become more and more nearly equivalent to the circle, — the first correct attack on this formidable problem. Bryson took the important further step of employing both in- scribed and circumscribed polygons, making however the not un- natural assumption that the area of the circle may be considered the arithmetical mean between them. Another great step in the development of the theory of the circle was accomplished by Hippocrates of Chios, who had rela- tions with the now dispersed Pythagoreans during the latter half of the fifth century and came to Athens in later life after financial reverses. He is said in the register of mathe- maticians to have written the first Elements or textbook of mathematics, in which he made effective use of the reductio ad ahsurdum as a method of relating one proposition to another. To Hippocrates is due the theorem that the areas of circles are proportional to the squares on their diameters. He appears to have employed geometrical figures with letters at the vertices, in the modern fashion. From the theorem in regard to areas of circles follows naturally a general theorem for similar segments and sectors of circles. His work on lunes is remarkable. Start- 68 A SHORT HISTORY OF SCIENCE ing with an isosceles right triangle, he describes a semicircle on each of the three sides. By the theorem just quoted the semi- circle on the hypotenuse is equal in area to the sum of the other two. If the larger semicircle is taken away from the entire figure, two equal lunes remain; if the two smaller semi- circles are taken away, the triangle remains. Therefore the two lunes are together equivalent to the triangle, and the area of each may be determined. The gulf between rectilin- ear and curvilinear figures has at last been successfully crossed. A second attempt employs three equal chords instead of two, and incidentally the theorem that the square on the side of a triangle is greater than the sum of the squares on the other two sides when the angle opposite the first side is greater than a right angle. Other interesting and still more complicated attempts are pre- served. A third classical problem was that of the so-called " duplication of the cube." One of the older Greek tragedians attributed to King Minos the words referring to a tomb erected at his order : Too small thou hast designed me the royal tomb. Double it, yet fail not of the cube. At a somewhat later period it is related that the Delians, suf- fering from a disease, were bidden by the oracle to double the size of one of their altars, and invoked the aid of the Athenian geom- eters. Hippocrates transformed the problem of solid geometry into one in two dimensions by observing that it is equivalent to that of inserting two geometrical means between given extremes. In our modern algebraic notation, the continued proportion x:y = y:z = z:a leads to the equations y^ = xz, ^ = ya, whence, ehmmatmg z, t/^ = oa;^ y = a^x' ; y and z are the desired means between x and a, and by putting a = 2x the problem is solved. No such algebraic notation existed at this time, however, and the geometrical methods invented by later Greek mathema- ticians were necessarily very complicated, as will appear below. THE GOLDEN AGE OF GREECE 69 Plato and the Academy. — One of the greatest names in the history of philosophy is that of Plato, and yet with Plato philosophy enters upon a new phase in which it almost parts company with science. Before Plato philosophy was almost wholly devoted to inquiries or speculations touching the earth, the heavens, and the universe, and hence was substantially "nature" or "natural" philosophy. But with Plato and ever since his time the larger part of philosophy has been devoted to observation and specu- lation upon the human mind and its products, and has accordingly often been called " mental" or "moral" as contrasted with "nat- ural" philosophy. It is therefore Thales and Pythagoras, Democ- ritus and Aristotle, rather than Plato and his disciples, who are the protagonists of science as the word is used to-day. As a disciple of Socrates, Plato found it expedient to leave Athens after the death of his master, and during the following eleven years he travelled widely in the Mediterranean world, doubtless familiarizing himself with the learning of Egypt and of the Greek Ptolemies. After having been sold as a slave, re- deemed and set free, Plato returned to his native city, and es- tablished himself as a philosopher. While primarily a philosopher rather than a mathematician, Plato, unlike his master Socrates, — who desired only enough mathematics for daily needs, — rated highly the importance of mathematics and rendered services of the greatest value in its development. This was doubtless due in part to the influence of Archytas, a friend of the Pythagoreans, with whom he had associated during his prolonged exile. The register proceeds : "Plato . . . caused mathematics in gen- eral, and geometry in particular, to make great advances, by reason of his well known zeal for the study, for he filled his writings with mathematical discourses, and on every occasion exhibited the remarkable connection between mathematics and philosophy." "Let no one ignorant of geometry enter under my roof" was the injunction which confronted Plato's would-be disciples. His respect for mathematics finds interesting expression in the re- marks he puts into the mouth of Socrates in the Dialogues, and to him it is largely indebted for its place in higher education. 70 A SHORT HISTORY OF SCIENCE In the Laws he advises the study of music or the lyre to last from the age of 13 years to 16, followed by mathematics, weights and measures, and the astronomical calendar until 17. For a few picked boys on the other hand in the Republic, he recommends before they are 18, abstract and theoretical mathematics, theory of numbers, plane and solid geometry, kinetics, and harmonics. Of arithmetic he says, "Those who are born with a talent for it are quick at learning, while even those who are slow at it have their general intelligence much increased by studying it." "No branch of education is so valuable a preparation for household management and politics and all arts and crafts, sciences and professions, as arithmetic; best of all by some divine art, it arouses the dull and sleepy brain, and makes it studious, mindful, and sharp." The geometrical Greek view of numbers, exemplified in our use of square and cube in algebra, is well illustrated by Thesetetus, who says to Socrates that his teacher was giving us a lesson in roots, with diagrams, showing us that the root of 3 and the root of 5 did not admit of linear measurement by the foot (that is, were not rational). He took each root separately up to 17. There as it happened he stopped, so the other pupil and I de- termined, since the roots were apparently infinite in number, to try to find a single name which would embrace all these roots. We di- vided all numbers into two parts. The number which has a square root we likened to the geometrical square, and called 'square and equilateral' {e.g. 4, 9, 16). The intermediate numbers, such as 3 and 5 and the rest which have no square root, but are made up of unequal factors, we likened to the rectangle with unequal sides, and called rectangular numbers. Under Plato's influence mathematics first acquired its unified significance, as distinguished from geometry, computation, etc. Accurate definitions were formulated, questions of possibility considered, methods of proof criticized and systematized, logi- cal rigor insisted upon. The philosophy of mathematics was begun. The point is the boundary of the line ; the line is the boundary of the surface ; the surface is the boundary of the solid. THE GOLDEN AGE OF GREECE 71 Such axioms as "Equals subtracted from equals leave equals" date from this period. The analytical method is developed, con- necting that which is to be proved with that which is already known. Another principle carefully observed is to isolate the problem by removing all non-essential elements, and a third con- sists in proving that assumptions inconsistent with that which is to be proved are impossible. The Analytic Method. — The' analytic method, proceed- ing from the unknown to the known, depends for its validity on the reversibility of the steps ; the synthetic method on the contrary proceeds from the known to the unknown, with unimpeachable validity. It was characteristic of the Greek geometers to aim at this form for their demonstrations, even if the results had been first obtained analytically. The two methods are well illustrated by the following : — A circle is given and two external points A and B. It is required to draw straight lines AC and BC meeting the circle in C, D, and E so that DE shall be parallel to AB. It is ^ shown that if the construction can be made, ,'^ ~''i^ \^ the tangent to the circle at D will meet AB / / \ v / /'^ (produced if necessary) in a point F which ( / ^7(0 ' will lie on a new circle passing through A, C, \ / / ' \ I and Z). This analysis of consequences is the \/ // X! desired clue on which the following synthesis F "" /A B of the construction is then based. Starting again with A, B and the circle, we locate F so that BA X BF = BC X BD = square of the tangent BG from B. Then drawing a tangent from F to the circle, D is determined and with it the re- quired line DE. A solution of the " duplication of the cube " problem is also attributed to Plato, though the mechanical process employed is so much at variance with his usual teachings that the correctness of the attribution is seriously questioned. SPQR is a frame in which SPQ and PQR are always right angles, while PQ may be varied, and SQ and PR can be revolved about Q 72 A SHORT HISTORY OF SCIENCE and P respectively. They are to be so revolved if possible that they shall cross at right angles at T, and that S T and TR shall be respect- RK ively equal to the lengths between which mean pro- portionals are to be inserted. Then by similar triangles ST:PT =PT:QT =QT:RT S PT and QT are the required mean proportionals. If Sr is taken equal to twice TR the special case of the duplication of the cube is represented. To Plato is attributed a systematic method for finding numbers which may be sides of right triangles, his method being essentially an extension of the Pythagorean already described. Plato's Timaeus dialogue is indeed an important soiu-ce of our information in regard to Pythagorean mathematics. Plato speaks with em- phatic scorn of the shameful ignorance of mensuration on the part of his countrymen. He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side. While predominantly interested in geometry, Plato's arithmetical attainments were considerable for his time. He made, for ex- ample, a correct statement about the 59 divisions of 5040. Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and if visible or tangible ob- jects are obtruding upon the argument, refusing to be satisfied. — Plato, Republic. . . It would be proper then, Glaucon, to lay down laws for this branch of science and persuade those about to engage in the most important state-matters to apply themselves to computation, and study it, not in the common vulgar fashion, but with the view of arriving at the contemplation of the nature of numbers by the in- tellect itself, — not for the sake of buying and selling as anxious merchants and retailers, but for war also, and that the soul may acquire a facility in turning itself from what is in the course of gen- eration to truth and real being. — Plato, Republic. THE GOLDEN AGE OF GREECE 73 But the mathematical doctrines concerning the parts and ele- ments of the Universe are put forward by Plato, not so much as as- sertions concerning physical facts, of which the truth or falsehood is to be determined by a reference to nature herself. They are rather propounded as examples of a truth of a higher kind than any refer- ence to observation can give or can test, and as revelations of prin- ciples such as must have prevailed in the mind of the Creator of the universe ; or else as contemplations by which the mind of man is to be raised above the region of sense, and brought nearer to the Divine Mind. — Whewell. Platonic Cosmology. — The spherical figure of the earth was now generally accepted in Greece, and the older fanciful cos- mogonies gradually disappeared. To Plato, whose interest in physical science was indeed but secondary, the earth was a sphere at the centre of the universe, requiring no support. He supposes the distances of the heavenly bodies to be proportional to the numbers: Moon 1, Sun 2, Venus 3, Mercury 4, Mars 8, Jupiter 9, Saturn 27, — these numbers being obtained by com- bining the arithmetic and geometric progressions, 1, 2, 4, 8 and 1, 3, 9, 27. Plato accepts as a principle that the heavenly bodies move with a uniform and regular circular motion ; he then proposes to the mathe- maticians this problem : ' What are the uniform and regular circular motions which may properly be taken as hypotheses in order that we may save the appearances presented by the planets ? ' His general conception of the world as expressed in the Timseus and in the tenth book of the Republic is decidedly mystical. In the latter a soul returning to its body after 12 days in the other world relates its experiences in imaginative language : — Everyone had to depart on the eighth day and to arrive at a place on the fourth day after, whence they from above perceived ex- tended through the whole heaven and earth a light as a pillar, mostly resembling the rainbow, only more splendid and clearer, at which they arrived in one day's journey ; and there they perceived in the neighborhood of the middle of the light of heaven, the extremities of 74 A SHORT fflSTORY OF SCIENCE the ligatures of heaven extended; for this Hght was the band of heaven, like the hawsers of triremes, keeping the whole circumference of the miiverse together. Aristotle sums up Plato's theories — not too clearly — in the words : In a similar manner the Timceus shows how the soul moves the body because it is interwoven with it. For consisting of the elements and divided according to the harmonic numbers, in order that it might have an innate perception of harmony and that the universe might move in corresponding movements, He bent its straight line into a circle, and having by division made two doubly joined circles out of the one circle, He again divided one of them into seven circles in such a maimer that the motions of the heavens are the motions of the soul. Plato probably had no real knowledge of those deviations of the planets from uniform circular motion, which were to engross the attention of succeeding philosophers and astronomers. His system is consistently geocentric, and assumes a stationary earth. According to Plutarch : — Theophrastus states that Plato, when he was old, repented of having given the earth the central place in the imiverse which did not belong to it, this presumably indicating an inclination towards the theories of the later Pythagoreans. Plato adopts the Pythagorean or Empedoclean hypothesis of the four elements, the component particles being assumed to have respectively the shapes of the cube (earth), icosahedron (water), octahedron (air), and tetrahedron (fire). All the heavenly bodies are looked on as divine beings, the first of all living creatures, the perfection of whose minds is reflected in their orderly motions. Summing up an extended discussion of Plato's astronomical theories, Dreyer says : — THE GOLDEN AGE OF GREECE 75 There is absolutely nothing in his various statements about the construction of the universe tending to show that he had de- voted much time to the details of the heavenly motions, as he never goes beyond the simplest and most general facts regarding the revo- lutions of the planets. Though the conception of the world as Cos- mos, the divine work of art, into which the eternal ideas have breathed life, and possessing the most godlike of all souls, is a leading feature in his philosophy, the details of scientific research had probably no great attraction for him, as he considered mathematics inferior to pure philosophy in that it assumes certain data as self-evident, for which reason he classes it as superior to mere opinion but less clear than real science. Through his widely read books he helped greatly to spread the Pythagorean doctrines of the spherical figme of the earth and the orbital motion of the planets from west to east. The conjunction of philosophical and mathematical activity such as we find, beside Plato, only in Pythagoras, Descartes and Leibnitz, has always borne the finest fruits for mathematics. To the first we owe scientific mathematics in general. Plato discovered the analytical method, through which mathematics was raised above the standpoint of the Elements, Descartes created analytic geometry, our own celebrated countryman Leibnitz the infinitesimal calculus, — and these are the four greatest steps in the development of mathe- matics. — Hankel. Aechytas. — To Archytas, a late Pythagorean, with whom Plato had had close relations, was due the earliest solution of the duplication problem. This very interesting and somewhat elab- orate solution involves a combination of three services, a cone of revolution, a cylinder having the vertex of the cone in the cir- cumference of its base, and a surface generated by revolving a semicircle about an axis passing through one end of its diam- eter. It shows remarkable mastery of elementary geometry, both plane and solid, and an interesting tendency to employ a wider range of methods, including motion, which might, but for adverse tendencies, have had important results in connecting mathematics with its possible applications to mechanics, etc. The influence of Plato in avoiding such connections and asso- 76 A SHORT fflSTORY OF SCIENCE elating geometry with abstract logic and philosophy, undoubtedly had compensating advantages in promoting elegance and scien- tific rigor, — crystallizing out a more refined product. Archytas is said also to have invented the screw and the pulley and to have been the first to give a systematic treatment of mechanics, employ- ing geometrical theorems for this purpose. MenjEchmus : Conic Sections. — Even more interesting in its foreshadowing of future mathematical developments are the solutions of the duplication problem by Menaechmus. The problem which we should express in modern algebraic notation by the continued proportion a:x: -.x-.y: :y:b, Menaechmus, with- out any such notation or any system of coordinate geometry, shows to be equivalent to that of determining the intersection either of a parabola and a hyperbola, corresponding to the two proportions a:x: :x:y and a:x: :y:h, or to the intersection of two parabolas, in case the second pro- portion is replaced by x:y: :y:b. The construction of either parabola or the hyperbola naturally required some mechanical device. The Greeks of this period distinguished three types of the cone formed by the rotation of the right triangle about one of its sides, according as the angle formed by that side with the hypotenuse was less than, equal to, or greater than half a right angle. A plane perpendicular to an element would cut a cone of the first kind in an ellipse, the second in a parabola, the third in a hyperbola. These curves were named accordingly sections of the acute-angled, the rjght-angled, the obtuse-angled cone. The discovery of the conic sections . . . first threw open the higher species of form to the contemplation of geometers. But for this discovery, which was probably regarded ... as the unprofitable amusement of a speculative brain, the whole course of practical phi- losophy of the present day, of the science of astronomy, of the theory of projectiles, of the art of navigation, might have run in a different channel ; and the greatest discovery that has ever been made in the history of the world, the law of universal gravitation, with its in- THE GOLDEN AGE OF GREECE 77 numerable direct and indirect consequences and applications to every department of human research and industry, might never to this hour hav« been elicited. — Sylvester. Many of Plato's followers and disciples in the Academy con- tinued the development of mathematics. To Xenocrates, for example, is attributed the determination of the number of all possible syllables as 1,002,000,000,000, a result obtained by some unknown method. This whole period is one of great pro- ductivity and importance in the history of mathematics. New theorems and new methods are discovered, former methods are critically scrutinized, loci problems are investigated, these and the study of the three classical problems leading to the introduction of new curves and a general extension of geo- metrical knowledge. Geometry, with emphasis, indeed, on its philosophical side, predominates over the theory of numbers, and even the latter is given so geometrical a form that mathe- matics is unified. A New Cosmology. ^ Eudoxus of Cnidos (408 ?-355 B.C.) was a student both of Archytas and, for a time, of Plato. He was not only mathematician and astronomer, but also physician. In mathematics he is almost a new creator of the science, developing the theory of proportion, making a special study of the "golden section," already mentioned in connection with the regular poly- gons, and obtaining important results in solid geometry. In the words of the register, "Eudoxus of Cnidos . . . first increased the number of general theorems, added to the three propor- tions three more, and raised to a considerable quantity the learn- ing begun by Plato on the subject of the (golden) section, to which he applied the analytical method." To him was formerly attributed the proof that the volume of a pyramid is one third that of the prism having the same base and altitude, as well as the corresponding theorem for cones and cylinders. A recently discovered manuscript of Archimedes shows, however, that for this Democritus deserves the credit. The method of exhaustion, so-called, employed in proving these theo- 78 A SHORT HISTORY OF SCIENCE rems was expressed In the auxiliary theorem: "When two volumes are unequal, it is possible to add their difference to itself so many times that the result shall exceed any assigned finite volume." This exceedingly useful and important princi- ple, avoiding the difficulties of infinitesimals, was expressed in several approximately equivalent forms, and was already im- plied in the work of Antiphon and Bryson. A solution of the duplication problem which gained Eudoxus the appellation "godlike" has been entirely lost. There appear to have been no astronomical instruments at this time except the simple gnomon and sun-dial, but the more obvious irregularities of the planetary motions were beginning to attract attention, and under Eudoxus led to the development of a new and important theory. Nearest to the central earth is the moon, carried on the equator of a sphere revolving from west to east in 27 days. The poles of this sphere are themselves car- ried on a second sphere, which turns in about I85 years about the axis of the zodiac. The angle between the axes of these two spheres corresponds with the moon's variation in latitude. A third outer sphere gives the daily east to west motion. Simi- larly there are three spheres for the sun. For each of the five planets a fourth sphere is necessary to account for the stations and retrogressions of its apparent orbital motion — thus making with the single sphere of the stars 27 spheres, all having their common centre at the centre of the earth. How far these spheres were regarded as having concrete exis- tence, how far they merely expressed in convenient geometrical form the observed relations and motions, we cannot determine from extant evidence. The amount of observational data available was entirely inadequate to serve as a basis for any quantitatively correct theory. The third sphere of the sun was based on an erroneous hypothesis as to its motion. For Mercury, Jupiter and Saturn the theory was reasonably adequate, for Venus less so, and for Mars quite defective. Calippus, a follower of Eudoxus, endeavored with some degree of success to remedy these defects by adding a fifth sphere for THE GOLDEN AGE OF GREECE 79 each of the refractory planets, and at the same time a fourth and fifth for the sun, in order to account for the recently discovered inequality in the length of the four seasons. Reviewing the development of this interesting theory, Dreyer says : — But with all its imperfections as to detail, the theory of homo- centric spheres proposed by Eudoxus demands our admiration as the first serious attempt to deal with the apparently lawless motions of the planets. . . . Scientific astronomy may really be said to date from Eudoxus and Calippus, as we here for the first time meet that mutual influence of theory and observation on each other which characterizes the development of astronomy from century to century. Eudoxus is the first to go beyond mere philosophical reasoning about the con- struction of the universe ; he is the first to attempt systematically to account for the planetary motions. When he has done this the next question is how far this theory satisfies the observed phenomena, and Calippus at once supplies the observational facts required to test the theory, and modifies the latter imtil the theoretical and observed motions agree within the limits of accuracy attainable at that time. Philosophical speculation unsupported by steadily pursued obser- vations is from henceforth abandoned : the science of astronomy has started on its career. Eudoxus made the first known proposal for a leap-year, and for a star catalogue. A marble celestial globe in the national museum at Naples is perhaps a copy of one made by him. Aeistotle, 384-322 B.C., "the master of those who know," the son of a physician, a student in Plato's Academy, and tutor of Alexander the Great, exercised a mighty and lasting influence on the development of Greek science and philosophy. His tenden- cies were mainly non-mathematical, but the theorem that the sum of the exterior angles of a plane polygon is four right angles is ascribed to him. He distinguishes sharply between geodesy as an art and geometry as a science ; he considers the plane sections of the circular cyclinder; he recognizes the physical reason for the adoption of ten as the base number of arithmetic ; he designates unknown quantities by letters. Continuity — an idea so impor- 80 A SHORT HISTORY OF SCIENCE tant in modern mathematical and physical science — he defines by saying : — A thing is continuous when of any two successive parts, the limits, at which they touch, are one and the same, and are, as the word im- plies, held together. Aristotle's Mechanics. — In mechanics Aristotle seems almost to recognize the principle of virtual velocities. He dis- cusses the composition of motions at an angle with each other. He enunciates the correct relation between the length of the arms of a lever and the loads which will balance each other upon it. He even deals with the central and tangential com- ponents of circular motion. He asks such questions as: "Why are carriages with large wheels easier to move than those with small ? " " Why do objects in a whirlpool move toward the center?" etc. He is convinced that the speed of falling bodies is proportional to their weight — a belief credulously accepted until Galileo's experiment nineteen centuries later. He illustrates his discussions by geometrical figures, and states correctly : — If a be a force, /3 the mass to which it is applied, y the distance through which it is moved, and 5 the time of the motion, then o will move 5 P through 2 y in the time 5, or through y in the time | 8. He adds erroneously, however : — It does not follow that | a will move j3 through | y in the time 5, be- cause I a may not be able to move /3 at all ; for 100 men may drag a ship 100 yards, but it does not follow that one man can drag it one yard. Of the bearing of Aristotle's physical theories Duhem says : — Incapable of any alteration, inaccessible to any violence, the celestial essence could manifest no other than its own natural motion, and that was uniform rotation about the centre of the universe. Aristotle is the author of eight books on Physics, four on the Heavens, and four on Meteorology. In physics he explains the rainbow, attributes sound to atmospheric motion, and discusses THE GOLDEN AGE OF GREECE 81 refraction mathematically. While he undertakes to deal with motion, space and time — i.e. with the subject-matter of me- chanics — his treatment is too metaphysical to have much real value. He declares for example that : — The bodies of which the world is composed are solids, and therefore have three dimensions. Now, three is the most per- fect number, — it is the first of numbers, for of one we do not speak as a number, of ttvo we say both, three is the first number of which we say all. Moreover, it has a beginning, a middle, and an end. Francis Bacon in the seventeenth century remarks of Aristotle : — Nor let any one be moved by this ; that in his books Of Animals, and in his Problems and in others of his tracts, there is often a quoting of experiments. For he had made up his mind beforehand ; and did not consult experience in order to make right propositions and axioms, but when he had settled his system to his will, he twisted ex- perience round, and made her bend to his system ; so that in this way he is even more wrong than his modern followers, the Schoolmen, who have deserted experience altogether. Aristotelian Astronomy. — Only the second of the four books on the Heavens is devoted to astronomy. He considers the universe to be spherical, the sphere being the most perfect among solid bodies, and the only body which can revolve in its own space. Rotation from east to west is more honorable than the reverse. He holds that the stars are spherical in form, that they have no individual motion, being merely carried all together by their one sphere. ' Ftu-thermore, since the stars are spherical, as others maintain and we also grant, because we let the stars be produced from that body, and since there are two motions of a spherical body, rolling along and whirling, then the stars, if they had a motion of their own, ought to move in one of these ways. But it appears that they move in neither of these ways. For if they whirled (rotated), they would re- main at the same spot and not alter their position, and yet they manifestly do so, and everybody says they do. It would also be 82 A SHORT HISTORY OF SCIENCE reasonable that all should be moved in the same motion, and yet among the stars the sun only seems to do so at its rising or setting, and even this one not in itself but only owing to the distance of our sight, as this when turned on a very distant object from weakness becomes shaky. This is perhaps also the reason why the fixed stars seem to twinkle, while the planets do not twinkle. For the planets are so near that the eyesight reaches them in its full power, but when turned to the fixed stars it shakes on account of the distance, be- cause it is aimed at too distant a goal ; now its shaking makes the motion seem to belong to the star, for it makes no difference whether one lets the sight or the seen object be in motion. But that the stars have not a rolling motion is evident ; for whatever is rolling must of necessity be turning, while of the moon only what we call its face is visible.' — Dreyer. Aristotle adopts the system of spheres of Eudoxus and Calippus, but seems to suppose these spheres to be concrete, and not a merely geometrical device for interpreting the phenomena or determining the positions. In order however to secure what he conceives to be the necessary relation between the motions of the spheres, he is obliged to increase their total number from 33 to not less than 55. The earth is fixed at the centre of the uni- verse. That the earth is a sphere is shown logically, and is also evident to the senses. During eclipses of the moon, namely, the boundary line, which shows the shadow of the earth, is always curved. ... If we travel even a short distance south or north, the stars over our heads show a great change, some being visible in Egypt, but not in more northern lands, and stars are seen to set in the south which never do so in the north. It seems therefore , not incredible that the vicinity of the pillars of Hercules is con- ' nected with that of India, and that there is thus but one ocean, i The bulk of the earth he considers to be " not large in compar- ison with the size of the other stars." The estimated circumfer-' ence of 400,000 stadia — about 39,000 miles — is the earliest known estimate of the size of the earth, and is of unknown origin, but may quite likely be due to Eudoxus. While the heavens proper are characterized by fixed order and circular motion, the space THE GOLDEN AGE OF GREECE 83 belowthe moon's sphere is subject to continual change, and motions within it are in general rectilinear — a theory destined long to block progress in mechanics. Of the four elements, earth is near- est the centre, water comes next, fire and air form the atmosphere, fire predominating in the upper part, air in the lower. In this region of fire are generated shooting stars, auroras, and comets, the latter consisting of ignited vapors, such as constitute the Milky Way. Against any orbital motion of the earth Aristotle urges the ab- sence of any apparent displacement of the stars. Reviewing his astronomical theories, Dreyer says : — His careful and critical examination of the opinions of previous phi- losophers makes us regret all the more that his search for the causes of iphenomena was often a mere search among words, a series of vague and loose attempts to find what was ' according to nature ' and what was not; and even though he professed to found his speculations on facts, he failed to free his discussion of these from purely metaphys- ical and preconceived notions. It is, however, easy to understand the great veneration in which his voluminous writings on natural science were held for so many centimes, for they were the first, and for many centuries the only, attempt to systematize the whole amount of knowledge of nature , accessible to mankind ; while the tendency to seek for the principles of natural philosophy by con- sidering the meaning of the words ordinarily used to describe the phenomena of nature, which to us is his great defect, appealed strongly to the mediseval mind, and, unfortunately, finally helped to retard the development of science in the days of Copernicus and Galileo. At times Aristotle shows consciousness that his theories are based on inadequate knowledge of facts. ' The phenomena are not yet sufficiently investigated. When they once shall be, then one must trust more to observation than to spec- ulation, and to the latter no farther than it agrees with the phe- nomena.' 'An astronomer' he says 'must be the wisest of men; his mind must be duly disciphned in youth ; especially is mathematical study necessary; both an acquaintance with the doctrine of number. 84 A SHORT HISTORY OF SCIENCE and also with that other branch of mathematics, which, closely con- nected as it is with the science of the heavens, we very absurdly call geometry, the measurement of the earth.' Aristotle's writings include not merely works on scientific sub- jects, but treatises of the very first importance On Poetry, On Rhetoric, On Metaphysics, On Ethics, and On Politics. Besides his scientific works mentioned above, there are others entitled On Generation and Destruction, On the Parts of Animals, On Generation of Animals, Researches about Animals, On the Locomotion of Animals. One of the most important of his many services to science is the encyclopedic character of his writings, since from time to time he reviews in them the opinions of his predecessors whose works are sometimes known to us chiefly through his references to them. While standing thus upon the shoulders of the past, he shows at the same time both vast learning and much originality. He may be truly called the founder of zoology. Of Aristotle's contributions to science, the greatest was un- questionably that spirit of curiosity, of inquiry, of scepticism, and of veracity which he brought to bear on everything about him and within him. His observations are often poor, his conclu- sions often erroneous, but his interest, his curiosity, his zeal are indefatigable. Theophhastus. — One of Aristotle's principal pupils, and his successor in his School, was Theophrastus (372-287 B.C.) notable in the history of science chiefly as an early student of plants, and writer of the most important treatises of antiquity on botany. These were two large works, one of ten books and the other of eight. On the History of Plants, and On the Causes of Plants, respectively. In these, more than 500 species of plants are de- scribed, chiefly with reference to their medicinal uses. It is es- pecially interesting to note that Theophrastus recognized the existence of sex in plants, though he does not appear to have known the sex organs. Epicurus and Epicureanism. — A few words may be said of another philosopher of the fourth century, a follower to some THE GOLDEN AGE OF GREECE 85 extent of Democritus and the forerunner and exemplar of the Roman Lucretius. This was Epicurus (342-270 B.C.), who, born in Samos and educated in Athens and Asia Minor, became a famous teacher and the head of a remarkable community "such as the ancient world had never seen." The mode of life in this community was not that of the so-called "epicures" of to-day, but very plain, — water the general drink, and barley bread the general food. The magnetic personality of Epicurus held the community together, and his chief work was a treatise on Nature in thirty-seven books. Epicureanism is of interest in the history of science chiefly because of its effect on its Roman exponent, the poet Lucretius. Much of it was even a negation of science and the scientific spirit. Heraclides. Rotation of the Earth. — To Heraclides of Pontus in the fourth century B.C. belongs the distinction of teach- ing that the earth turns on its own axis from west to east in 24 hours. He had been connected with the Pythagoreans, and with the schools of Plato and Aristotle. His work is known to us only indirectly, none of his own writings having survived. He is said also to have advanced the hypothesis that Venus and Mercury revolve about the sun, being therefore at a distance from the earth sometimes greater than the sun, sometimes less. Geminus writing in the first half of the first century B.C. of the different fields and points of view of astronomers and physicists, remarks : — For why do sun, moon and planets appear to move unequally? Because, when we assume their circles to be excentric, or the stars to move on an epicycle, the appearing anomaly can be accounted for, and it is necessary to investigate in how many ways the phenomena can be represented, so that the theory of the wandering stars may be made to agree with the etiology in a possible manner. Therefore also a certain Heraclides of Pontus stood up and said that also when the earth moved in some way and the sun stood still in some way, could the irregularity observed relatively to the sun be accounted for. In general it is not the astronomer's business to see what by its nature is immovable and of what kind the moved things are, but framing hypotheses as to some things being in motion and others being fixed, 86 A SHORT HISTORY OF SCIENCE he considers which hypotheses are in conformity with the phenomena in the heavens. He must accept as his principles from the physicist, that the motions of the stars are simple, uniform, and regular, of which he shows that the revolutions are circular, some along parallels, some along oblique circles. This contrast between the physical phenomena and the mathe- matical theory which corresponds with them, without being true or perhaps even possible in all respects, is of continued and in- creasing importance in the history of science, as a larger stock of facts was accumulated and as theories still imperfect were more frequently subjected to critical comparison with observed data, instead of being accepted on purely philosophical or metaphysical grounds. Heraclides is not credited with any conception of orbital or progressive motion of the earth. References for Reading Allman. Greek Geometry, Chapters III-IX. Aristotle. On the Parts of Animals, On Generation, etc. Ball. History of Mathematics, Chapter III. Butcher, S. H. Aspects of the Greek Genius, Chapter I. Berry. History of Astronoviy, Chapter II, pp. 26-33. Dreyer. Planetary System., Chapters III-V. Freeman, K. E. Schools of Hellas. Garrison, F. H. A History of Medicine. (For Hippocrates of Cos.) GoMPERZ. Greek Thinkers, Vol. I. Gow. History of Greek Mathematics, Chapter VI, Articles 97-116. Lewes, G. H. Aristotle, a Chapter in the History of Science. CHAPTER V GREEK SCIENCE IN ALEXANDRIA There is an astonishing imagination, even in the science of mathe- matics. . . . We repeat, there was far more imagination in the head of Archimedes than in that of Homer. — Voltaire. If the Greeks had not cultivated Conic Sections, Kepler could not have superseded Ptolemy; if the Greeks had cultivated Dynamics, Kepler might have anticipated Newton. — Whewell. If we compare a mathematical problem with an immense rock, whose interior we wish to penetrate, then the work of the Greek mathematicians appears to us like that of a robust stonecutter, who, with indefatigable perseverance, attempts to demolish the rock gradually from the outside by means of hammer and chisel ; but the modern mathematician resembles an expert miner, who first con- structs a few passages through the rock and then explodes it with a single blast, bringing to light its inner treasures. — Hankel. The Museum at Alexandria. — The subjugation of Greece by Alexander the Great in 330 B.C. cheeked the further develop- ment of Greek civilization on its native soil. After Alexander's death in 323, his vast empire was divided among his generals, and Alexandria, the new Egyptian capital, fell to the lot of Ptolemy. The city as such was then barely ten years old, but very soon became, under the rule of the Ptolemies, the centre of the learned world. By 300 B.C. the Museum (Seat of the Muses) was founded, becoming in effect a veritable university of Greek learning. To this were attached a great library, a dining hall, and lecture- rooms for professors. Here for the next 700 years Greek science had its chief abiding place. The fame of Alexandria soon out- shone and eventually eclipsed that of Athens, while Romans journeyed from Rome — never important in ancient times as a 87 88 A SHORT HISTORY OF SCIENCE scientific centre — to study at Alexandria the healing art, anatomy, mathematical science, geography, and astronomy. Neither Athens, Rome, Carthage, nor any other city of the ancient world can boast similar distinction as a home of science. Euclid. — Three centuries after Thales had introduced the rudiments of Egyptian mathematics into Greece, the focus of mathematical activity was again transferred to that ancient land, but its spirit and aims remained there still for centuries essentially Greek. Continuing the ancient register, Proclus writes : — Not much younger than these (the Aristotelians) is Euclid, who brought the elements together, arranged much of the work of Eudoxus in complete form, and brought much which had been begun by Thesetetus to completion. Besides he supported what had been only partially proved by his predecessors with irrefragable proofs. . . . It is related that King Ptolemy asked him once if there were not in geometrical matters a shorter way than through the Elements : to which he replied that in geometry there is no straight path for kings. . . . As a recent writer has well said : " There are royal roads in science ; but those who first tread them are men of genius and not kings." Euclid's period of activity was about 300 B.C. ; his place of birth and even his race are unknown; he is said to have been of a mild and benevolent disposition, and to have appreciated fully the scientific merits of his predecessors. While we know next to nothing of his life and personality, his writings have had an influence and a prolonged vitality almost, if not quite, unpar- alleled. Euclid's "Elements." — Scientifically, Euclid is attached to the Platonic philosophy. Thus he makes the goal of his Elements the construction of the so-called "Platonic bodies" i.e., the five regular polyhedrons. This treatise, which served as the basis of practically all elementary instruction for the following 2000 years, is naturally his best-known work, and appears to have been ac- cepted in the Greek world after many previous attempts as a GREEK SCIENCE IN ALEXANDRIA 89 finality. It consisted of thirteen books, of which only the first six are ordinarily included in modern editions. The whole is essentially a systematic introduction to Greek mathematics, consisting mainly of a comparative study of the properties and relations of those geometrical figiu-es, both plane and solid, which can be constructed with ruler and compass. The comparison of unequal figures leads to arithmetical discussion, including the consideration of irrational numbers corresponding to incommensu- rable lines. The contents may be briefly siunmarized as follows : Book I deals with triangles and the theory of parallels : Book II with applications of the Pythagorean theorem, many of the prop- ositions being equivalent to algebraic identities, or solutions of quadratic equations, which seem to us more simple and obvious than to the Greeks. It should be noted however that the geomet- rical treatment is relatively advantageous for oral presentation. Book III deals with the circle, Book IV with inscribed and cir- cumscribed polygons. These first four books thus contain a general treatment of the simpler geometrical figures, together with an elementary arithmetic and algebra of geometrical magni- tudes. In Book V, for lack of an independent Greek arithmetical analysis, a theory of proportion (which has thus far been avoided) is worked out, with the various possible forms of the equation - = -.■ The results are applied in Book VI to the com- b a parlson of similar figures. This contains the first known problem in maxima and minima, — the square is the greatest rectangle of given perimeter, — also geometrical equivalents of the solution of quadratic equations. The next three Books are devoted to the theory of numbers, including for example the study of prime and composite numbers, of numbers in proportion, and the determina- tion of the greatest common divisor. He shows how to find the sum of a geometrical progression, and proves that the number of prime numbers is infinite. If there were a largest prime number n then the product 1 X 2 X 3 ... X n increased by 1 would always leave a remainder 1 when di- vided by n or by any smaller number. It would thus either be prime 90 A SHORT HISTORY OF SCIENCE itself, or a product of prime factors greater than n, either of which suppositions is contrary to the hypothesis that n itself is the greatest prime number. Book X deals with the incommensurable on the basis of the theorem : If two unequal magnitudes are given, and if one takes from the greater more than its half, and from the remainder more than its half and so on, one arrives sooner or later at a remainder which is less than the smaller given magnitude. Books XI, XII, and XIII are devoted to solid geometry, leading up to our familiar theorems on the volume of prism, pyramid, cylinder, cone, and sphere, but in every case without computation, emphasizing the habitual distinction between geometry and geodesy or mensura- tion ... a distinction expressed by Aristotle in the form : " One cannot prove anything by starting from another species, for ex- ample, anything geometrical by means of arithmetic. Where the objects are so different as arithmetic and geometry one cannot apply the arithmetical method to that which belongs to magni- tudes in general, unless the magnitudes are numbers, which can happen only in certain cases." Book XIII passes from the regular polygons to the regular polyhedrons, remarking in con- clusion that only the known five are possible. The extent to which Euclid's Elements represent original work rather than compilation of that of earlier writers cannot be deter- mined. It would appear, for example, that much of Books I and II is due to Pythagoras, of III to Hippocrates, of V to Eudoxus, and of IV, VI, XI, and XII, to later Greek writers ; but the work as a whole constitutes an immense advance over previous similar attempts. Proclus (410-485 a.d.) is the earliest extant source of informa- tion about Euclid. Theon of Alexandria edited the Elements nearly 700 years after Euclid, and until comparatively recent times modern editions have been based upon his. Like other Greek learning, Euclid has come down to later times through Arab channels. There is a doubtful tradition that an English monk, Adelhard of Bath, surreptitiously made a Latin GREEK SCIENCE IN ALEXANDRIA 91 translation of the Elements at a Moorish university in Spain in 1120. Another dates from 1185, printed copies from 1482 on- ward, and an English version from 1570. After Newton's time it found its way from the universities into the lower schools. Different versions vary widely as to the axioms and postulates on which the work as a whole is based. It is believed that Euclid originally wrote five postulates, of which the fourth and fifth are now known as Axioms 11 and 12, — "All right angles are equal " ; and the famous parallel axiom : — "If a straight line meets two straight lines, so as to make the two interior angles on the same side of it together less than two right angles, these straight lines will meet if produced on that side." The necessarily unsuccess- ful attempts which have since been made to prove this as a proposition rather than a postulate constitute an important chapter in the history of mathematics, leading in the last century to the invention of the generalized geometry known as non- Euclidean, in which this axiom is no longer valid. Influence of Euclid. — The Elements of Euclid have exerted an immense influence on the development of mathematics, and particularly of mathematical pedagogy. Aside from their sub- stance of geometrical facts, they are characterized by a strict conformity to a definite logical form, the formulation of what is to be proved, the hypothesis, the construction, the progressive reasoning leading from the known to the unknown, ending with the familiar Q.E.D. There is a careful avoidance of whatever is not geometrical. No attempt is made to develop initiative or invention on the part of the student ; the manner in which the results have been discovered is rarely evident and is even some- times concealed ; each proposition has a degree of completeness in itself. This treatise translated into the languages of modern Europe has been a remarkable means of disciplinary training in its special form of logic. No other science has had any such single permanently authoritative treatise. Criticism of Euclid. — On the other hand, its narrowness of aim, its deliberate exclusion of the concrete, its laborious methods of dealing with such matters as infinity, the incommensurable or 92 A SHORT HISTORY OF SCIENCE irrational, its imperfect substitutes for algebra, as in the theory of proportion, have diminished its usefulness, and have in com- paratively recent times (in English-speaking countries) led to the substitution of modernized texts. Still, no other mathematical treatise has had even approximately the deservedly far-reaching influence of EucUd. Its subject-matter is so nearly complete that its author's name is still a current synonym for elementary geometry. His elements are particularly admired for the order which con- trols them, for the choice of theorems and problems selected as funda- mental (for he has by no means inserted all which he might give, but only those which are really fmidamental), and for the varied argmnentation, producing conviction now by starting from causes, now by going back to facts, but always irrefutable, exact and of most scientific character. . . . Shall we mention the constantly main- tained invention, economy and orderliness, the force with which he establishes every point? If one adds to or takes from it, one will recognize that he departs thereby from science, tending towards error or ignorance. . . . Elsewhere Proclus : — It is difficult in every science to choose and dispose in suitable order the elements from which all the rest may be derived. Of those who have attempted this some have increased their collection, others have diminished it ; some have employed abridged demonstrations, others have expanded their presentation indefinitely, etc. In such a treatise it is necessary to avoid everything superfluous ... to combine all that is essential, to consider principally and equally clearness and brevity, to give theorems their most general form, — for the detail of teaching particular cases only makes the acquisition of knowledge more difficult. From all these points of view, Euclid's Elements will be found superior to every other. In a recent interesting discussion of Euclid's Elements, F. Klein {Elementar-MatJiematik mm Hoheren Standpunlct aus. II) says in substance: "A false estimation of the Elements finds its source in the general misunderstanding of Greek genius which long pre- GREEK SCIENCE IN ALEXANDRIA 93 vailed and still finds popular acceptance, namely that Greek culture was confined to relatively few fields, but in them reached a high degree of perfection and finality. The fact is, however, that the Greeks occupied themselves with the greatest versatility in all directions, and made in all directions wonderful progress. Never- theless, from our modern standpoint, they fell short of the pos- sibly attainable in all, and in some directions made only a begin- ning. " In mathematics, for example, it has become a tradition that Greek geometry reached unique development, while in reality many other branches of mathematics were successfully cultivated. The development of Greek mathematics was particularly ham- pered by the lack of a convenient number-system and notation as a basis for an independent arithmetic, and by ignorance of negative and imaginary numbers. Euclid's intention in the Ele- ments was by no means to write an encyclopedia of current geom- etry, which must have included conic sections and other curves, but rather to write for mature readers an introduction to mathe- matics in general, the latter being regarded in its turn, in the Platonic sense, as necessary preparation for general philosophic studies. Hence the emphasis on formal order and logical method, as well as the omission of all practical applications. He aims at the flawless logical derivation of all geometrical theorems from premises completely stated in advance." Allowing for grave uncertainties of text, Klein's view is summed up as follows : " (1) The great historical significance of Euclid's Elements consists in the fact that through it the ideal of a flawless logical treatment of geometry was first transmitted to future times. " (2) As to the execution, much is very finely done, but much remains fundamentally imperfect from our present standpoint. " (3) Numerous details of importance, especially at the begin- ning, remain completely doubtful on account of uncertainties of the text. "(4) The whole development is often needlessly clumsy, as Euclid has no arithmetic ready to his hand. 94 A SHORT HISTORY OF SCIENCE "(5) In general the one-sided emphasis on the logical makes it difBcult to understand the subject-matter as a whole, and its internal relations." The Elements of the great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively per- ceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed, one must turn to the Elements of Euclid. — Hankel. Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be pro- duced beyond this segment. He never thinks of the line as an entity given once for all as a whole. This careful definition and limitation, so as to exclude an infinity not immediately apparent to the senses, was very characteristic of the Greeks in all their many activities. It is enshrined in the difference between Greek architecture and Gothic architecture, and between the Greek religion and the modern religion. The spire on a Gothic cathedral and the importance of the unbounded straight line in modern geometry are both emblematic of the trans- formation of the modern world. — Whitehead. The universally admired perfection of the work of Euclid is re- vealed to the historians as the natural product of a long criticism which was developed in the constructive period of rational geometry, from Pythagoras to Eudoxus. Then commenced to appear the signifi- cation of those methods and principles by means of which the Greeks themselves attempted to interpret and conquer the paradoxes con- cerning infinity. These are the same difficulties which reappeared at the time the infinitesimal calculus was founded, and are now again asserting themselves in the most refined analysis. — Enriques. Other Works op Euclid. —Besides the "Elements" Euclid wrote several other mathematical treatises, including one on Porisms, a special type of geometrical proposition; and one on Data, containing such theorems as the following : GREEK SCIENCE IN ALEXANDRIA 95 Given magnitudes have a given ratio to each other. When two lines given in position cut each other their point of intersection is given. When in a circle of given magnitude a line of given magnitude is given, it bounds a segment which contains a given angle. A work on Fallacies is designed to safeguard the student against erroneous reasoning. Still other treatises are devoted to Division of Figures, Loci, and Conic Sections ; finally there are works on Phenomena, on Optics, and on Catoptrics dealing with applica- tions of geometry. The Phenomena gives a geometrical theory of the imiverse, the Optics is an unsuccessful attempt to deal with problems of vision on the hypothesis that light proceeds from the eye to the object seen. The fundamental assumptions are, for example : " Rays emitted from the eye are carried in straight lines, distant by an interval from one another, " etc. The Catoptrics deals in 31 propositions with reflections in plane, concave, and convex mirrors. It is remarked that a ring placed in a vase so as to be invisible from a certain position, may be made visible by filling the vase with water. The authenticity of this work is however questionable. These two works constitute the earliest known attempt to apply geometry systematically to the phenomena of light-rays. The law of reflection is correctly applied. Just as geometry is based on a definite list of axioms, so Euclid makes his optics depend on eight fundamental facts of experience. For example, the light rays are straight lines. The figure inclosed by the rays is a cone with its vertex at the eye, while the boundary of the object corresponds to the base, etc. This work, though in very imperfect form, continued in use until Kepler's time. Archimedes. — The second great name in the Alexandrian school and one of the greatest in the whole history of science is that of Archimedes. He was both geometer and analyst, mathe- matician and engineer. He enriched even the highly developed Euclidean geometry, made important progress in algebra, laid the foundations of mechanics, and even anticipated the infinitesimal 96 A SHORT HISTORY OF SCIENCE calculus, reaching thus a level which was not surpassed for 2000 years. Born in Syracuse, probably 287 B.C., the greater part of his life was spent in his native city, to which he rendered on oc- casion invaluable services as a military engineer. According to Livy it was due to the efforts of Archimedes that the Romans under Marcellus were held in check during the protracted siege of Syracuse. On the fall of the city in 212 B.C. the venerable mathematician, absorbed in a geometrical problem, was killed by a Roman soldier, much to the regret of Marcellus, who appre- ciated and would have spared him. The conqueror carried out the wish of Archimedes by erecting a monument with a mathe- matical figure, and this was with some difficulty rediscovered and put in order by Cicero, during his official residence in Sicily, 75 B.C. Nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him commanded him to follow to Marcellus, which he declined to do be- fore he had worked out his problem to a demonstration ; the soldier, enraged, drew his sword and ran him through. Others write, that a Roman soldier, running upon him with a drawn sword, offered to kill him ; and that Archimedes, looking back, earnestly besought him to hold his hand a Uttle while, that he might not leave what he was at work upon inconclusive and imperfect ; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate, that as Archimedes was carrying to Marcellus mathematical instru- ments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and think- ing that he carried gold in a vessel, slew him. Certain it is, that his death was very afflicting to Marcellus ; and that Marcellus ever after regarded him that killed him as a murderer ; and that he sought for his kindred and honored them with signal favours. — Plutarch. The known works of Archimedes include the following : two books on the Equilibrium of Planes, with an interpolated treatise GREEK SCIENCE IN ALEXANDRIA 97 on the Quadrature of the Parabola, two books on the Sphere and the CyHnder, the Circle Measurement, the Spirals, the book of Conoids and Spheroids, the Sand Number, two books on Floating Bodies, Choices. Unlike Euclid's Elements, these are for the most part original papers on new mathematical discoveries, which were also often communicated to his contemporaries in the form of letters. Pappus quotes Geminus as saying of Archimedes : "He is the only man who has known how to apply to all things his varied natural gifts and inventive genius." Archimedes and Euclid. — In contrasting the limitations of Euclid's Elements with the broad range of Greek mathematics, Klein characterizes the work of Archimedes somewhat as follows : (1) Quite in contrast to the spirit controlling Euclid's Elements, Archimedes has a strongly developed sense for numerical com- putation. One of his greatest achievements indeed is the calcu- lation of the ratio ir of the circumference of a circle to its diameter, by approximations with regular polygons. There is no trace of interest for such numerical results with Euclid, who merely men- tions that the areas of two circles are proportional to the squares of the radii, two circumferences as the radii, regardless of the actual proportionality factor. (2) A far-reaching interest in applications of all sorts is char- acteristic of Archimedes, including the most varied physical and technical problems. Thus he discovered the principles of hydro- statics and constructed engines of war. Euclid on the contrary does not even mention ruler or compass, merely postulating that a straight line can be drawn through two points, or a circle de- scribed about a point. Euchd shares the view of certain ancient schools of philosophy, — a view unfortunately extant in certain quarters, — that the practical application of a science is something mechanical and unworthy. The very greatest mathematicians, Archimedes, Newton, Gauss, have combined theory and applica- tions consistently.^ ' Plutarch, however, saya : "Archimedes possessed so high a spirit, so profound '' a soul, and such treasures of highly scientific knowledge, that though these inven- f' tions (used to defend Syracuse against the Romans) had now obtained him the re- 98 A SHORT HISTORY OF SCIENCE (3) Finally, Archimedes was a great investigator and pioneer, who in each of his works carries knowledge a step forward. This affects materially the form of presentation. In a most recently discovered manuscript, the procedure is essentially modern as contrasted with the rigid formalism of the Elements. Circle Measukement. — In this Archimedes proves three theorems. (1) Every circle is equivalent to a right triangle having the sides adjacent to the right angle equal respectively to the radius and circumference of the circle. (2) The circle has to the square on its diameter approximately the ratio 11 :14. (3) The circumference of any circle is three times as great as the diameter and somewhat more, namely less than y but more than y^. He proves the first theorem by showing that the assumption that the circle is either larger or smaller than the triangle leads to a contradiction. The second he bases on the third, at which he arrives by computing successively the perimeters of both inscribed and circumscribed polygons of 3, 6, 12, 24, 48 and 96 sides. All this is contrary to the spirit of Euclid and essentially modern in its method of successive approximation. The difficulty of the achievement in view of the imperfect arithmetical notation avail- able can hardly be overrated. Quadrature of the Parabola. — Of special interest Is his quadrature of the parabola. A segment is formed by drawing any chord PQ of the parabola : it is known that if a line is drawn from the middle point R of the chord parallel to the axis of the parabola, the tangent at the point S where this line meets the curve will be parallel to the chord, and the perpendicular from S to the chord is greater than any other which can be drawn from a point of the nown of more than human sagacity ; he yet would not deign to leave behind him any commentary or writing on such subjects ; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life ; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration." GREEK SCIENCE IN ALEXANDRIA 99 arc. The triangle formed by joining the same point S to the ends of the original chord being wholly contained within the segment, the area of the latter will be greater than that of the triangle and less than that of a parallelogram having the same base and alti- tude. Now the segment exceeds the triangle by two smaller segments, in each of which triangles STQ and SPU are again inscribed. It is a known property of the parabola that each of these triangles has one-eighth the area of the triangle PSQ. The area of each of the two smaller segments is therefore greater than one-eighth and less than one-fourth that of the triangle PSQ. The area of the original segment therefore is less than three-halves and greater than five-fourths that of triangle PSQ. The construc- tion may evidently be repeated any number of times, and the ratio of the segment to the triangle will lie between numbers which converge towards four-thirds. Archimedes also succeeded in determining the area of the ellipse. Spirals. — The discussion of spirals is based on the definition, "If a straight line moves with uniform velocity in a plane about one of its extremities which remains fixed, until it returns to its original position, and if at the same time a point moves with uni- form velocity starting at the fixed point, the moving point de- scribes a spiral." With the simple resources at his command, he also succeeds in obtaining the quadrature of this spiral, and in drawing a tangent at any point. In these quadratures he approx- imates the summation principle of the modern integral calculus. Supplementing Euclid's treatment of the regular polyhedrons, Archimedes investigates the semi-regular solids formed by com- bining regular polygons of more than one kind. Of these he finds 13, ten of which have two kinds of bounding polygons, the others three kinds. Sphere and Cylinder. — In his important treatise on " The Sphere and the Cylinder" he derives three new theorems : (1) That the surface of a sphere is four times the area of its great circle. 100 A SHORT HISTORY OF SCIENCE (2) That the convex surface of a segment of a sphere is equal to the area of a circle whose radius is equal to the straight line from the vertex of the segment to any point in the perimeter of its base. (3) That the cylinder having a great circle of the sphere for its base and the diameter of the sphere for its altitude exceeds the sphere by one-half, both in volume and in surface. It was the figure for this last proposition which was at his wish carved upon his tombstone. In attempting to solve the problem of passing a plane through a sphere so that the segments thus formed shall have either their surfaces or their volumes in an assigned ratio, he is led to a cubic equation ; he appears to have given both a solution and a criterion for the existence of a positive root, but the work is lost. In his Conoids and Spheroids he deals with the bodies formed by the revolution of the ellipse, parabola, and hyperbola, by means of plane cross-sections, ascertains the volume of these solids by comparing the portion between two neighboring planes with an inscribed and a circumscribed cylinder, — much in the modern manner. It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations (than those given by Archimedes). Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlabored results. No amount of inves- tigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required. — Plutarch. In other branches of mathematical science than geometry the work of Archimedes was relatively even more important. The so-called Cattle Problem, for example, is a notable per- formance in the algebra of linear equations. "The sun had a herd of bulls and cows, all of which were either GREEK SCIENCE IN ALEXANDRIA 101 white, gray, dun, or piebald; the number of piebald bulls was less than the number of white bulls by (| + i) of the number of gray bulls, it was less than the number of gray bulls by (j + f ) of the number of dun bulls, and it was less than the number of dun bulls by {i + -f ) of the number of white bulls. The number of white cows was (^ + |) of the number of gray cattle (bulls and cows), the number of gray cows was (j + i) of the number of dun cattle, the number of dun cows was (i + i) of the number of piebald cattle, and the number of piebald cows was (i + t) of the number of white cattle." The seven equations are insufiBcient to determine the eight unknown quantities. The solution attributed to Archimedes consists of numbers of nine figures each. Again he succeeds Ln summing the series of squares : 1,4, 9, 15, 25, 36, etc., to n terms, expressing the result in geometrical form. Both proof and formulation are of course much more complicated by reason of the entire lack of an algebraic symbolism, the same remark naturally applying also to the preceding cattle problem and to the cubic equation referred to above. This last was in- deed to Archimedes not primarily an equation at all, but a pro- portion a — X : 6 : : f a^ : a;^. In his Circle Measurement already outlined, he showed mastery of square root, and the comparison of irrational numbers with fractions, showing for example that 135 1 ^ -v/o ^ 265 How these fractions were obtained cannot be certainly deter- mined, but it was presumably by a process analogous at least to the modern method of continued fractions, though such fractions themselves could not have been known to him. In the Sand Counting, Archimedes undertakes to give a number which shall exceed the number of grains of sand in a sphere with a radius equal to the distance from the earth to the starry firma- ment. The treatise begins : "Many people believe. King Gelon, that the number of sand grains is infinite. I mean not the sand 102 A SHORT HISTORY OF SCIENCE about Syracuse, nor even that in Sicily, but also that on the whole mainland, inhabited and uninhabited. There are others again who do not indeed assume this number to be infinite, but so great that no number is ever named which exceeds this. ... I will attempt to show however by geometrical proofs which you will accept that among the numbers which I have named . . . some not only exceed the number of a sand-heap of the size of the earth, but also of that of a pile of the size of the universe." He assumes that 10,000 grains of sand would make the size of a poppy-seed, that the diameter of a poppy-seed is not less than one-fortieth of a finger-breadth, that the diameter of the earth is less than a million stadia, that the diameter of the universe is less than 10,000 di- ameters of the earth. To express the vast number which results from these assumptions — 10^^ in our notation — he employs an ingenious system of units of higher order comparable with the modern use of exponents, an immense advance on current arith- metical symbolism. Mechanics of Archimedes. — In mechanics Archimedes is a pioneer, giving the first mathematical proofs known. In two books on Equlponderance of Planes or Centres of Plane Gravities, he deals with the problem of determining the centres of gravity of a variety of plane figures, including the parabolic segment. A treatise on levers and perhaps on machines in general has been lost, as also a work on the construction of a celestial sphere. A sphere of the stars and an orrery constructed by him were long preserved at Rome. He describes an original apparatus for deter- mining the angular diameter of the sun, discussing its degree of accuracy. The lever and the wedge had been practically known from remote antiquity, and Aristotle had discussed the practice of dishonest tradesmen shifting the fulcrum of scales towards the pan in which the weights lay, but no previous attempt at exact mathematical treatment is known. Archimedes assumes as evident at the outset : (1) Magnitudes of equal weight acting at equal distances from their point of support are in equilibrium ; GREEK SCIENCE IN ALEXANDRIA 103 (2) Magnitudes of equal weight acting at unequal distances from their point of support are not in equilibrium, but the one acting at the greater distance sinks. From these he deduces : (3) Commensiu-able magnitudes are in equilibrium when they are inversely proportional to their distances from the point of support. In a work on Floating Bodies, extant in a Latin version by Tartaglia, Archimedes defines a fluid as follows : " Let it be assumed that the nature of a fluid is such that, all its parts lying evenly and continuous with one another, the part subject to less pressure is expelled by the part subject to greater pressure. But each part is pressed perpendicularly by the fluid above it, if the fluid is falling or under any pressure." " Every solid body lighter than a liquid in which it floats sinks so deep that the mass of liquid which has the same volume with the submerged part weighs just as much as the floating body." The specific gravity of heavier bodies was of course employed in his solution of the crown problem, which with his achievements as a military engineer gave him a great reputation among his contemporaries. Vitruvius in his De Architedura says: Though Archimedes discovered many curious matters that evinced great intelligence, that which I am about to mention is the most extraordinary. Hiero, when he obtained the regal power in Syracuse, having, on the fortunate turn of his affairs, decreed a votive crown of gold to be placed in a certain temple to the immortal gods, commanded it to be made of great value, and assigned for this purpose an appropriate weight of the metal to the manufacturer. The latter, in due time, presented the work to the king, beautifully wrought; and the weight appeared to correspond with that of the gold which had been assigned for it. But a report having been circulated, that some of the gold had been abstracted, and that the deficiency thus caused had been sup- plied by silver, Hiero was indignant at the fraud, and, unacquainted with the method by which the theft might be detected, requested Archimedes would imdertake to give it his attention. Charged with 104 A SHORT HISTORY OF SCIENCE this commission, he by chance went to a bath, and on jumping int( the tub, perceived that, just in the proportion that his body becami immersed, in the same proportion the water ran out of the vessel Whence, catching at the method to be adopted for the solution of thi proposition, he immediately followed it up, leapt out of the vessel ii joy, and returning home naked, cried out with a loud voice that h( had found that of which he was in search, for he continued exclaiming ' I have found it, I have found it !' — Vitruvius. Archimedes, who combined a genius for mathematics with a physica insight, must rank with Newton, who lived nearly two thousanc years later, as one of the founders of mathematical physics. . . . Th( day (when having discovered his famous principle of hydrostatic he ran through the streets shouting Eureka ! Eureka !) ought to b( celebrated as the birthday of mathematical physics ; the science cam( of age when Newton sat in his orchard. — Whitehead. The recently discovered New Manuscript ^ of Archimedes throws a very interesting light on his methods of attacking prob- lems in mechanics, as well as on his use of mechanical method; for geometrical problems. Naturally his mathematical methods are highly developed in comparison with the relatively simpl( problems of mechanics with which he deals. ' ' Certain things first became clear to me by a mechanical method although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actua demonstration. But it is of coiu'se easier, when we have previouslj acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. . apprehend that some, either of my contemporaries or of my succes sors, will, by means of the method when once established, be able t( discover other theorems . . . which have not yet occurred to me.' Our admiration of the genius of the greatest mathematician o antiquity must surely be increased, if that were possible, by a perusa of the work before us. — HeatKy, Archimedes as an Engineer. — His engineering skill, whicl has gained from an eminent German historian the appellation o ' See Reviews by C. S. Sliohter and D. E. Smith. Bulletin, American Math( matical Society, May, 1908, Feb. 1913. GREEK SCIENCE IN ALEXANDRIA 105 "the technical Yankee of antiquity," may be inferred from Plu- tarch's account of the siege of Syracuse : — Now the Syracusans, seeing themselves assaulted by the Romans, both by sea and by land, were marvellously perplexed, and could not tell what to say, they were so afraid ; imagining it was impossible for them to withstand so great an army. But when Archimedes fell to handling his engines, and set them at liberty, there flew in the air infinite kinds of shot, and marvellous great stones, with an incredible noise and force on the sudden, upon the footmen that came to assault the city by land, bearing down, and tearing in pieces all those which came against them, or in what place soever they lighted, no earthly body being able to resist the violence of so heavy a weight ; so that all their ranks were marvellously disordered. And as for the galleys that gave assault by sea, some were sunk with long pieces of timber like unto the yards of ships, whereto they fasten their sails, which were suddenly blown over the walls with force of their engines into their galleys, and so sunk them by their over great weight. These machines (used in the defense of the Syracusans against the Romans under Marcellus) he (Archimedes) had ■ designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with king Hiero's desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theo- retic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of me- chanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and dia- grams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had re- course to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato's indignation at it, and his invectives against it as the mere corruption and annihila- tion of the one good of geometry, — which was thus shamefully turn- ing its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base super- 106 A SHORT HISTORY OF SCIENCE visions and depravation) from matter ; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art. One of his most famous inventions was the water-screw used for irrigation, in Egypt, and for pumping. On occasion of diffi- culty in the launching of a certain ship he successfully applied a cogwheel apparatus with an endless screw. Archimedes . . . had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amaze- ment at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king's arsenal, which could not be drawn out of the dock without great labor and many men ; and, loading her with many passengers and a full freight, sitting himself the while far off with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly, as if she had been in the sea. The king, astonished at this, and con- vinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege . . the apparatus was, in most opportune time, ready at hand for the Syracusans, and with it also the engineer himself. — Plutarch. In astronomy his orrery has been mentioned ; he also attempted to determine the length of the year more closely. To the critical estimates already cited may be added as typical of countless others : — Whoever gets to the bottom of the works of Archimedes will admire the discoveries of the moderns less. — Leibnitz. His discoveries are forever memorable for their novelty and the difficulty which they presented at that time, and because they are the germ of a great part of those which have since been made, chiefly in all branches of geometry which have for their object the measiure- GREEK SCIENCE IN ALEXANDRIA 107 ment of the dimensions of lines and curved surfaces and which require the consideration of the infinite. — Mach. The genius of Archimedes created the theory of the composition of parallel forces, of centres of gravity, and of equilibrium of float- ing bodies. But antiquity went no farther ; not only were the first principles of dynamics unsuspected, but the statistical composition of concurrent forces was unknown, and the explanation of machines was confined to extension of the principles of the lever, which is the starting-point of the works of Archimedes, but may nevertheless have been recognized before him. — Tannery. Alexandrian Geography : Earth Measurement. — The far reaching conquests of Alexander and the resulting migrations and colonizations naturally gave a powerful stimulus to geography as a branch of descriptive knowledge. Chaldean records became accessible to the Alexandrian Greeks and a more accurate system of time-measurement was introduced. Until about this period it had been customary to make appointments at the time when a person's shadow should have a certain length. Eratosthenes, — 275-194 B.C., librarian of the great library at TUexandria, making a systematic quantitative study of the data thus collected, laid the foundations of mathematical geography — a transformation quite analogous to that taking place in as- tronomy. After a historical review he gives numerical data about the inhabited earth, which he estimates to have a length of 78,000 stadia and a breadth of 38,000. In connection with this he gives also a remarkably successful determination of the circumference of the earth. This was based on his observation that a gnomon at Syene (Assouan) threw no shadow at noon of the summer solstice, while at Alexandria the zenith distance of the sun at noon was ■^ of the circumference of the heavens. Assuming the two places to lie on the same meridian and taking their distance apart as 5000 stadia, he infers that the whole circumference must be 250,000 stadia. He or some successor afterwards substituted 252,000, perhaps in order to obtain a round number, 700 stadia, for the length of one degree. This result, subject to some uncertainty as to the length of 108 A SHORT HISTORY OF SCIENCE the stadium, was a close approximation to the real circumference but we may suppose that this degree of accuracy was to somi extent a matter of accident. Posidonius, a noted Stoic philosopher born in 136 B.C., stated that the bright star Canopus culminatec just on the horizon at Rhodes, while its meridian altitude a Alexandria was " a quarter of a sign, that is, one forty-eighth par of the zodiac." This would correspond with a circumference o 240,000 stadia, the method being quite inferior in accuracy to tha of Eratosthenes, on account of the impossibility of determininj when a star is just on the horizon. Eratosthenes is also creditec with measiu-ing the obliquity of the ecliptic with an error of bu about seven minutes. A student of the Athenian Platonists and a man of extraor dinary versatility, philosopher, philologian, mathematician, ath lete, Eratosthenes wrote on many subjects. He may well hav( been responsible for the introduction of leap-year into the Egyp tian calendar by the " Decree of Canopus " in 238 B.C., in order that the seasons may continually render their service accord ing to the present order and that it may not happen that some o the public festivals which are celebrated in the winter come to bi observed sometimes in the summer. . . . He invented a method and a mechanical apparatus for duplicat ing the cube.^ Such a mechanical solution is naturally obnoxiou; to the principles of Plato and Euclid. His so-called "sieve" is a method for systematically separating out the prime numbers by arranging all the natural number; in order, and then striking out first all multiples of 2, then of 3 and so forth, thus sifting out all but the primes 1, 2, 3, 5, 7, 11 13, 17, etc. Apollonius of Perga, about 260-200 B.C., "the great geom eter," was the last of this famous Alexandrian group of mathema ticians, and owes his reputation to his important work on the conii sections. His predecessors had in general recognized only those sec tions formed from right circular cones by planes normal to an ele ' See Gow, p. 245. GREEK SCIENCE IN ALEXANDRIA 109 ment. Archimedes, indeed, and Euclid obtained ellipses by passing other planes through right cones, but Apollonius first showed that any cone and any section could be taken, and introduced the names ellipse, parabola, and hyperbola. In the prefatory letter to Book I, Apollonius says to the friend to whom it is addressed : — 'Apollonius to Eudemus, greeting. When I was in Pergamum with you, I noticed that you were eager to become acquainted with my Conies ; so I send you now the first book with corrections and will forward the rest when I have leisure. I suppose you have not for- gotten that I told you that I undertook these investigations at the request of Naucrates the geometer, when he came to Alexandria and stayed with me; and that, having arranged them in eight books, I let him have them at once, not correcting them very carefully (for he was on the point of sailing) but setting down everything that occiKred to me, with the intention of returning to them later. Where- fore I now take the opportunity of publishing the needful emendations. But since it has happened that other people have obtained the first and second books of my collections before correction, do not wonder if you meet with copies which are different from this.' — Gow. Of the eight books, the first four are devoted to an elementary introduction. In Book I he defines the cone as generated by a straight line passing through a point on the circumference of a circle and a fixed point not in the same plane ; he fixes the manner in which sections are to be taken and defines diameters and ver- tices of the curves, also the latus rectum and centre, conjugate diameters and axes. The other branch of the hyperbola is taken due account of for the first time. In Book II asymptotes are defined by the statement : " One draws a tangent at a point of the hyperbola, measures on it the length of the diameter parallel to it, and connects the point thus determined with the centre of the hyperbola." Book III contains numerous theorems on tangents and secants and introduces foci with the definition : "A focus is a point which divides the major axis into two parts whose rectangle is one-fourth that of the latus rectum and the major axis," or the square on the minor axis. The focus of the parabola however is not recognized, nor has he any knowledge of the directrix of a 110 A SHORT HISTORY OF SCIENCE conic section, these omissions being first filled by Pappus in th. third century a.d. It is shown that the normal makes equal angle with the focal radii to the point of contact, and that the latte have a constant sum for the ellipse, a constant difference for th( hyperbola. This book, he says in the letter quoted above, " contain; many curious theorems, most of them are pretty and new, usefu for the synthesis of solid loci. ... In the invention of these, I ob served that Euclid had not treated synthetically the locus . . but only a certain small portion of it, and that not happily, nor in- deed was a complete treatise possible at all without my discoveries.' These three books, which are indeed based largely on the earlie: work of Euclid and others, contain most of the properties of conic sections discussed in modern text-books on analytic geometry, Book IV discusses the intersections of conies, treating tangencj correctly as equivalent to two ordinary intersections. In Book 'V ApoUonius even undertakes the difficult problem of determining the longest and shortest lines which can be drawn from a given point to a conic, identifying this with the problem of drawing normals from a given point. He succeeds in discovering the points for which two such normals coincide, i. e. what we call the centre of curvature. Book VI deals with equal and similar conies, reach- ing the problem of passing through a given cone a plane which shall cut out a given ellipse. Book VII deals with conjugate diameters and the complementary chords parallel to them. Book VIII is lost. On the whole, in this remarkable work of some 4O0 propositions he achieved nearly all the results which are included in our modern elementary analytic geometry, even approximating the introduction of a system of coordinates by his use of lines parallel to the principal axes. It is noteworthy that Fermat, one of the inventors of modern analytic geometry, was led to it by attempting to restore certain lost proofs of ApoUonius on loci. Of his other mathematical writings little more than the titles are known. Among these are one on burning mirrors, one on stations and retrogressions of the planets, and one on the use and theory of the screw. In astronomy he is believed to have sug- GREEK SCIENCE IN ALEXANDRIA 111 gested expressing the motions of the planets by combining uniform circular motions, an idea afterwards elaborated by Hipparchus and Ptolemy. How far his mathematical results were new, how far he merely compiled and coordinated the work of others, notably Euclid and Archimedes, cannot be precisely determined, but the proportion of original work is certainly very large. On the arithmetical side he obtained a closer approximation than Archimedes for the value of ir, invented an abridged method of multipHcation, and employed numbers of higher order in the manner of Archimedes. This last experiment if followed out to its logical conclusions might have had fundamental significance for the future development of computation. In the words of Gow: — he, as well as Archimedes, lost the chance of giving to the world once for all its numerical signs. That honor was reserved by the irony of fate for a nameless Indian of an unknown time, and we know not whom to thank for an invention which has been as important as any to the general progress of intelligence. Apollonitjs and Archimedes. — With Apollonius and Archi- medes the ancient mathematics had accomplished whatever was possible without the resources of analytic geometry and infinitesi- mal calculus, which, though already foreshadowed, were not fully realized until the seventeenth century. It is not only a decided preference for synthesis and a complete denial-of general methods which characterize the ancient mathematics as against our newer science (modern mathematics) : besides this external formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two as- sumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the philosophic school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phoronomically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations 112 A SHORT HISTORY OF SCIENCE and proceeded to investigate the variations which an algebraic ex- pression undergoes when one of its variables assumes a continuous succession of values. — Hankel. In one of the most briUiant passages of his Aperfu historiqui Chasles remarks that, while Archimedes and Apollonius were the most able geometricians of the old world, their works are distinguished by a contrast which runs through the whole subsequent history ol geometry. Archimedes, in attacking the problem of the quadrature of curvilinear areas, established the principles of the geometry whici rests on measurements ; this naturally gave rise to the infinitesimal calculus, and in fact the method of exhaustions as used by Archimedes does not differ in principle from the method of limits as used bj Newton. Apollonius, on the other hand, in investigating the proper- ties of conic sections by means of transversals involving the ratio oi rectilineal distances and of perspective, laid the foundations of the geometry of form and position. — Ball. The works of Archimedes and Apollonius marked the mos1 brilliant epoch of ancient geometry. They may be regarded, more- over, as the origin and foundation of two questions which have occu- pied geometers at all periods. The greater part of their works are connected with these and are divided by them into two classes so that they seem to share between them the domain of geometry The first of these two great questions is the quadrature of curvi- linear figures, which gave birth to the calculus of the infinite, con- ceived and brought to perfection successively by Kepler, Cavalieri Fermat, Leibnitz and Newton. The second is the theory of conic sections, for which were in vented first the geometrical analysis of the ancients, afterward: the methods of perspective and of transversals. This was the pre lude to the theory of geometrical curves of all degrees, and to tha considerable portion of geometry which considers, in the genera properties of extension, only the forms and situations of figures and uses only the intersection of lines or surfaces and the ratios o rectilineal distances. These two great divisions of geometry, which have each its pe culiar character, may be designated by the names of Geometry q Measurements and Geometry of Forms and Situations, or Geometr; of Archimedes and Geometry of Apollonius. — Chasles (Gow). GREEK SCIENCE IN ALEXANDRIA 113 Medical Science at Alexandria. Beginnings of Human Anatomy. — Alexandria is famous in the history of medicine for many reasons. It was here that human, — as contrasted with com- parative, — anatomy was first freely studied (probably favored by the Egyptian practice of disemboweling and embalming the dead) with the result that many of the grotesque errors of the earlier Greeks, including even Aristotle, were corrected. In this connection two names, and those of rivals, have come down to us as of chief importance, Herophilus and Erasistratus. The former, himself a student at Cos, was a close follower of the teach- ings of Hippocrates and regarded by the ancient world as his worthy successor. Erasistratus, on the contrary, opposed the Hippocratic doctrines. Both became distinguished anatomists. It is believed that the valves of the heart were first recognized and named by Erasistratus, who also studied and described the divisions, cavities and membranes of the brain, as well as the true origin and nature of the nerves. Herophilus like- wise studied the brain, the pulmonary artery and the liver, besides giving to the duodenum the name (twelve-inch) which it still bears. Physiology, meanwhile, made little or no progress, and Cicero, two centuries later, still speaks of the arteries as "air tubes." It appears also that vivisection as well as anatomy was practised at Alexandria, and probably even upon human beings. Pergamum, in Asia Minor, was for a time a rival centre of medical learning and medical education, but was eventually overshadowed by the more famous Alexandrian school. Of this last the most celebrated pupil was Galen (born 130 a.d.), the most noted medical man of the ancient Roman world. Galen was a native of Pergamum who, having first studied at home and at Smyrna, spent some years at Alexandria. He then returned to Pergamum, but soon went to Rome, where he became physician to the Emperor Commodus. Galen was an original and volu- minous writer on anatomy. That his name is still constantly linked with that of Hippocrates is probably the best evidence of his importance in the history of medical science. 114 A SHORT HISTORY OF SCIENCE References for Reading Ball. Chapter IV to page 84. Beret. Chapter II, Articles 31-36. Garrison, F. H. History of Medicine (On Galen, Herophilus, Erasistratus etc.). Gow. Chapter VII. Heath, T. L. Eyelid's Elements. The Works of Archimedes. ApoUonius o Perga. Mach, E. Science of Mechanics (on Archimedes). Mahaffy, J. Alexander's Empire. The three normals to the ellipse. ApoUomua. CHAPTER VI THE DECLINE OF ALEXANDRIAN SCIENCE The century which produced Euclid, Archimedes and Apollonius was . . . the time at which Greek mathematical genius attained its highest development. For many centuries afterwards geometry re- mained a favorite study, but no substantive work fit to be compared with the Sphere and Cylinder or the Conies was ever produced. One great invention, trigonometry, remains to be completed, but trigo- nometry with the Greeks remained always the instrument of astronomy and was not used in any other branch of mathematics, pure or applied. The geometers who succeed to Apollonius are professors who signalised themselves by this or that pretty little discovery or by some com- mentary on the classical treatises. The force of nature could go no further in the same direction than the ingenious applications of exhaustion by Archimedes and the por- tentous sentences in which Apollonius enunciates a proposition in conies. A briefer symbolism, an analytical geometry, an infinitesimal calculus were wanted, but against these there stood the tremendous authority of the Platonic and Euclidean tradition, and no discoveries were made in physics or astronomy which rendered them imperatively necessary. It remained only for mathematicians, as Cantor says, to descend from the height which they had reached and " in the descent to pause here and there and look around at details which had been passed by in the hasty ascent." The elements of planimetry were exhausted, and the theory of conic sections. In stereometry some- thing still remained to be done, and new curves, suggested by the spiral of Archimedes, could still be investigated. Finally, the arith- metical determination of geometrical ratios, in the style of the Meas- urement of the Circle, offered a considerable field of research, and to these subjects mathematicians now devoted themselves. — Gow. In the second century B.C. Hypsieles developed the theory of arithmetical progression and added two books of elements to Euclid's thirteen, but the chief mathematical work of this cen- 115 116 A SHORT HISTORY OF SCIENCE tury was due to Hipparchus, a great astronomer, and Hero, ai engineer. Orbital Motion of the Earth. Aeistarchus. — Befon dealing with Hipparchus and Hero, however, we have to conside the highly interesting and significant astronomical theories o Aristarchus of Samos (270 B.C.-?), who was the author of i treatise On the Dimensions and Distances of the Sun and Moon He endeavored to determine these distances relatively by ascer taining or estimating the angular distance between the two bodiei when the moon is just half illuminated, that is, when the linei joining sun, earth, and moon form a right angle at the moon — i method which may have been due to Eudoxus. The difficulties o; this determination are so serious, however, that no high degre( of accuracy could be attained, the actual result of Aristarchus f^ of a right angle — against the true ffs- — corresponding to s ratio of about 1 to 19 of the two distances. Aristarchus had nc trigonometry, and no other method of attacking this problem seems to have been known to the Greeks. In his Sand Counting already mentioned, Archimedes says oi Aristarchus, He supposes that the fixed stars and the sun are immovable, bul that the earth is carried round the sun in a circle which is in the middle of the course ; but the sphere of the fixed stars, lying with the sun rounc the same centre, is of such a size that the circle, in which he supposes the earth to move, has the same ratio to the distance of the fixed stars as the centre of the sphere has to the surface. But this is evidentlj impossible, for as the centre of the sphere has no magnitude, it follows that it has no ratio to the surface. It is therefore to be supposed thai Aristarchus meant that as we consider the earth as the centre of thi world, then the earth has the same ratio to that which we call the world as the sphere in which is the circle, described by the earth according to him, has to the sphere of the fixed stars. Aristarchus thus meets the objection that motion of the eartt would cause changes in the apparent positions of the stars by as suming that their distances are so great as to render the motion o DECLINE OF AI.EXANDRIAN SCIENCE 117 the earth a negligible factor. Another reference to Aristarchus, in Plutarch, mentions an opinion that he ought to be accused of impiety for moving the hearth of the world, as the man in order to save the phenomena supposed that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis. How far this remarkable anticipation of the Copernican theory was a conviction rather than a mere fortunate speculation cannot be known, but at any rate it failed of that acceptance necessary to its permanence. In the next century the rotation of the earth on its axis was indeed taught by Seleucus, an Asiatic astronomer, but it was 1700 years before these daring theories were again advanced. Seleucus also observed the tides, saying "that the revolution of the moon is opposed to the earth's rotation, but the air between the two bodies being drawn forward falls upon the Atlantic Ocean, and the sea is disturbed in proportion." Planetaky Irregularities. — The earlier theory of homocen- tric spheres, while accounting more or less successfully for the ap- parent motions of the heavenly bodies, had maintained each of them at a constant distance from the earth, and thus quite failed to explain the differences of brightness which were soon discovered, as well as the variations in the apparent size of the moon. The conception of motion in neither a straight line nor a circle was re- pugnant to the Greek philosophers, and the difficulty was therefore met, first by supposing the earth not to be exactly at the centre of the circular orbits about it, second by introducing subsidiary circles or epicycles. ExCENTEic Circular Orbits. — The complete planetary system according to the excentric circle theory was therefore as follows. In the centre of the universe the earth, round which moved the moon in 27 days, and the sun in a year, probably in concentric circles. Mercury and Venus moved on circles, the centres of which were al- ways on the straight fine from the earth to the sun, so that the earth was always outside these circles, for which reason the two planets are always within a certain limited angular distance of the sun, from 118 A SHORT HISTORY OF SCIENCE which the ratio of the radius of the excentric to the distance of it centre from the earth could easily be determined for either planet Similarly, the three outer planets moved on excentric circles, tb centres of which lay somewhere on the line from the earth to the sun but these circles were so large as always to surround both the sun anc the earth. — Dreyer. It seems probable that Aristarchus was led through this theorj to conceive of heliocentric orbits, and then to reflect that the earth too, might revolve about the sun as easily as the sun and planets round the earth. Epicycles. — Progress in observational astronomy increasec the number and magnitude of planetary irregularities beyond th( stationary points, retrograde motions, and variations, known tc Aristarchus, and apparently far beyond possible explanation bj the simple theory of excentric circles. The system was therefon superseded by, or combined with, that of epicycles, not necessarilj as physically realized, but as at least a geometrical working hy- pothesis, which should conform to and explain the observed phe- nomena. The system of epicycles consists in superimposing one circulai motion upon another, and repeating the process to any needfu] extent. The motion of the moon about the earth, for example, is explained by assuming first a circle (later called the deferent) or which moves the centre of a second smaller circle called the epi- cycle, on which the moon itself travels. By varying the dimensions of both circles and the velocities of the two motions, the observed changes, both of position and bright- ness of the moon, may be more or less satisfactorily accounted for and ever computed in advance. In particular the apparent retrograde motions ol the planets in certain parts of thai] orbits may be explained. In the figure E denotes the earth the large circle is the deferent of i planet, C the centre of the epicycle, Pi, P^, P3, P4 different pos DECLINE OF ALEXANDRIAN SCIENCE 119 sible positions of the planet in its epicycle. The distance of P from E obviously varies; the apparent motion of P being com- pounded of a forward motion of C and a backward motion at Pi is slower, at Ps faster, than the average. By suitable adjust- ment of the dimensions and velocities there may be retrogression for a certain length of arc near Pi, bounded by stationary points where the two motions seem to an observer at E to neutralize each other. How far this complicated scheme really departed from the original postulate of uniform circular motion is sufficiently in- dicated by Proclus' remark, "The astronomers who have pre- supposed uniformity of motions of the celestial bodies were ig- norant that the essence of these movements is, on the contrary, irregularity." While in point of fact the theory of epicycles and that of excentric circles have much in common, the former gradu- ally displaced the latter on account of its greater simplicity. Had Aristarchus worked out the earlier system in full detail, the history of astronomy might have been considerably modified. At the Museum of Alexandria a school of observers of whom Aristillus and Timocharis were notable members instituted sys- tematic astronomical observations with graduated instruments and made a small star catalogue. Thus was laid a foundation for the brilhant discoveries of Hipparchus and Ptolemy, while astronomy, which had in the work of Eudoxus assumed the character of true science, though with a too slender observational basis, now became an exact science, gradually shedding its encum- brances of speculation and vague generalization. Hipparchus. Star Catalogue. — The next great astronomer and much the greatest of antiquity is Hipparchus, probably a native of Bithynia, but long resident at Rhodes, a city which rivalled Alexandria itself in its intellectual activity. All his works but one are lost, but his great successor and disciple, Ptolemy, has based his famous Almagest on the work of Hipparchus and it is possible to determine in a general way how much is to be credited to each. Having at his disposal the primitive star catalogue of Aristillus and Timocharis, Hipparchus was profoundly impressed — 120 A SHORT HISTORY OF SCIENCE as was Tycho Brahe centuries later — by the sudden appearano in 134 B.C. in the supposedly changeless starry firmament of a ne^ star of the first magnitude. He accordingly set himself the heav; task of making a new catalogue, which ultimately included mon than 1000 stars, for the part of the sky visible to him, and "re mained, with slight alterations, the standard for nearly sixteei centuries." His list of constellations is the basis of our own. Precession op the Equinoxes. — While this great piece o routine work was deliberately planned by Hipparchus, not so mucl as an end in itself as a necessary basis for future investigators, i nevertheless led to his most remarkable discovery, that of the pre cession of the equinoxes. In comparing, namely, the positions o: certain stars with those observed about 150 years earlier, he de tected a change of distance from the equinoctial point — where the celestial equator and the ecliptic meet — amounting in one case to about 2°. By an inspiration of genius, he interpreted this correctly as due to a slight progressive shifting of the equinoctia points, corresponding to a slow rotation of the earth's axis, bj means of which the celestial pole in many thousand years describes a complete circle. His estimate of 36" per year was considerablj below the actual value, which is about 50". Other Astronomical Discoveries. Planetary Theory. — Striving always for greater accuracy and completeness of data, h( determined the length of the year within about six minutes. Ir attempting to explain the annual motion of the sun, he was aware that the change of direction is not uniform, and its distance fron the earth, as shown by its apparent size, not constant. He de- termined the length of spring as 94 days, that of summer as 92| and by a somewhat complicated calculation arrived at the value ^ as the eccentricity of the earth's position in the sun's orbit. These determinations were naturally very difficult and imperfect on ae count of the entire lack of accurate time measurement. Following Apollonius, Hipparchus devised a combination of uniform circulai motions which should account for the observed facts within the limits of probable error of observation, and in this undertaking he was successful, the degree of accuracy of his theory correspondinj to that of which his instruments were capable. DECLINE OF ALEXANDRIAN SCIENCE 121 With the more comphcated lunar theory he was naturally less successful. He is believed, however, to have discovered the more important irregularities of the moon's motion, supposing it to have a circular orbit in a plane making an angle of 5° with that of the sun's orbit — the ecliptic. The earth is not at the centre, but the latter revolves about the earth in a period of nine years. Extending his study of eclipses to the ancient records of the Chal- deans, he made substantial improvements in the theory of both solar and lunar eclipses, and obtained a close approximation for the distance of the moon. He estimated the sun's radius at about twelve times that of the earth, its distance from the earth at about 2550 earth-radii, the moon's radius i^ that of the earth, its distance about 60 earth-radii. The comparison of these figures with Ptolemy's and with the actual are (in earth-radii) — Hipparchus Ptolemy . Actual Sun's Radius 12 5.5 109. Sun's Distance 2550 1210 23,000 Moon's Radius .29 .29 .273 Moon's Distance 60 59 m Hipparchus realized that he had no adequate method for de- termining these numbers for the sun. The generally accepted order of the planets had now become Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, — an order adopted very early in Babylonia, and received as a more or less probable hypothesis from this time until that of Copernicus. In attempting to deal with the motions of the other planets as he had done with that of the sun and moon, Hipparchus was soon baffled by lack of adequate data, and set himself steadfastly to supply the need, resigning to more fortunate future astronomers the task of interpretation. Eudoxus, more than two centuries earlier, had developed a logical mathematical theory of the planetary motions. The more exact methods and data of Hipparchus brought out the entire inadequacy of existing theory to furnish anything better than a crude approxi- 122 A SHORT HISTORY OF SCIENCE mation to the motions of the planets, and showed the necessity both of a better theory and of more complete observational data. It is interesting to speculate on the consequences which might have resulted for astronomical science had the genius of Hipparchus adopted the daring heliocentric theories of Aristarchus instead ol adhering to the traditional geocentric ideas. Invention of Trigonometry. — Not least important among the services of Hipparchus to science was his laying the foundationj of trigonometry, by constructing for astronomical use a table oi chords, equivalent to our tables of natural sines. He gave also a method for solving spherical triangles. It is said that he firsl indicated position on the earth by latitude and longitude — the germ of coordinate geometry — Eratosthenes having merely giver the latitude by means of the height of the pole-star. For mapping the sky he used stereographic projection, for mapping the eartl orthographic. To sum up the chief work of Hipparchus : — he made very effec- tive use of extant records of earlier astronomers with critical con- sideration of their value; he made a prolonged and systematic series of observations with the best available instruments; he worked out a consistent mathematical theory of the motions of th( heavenly bodies so far as his data warranted; he made a ne^ catalogue of 1080 stars, with the classification by magnitude still ii use ; he discovered the precession of the equinoxes ; he laid th( foundations of trigonometry. Delambre, the great French historian of astronomy, says : — When we consider all that Hipparchus invented or perfected anc reflect upon the number of his works and the mass of calculation; which they imply, we must regard him as one of the most astonishini men of antiquity, and as the greatest of all in the sciences which are no purely speculative, and which require a combination of geometrica knowledge with a knowledge of phenomena, to be observed only b; diligent attention and refined instruments. In spite of these brilliant achievements, the position of ApoUoniu and Hipparchus had become relatively isolated under the prevalen DECLINE OF ALEXANDRIAN SCIENCE 123 Stoic philosophy, which was attended with a reversion to primitive cosmical notions. Even in Hipparchus a somewhat critical atti- tude, excellent in its immediate results, has been regarded by some as foreshadowing the period of decadence which actually followed. Astronomy is to remain nearly stationary for sixteen centuries. Inventions. Ctesibus and Hero. — In the period of civil war following the death of Alexander and followed in turn by Roman conquest, much attention was naturally devoted to the invention and improvements of military engines. Compressed air came into use as a motive power and the foundations of pneumatics were laid. Ctesibus, a barber of Alexandria, distinguished by his mechanical inventions, and his follower Hero (or Heron) who flourished in the latter part of the second century B.C., made notable inventions and some real contributions to mathematical science. The works attributed to Hero, on the basis of a great quantity of confused and doubtful material, include : — a Mechanics, treating of centres of gravity and of the lever, wedge, screw, pulley, and wheel and axle ; various works on military engines and mechanical toys, a Pneumatics — the oldest work extant on the properties of air and vapor — describing many machines, among others a fire-en- gine, a water-clock, organs, and in particular a steam-engine which we may regard as a remote precursor of our modern steam turbine. Many of the machines depend for their action on the flow of water into a vacuum, which Hero, having no conception of atmospheric pressure, attributed to nature's "abhorrence" of a vacuum. He arrived at the important law for the lever and the pulley : "The ratio of the times is equal to the inverse ratio of the forces applied." The Dioptra, a treatise on a kind of rudimentary theod- olite, discusses such engineering problems as finding differences of level, cutting a tunnel through a hill, sinking a vertical shaft to meet a horizontal tunnel, measuring a field without entering it, etc. The instrument employed is described as a straight plank, 8 or 9 feet long, mounted on a stand but capable of turning through a semicircle. It was adjusted by screws, turning cog-wheels. There was an eye-piece at each end and a water level at the side. 124 A SHORT HISTORY OF SCIENCE With it two poles, bearing disks, were used, exactly as by modei surveyors. A cyclometer for a carriage is also described, with series of cog-wheels and an index. In optics he shows that under the law of equal angles of incidenc and reflection, the path described by the ray is a minimum. Hero's Triangle Formula. — His Geodesy, — also the Dioptr — contains the well-known formula for the area of a triangle K -4 a + h + c a +h — c h_ _a c + a — b which, since it involves the multiplication of four lengths togethei is heterodox from the Euclidean standpoint. ABC is the given triangle of sides a, b, c, touching its inscribed circl at D, E, and F. Taking BJ = AD, we have CJ = l{a + b + c) am area ABC = twice area CJM Draw perpendiculars to CI at M and to CJ at B, meet ing in H. A semicircle on thi diameter CH will pass throug] both M and B. The sun of the angles CHB and CMl is 180°; the triangles BCL and MAD are therefore simi lar, so that BC:BH=AD: MD, or BC : BJ = BH : ME. Also the triangles BGH and EGM are similar, so that BH : ME = BG : EG and BC:BJ = BG: EG, whence BC + BJ : BJ =. BG +EG : EG, that is, CJ : BJ = BE -.EG and CJ'-.BJ X CJ =CE X BE: CE X EG, that is, CJ':BJXCJ = BEX CE:EM\ which is equivalent tc BE X CE-.CJ X EM = CJ X EM : BJ x CJ. K But CJ xEM =. — , 2' C£ = Ha + 6 -c), etc., whence 4. K'^ =(a + b + c) {a + b - c) {b + c - a) {c + a - b) DECLINE OP ALEXANDRIAN SCIENCE 125 A triangle with sides 13, 14, 15 is selected as an illustration. Its area is V21 X 6 X 7 X 8 = 84. This work seems to have become a standard authority for generations of surveyors, and thus in course of time to have lost much of its identity by successive changes. The whole spirit of the work is rather Egyptian than Greek, that of the practical engineer as distinguished from that of the mathematician, thus in a measure a reversion to the aims of the Ahmes manuscript. " Let there be a circle with circumference 22, diameter 7. To find its area. Do as follows. 7 X 22 = 154 and ^ = 38i That is the area." Some of Hero's methods indicate knowledge of the new trigo- nometry of Hipparchus and of the principle of coordinates. Thus he finds areas of irregular boundary by counting inscribed rec- tangles, a process corresponding to the use of coordinate paper. From Hero date such time-honored problems as that of the pipes. A vessel is filled by one pipe in time ti, by another in time <2. How long will it take to fill it when both pipes are used ? He defines spherical triangles and proves simple theorems about them : — for example, that the angle-sum lies between 180° and 540°. He determines the volume of irregular solids by measuring the water they displace. Having by a blunder introduced V— 63 he confuses it with v63. Indxtctive Arithmetic. Nicomachus. — As in the case of astronomy, progress in geometry now lags and finally ceases alto- gether. About 100 A. D. a final era of Greek mathematical science, predominantly arithmetical in character, begins with Nicomachus of Judea, whose work remained the basis of European arithmetic until the introduction of the Arabic arithmetic a thousand years later. He enunciates curious theorems about squares and cubes, for example : — In the series of odd numbers from 1, the first term is the first cube, the sum of the next two is the second, of the next three the third, etc., — doubtless simple observation and induction. He refers to proportion as very necessary to " natural science, music, spherical trigonometry and planimetry," and discusses various cases in great detail. 126 A SHORT fflSTORY OF SCIENCE Mathematics had passed from the study of the philosopher t the lecture-room of the undergraduate. We have no more the grav and orderly proposition, with its deductive proof. Nicomachu writes a continuous narrative, with some attempt at rhetoric, witl many interspersed allusions to philosophy and history. But more im portant than any other change is this, that the arithmetic of Nico machus is inductive, not deductive. It retains from the old geometrica style only its nomenclature. Its sole business is classification, am all its classes are derived from, and are exhibited in, actual numbers But since arithmetical inductions are necessarily incomplete, a genera proposition, though prima facie true, cannot be strictly proved savi by means of an universal symbolism. Now though geometry wa; competent to provide this to a certain extent, yet it was useless fo: precisely those propositions in which Nicomachus takes most interest The Euclidean symbolism would not show, for instance, that all th( powers of 5 end in 5 or that the square numbers are the sums of th( series of odd numbers. What was wanted, was a symbolism similai to the ordinary numerical kind, and thus inductive arithmetic lee the way to algebra. — Gow. Ptolemy and the Ptolemaic System. — With Claudiu! Ptolemy, in the second century of our era, Greek astronomj reaches its definitive formulation. In the 260 years which hac elapsed since Hipparchus no progress of consequence had beer made. Of Hipparchus, from whom he inherited so much, Ptolemj writes : — It was, I believe, for these reasons and especially because he hac not received from his predecessors as many accurate observations as he has left to us, that Hipparchus, who loved truth above everything only investigated the hypotheses of the sun and moon, proving thai it was possible to account perfectly for their revolutions by combi nations of circular and uniform motions, while for the five planets at least in the writings which he has left, he has not even com menced the theory, and has contented himself with collecting sys tematically the observations, and showing that they did not agrei with the hypotheses of the mathematicians of his time. He explainet in fact not only that each planet has two kinds of inequalities but als( DECLINE OF ALEXANDRIAN SCIENCE 127 that the retrogradations of each are variable in extent, while the other mathematicians had only demonstrated geometrically a single in- equality and a single arc of retrograde motion ; and he believed that these phenomena could not be represented by excentric circles nor by epicycles carried on concentric circles, but that, it would be necessary to combine the two hypotheses. — Dreyer. The instruments used by Ptolemy for his astronomical observa- tions included: — the "Ptolemaic rule," consisting of a rod with sights pivoted to a vertical rod, the angle at the junction being measured by the subtended chord ; the armillary circle, a copper or bronze ring marked in degrees and mounted in the meridian plane on a post. A second movable ring is fitted into this with pegs diametrically opposite each other, by means of which the sun's midday height could be measured ; the armillary sphere, similar in principle but somewhat more complicated; the astrolabe or as- tronomical ring for measuring either horizontal or vertical angles. Like the Chaldeans Ptolemy also used meridian quadrants of masonry. Time was still measured by the flow of water, with apparatus considerably improved by Ctesibus and Hero. The nmnerous observations of Ptolemy were made during the period 125-151 A.D. and he was in Alexandria in 139. One of his observations he describes as follows : In the 2d year of Antoninus, the 9th day of Pharmonthe, the sun being near setting, the last division of Taurus being on the meridian (that is, b^ equinoctial hours after noon), the moon was in 3 degrees of Pisces, by her distance from the sun (which was 92 de- grees, 8 minutes) ; and half an hour after, the sun being set, and the quarter of Gemini on the meridian, Regulus appeared, by the other circle of the astrolabe, 57| degrees more forwards than the moon in longitude. — Whewell. The Almagest. — In his celebrated Syntaxis, better known from Arabic translations as the Almagest, Ptolemy undertakes to present for the first time the whole astronomical science of his age. In Book I he reviews the fundamental astronomical data thus : — 128 A SHORT HISTORY OF SCIENCE The earth is a sphere, situated in the centre of the heavens; ii it were not, one side of the heavens would appear nearer to us than the other, and the stars would be larger there ; if it were on the celes- tial axis but nearer to one pole, the horizon would not bisect the equatoi but one of its parallel circles; if the earth were outside the axis, the ecliptic would be divided unequally by the horizon. The earti is but as a point in comparison to the heavens, because the stars appear of the same magnitude and at the same distances inter se, no mattei where the observer goes on the earth. It has no motion of translation, first, because there must be some fixed point to which the motions of the others may be referred, secondly, because heavy bodies descend to the centre of the heavens which is the centre of the earth. And if there was a motion, it would be proportionate to the great mass of the earth and would leave behind animals and objects thrown into the air. This also disproves the suggestion made by some, that the earth, while immovable in space, txirns round its own axis, which Ptolemy ac- knowledges would simplify matters very much.' Chapter IX explains the calculation of a table of chords. Start- ing with the chords of 60° and 72°, already known as sides of regular polygons, he devises ingenious geometrical methods for finding chords of differences and of half-angles. Thus he computes the chords for 12°, 6°, 3°, 1^°, and f °. Hipparchus had already com- puted such a table, but Ptolemy completes it by showing that f chord Ii° < chord 1° < f chord f° and thence deriving close approximations for the chords of I^ and J° and constructing a table for each half-degree up to 180° His results are expressed in sexagesimal fractions of the radius (ol which they are thus numerically independent) and are equivaleni in accuracy to five decimals in our notation. He also employs our present method of interpolation skilfully. This chapter v. the culmination of Greek trigonometry, which owed its furthei development to Indian and Arabic mathematicians. ' "For Ptolemy more geometer and astronomer than philosopher, the astro nome who seeks hypotheses adapted to save the apparent movements of the stars know no other guide than the rule of greatest simplicity : It is necessary as far as possibl to apply the simplest hypotheses to the celestial movements, but if they do no suffice, it is necessary to take others which fit better. " — Duhem. DECLINE OF ALEXANDRIAN SCIENCE 129 In Books III, IV, and V, Ptolemy discusses the apparent motions and distances of the sun and moon by means of excentrics and epicycles, his method for determining the moon's distance being substantially the same as the modern. Book V describes the con- struction and use of his chief instrument, the astrolabe. Book VI deals with eclipses, using a value of ir equivalent to our 3.1416. He determines the distance of the sun, following Hipparchus, by observing the breadth of the earth's shadow when the moon crosses it at an eclipse. Books VII and VIII contain a catalogue of 1028 stars based on that of Hipparchus, and a discussion of precession of the equinoxes, with a close determination of the unequal intervals between successive vernal and autumnal equi- noxes. The remainder of the treatise is devoted to the planets, containing Ptolemy's chief original contributions. While Ptolemy did not take advantage of the better data at his command to improve the theory of the sun's motion, he did make substantial progress with that of the moon, the discrepancies for which rarely exceed 10', which represented about the maximum precision of his instruments. Hipparchus had assumed the moon to have a motion representable by one circle with the earth as a centre and by an epicycle with its centre upon this. Discrepancies between observed and computed positions led Ptolemy, bound as he was by the Aristotelian dictum that celestial bodies can move only in circular paths, to modify this by making the first circle excentric to the earth, the line joining the centres of the circle and the earth being itself assumed to revolve. This theory, while giving results of sufficient accuracy for the observations at certain positions of the moon, exaggerated considerably the variation of its distance from the earth, making this at times almost twice as great as at others. For the five planets, or "wandering stars," he also assumed ex- centric deferents, and as a further means of accounting for dis- crepancies, an additional point, in line with the centres of earth and deferent, called the "equant," with respect to which the centre of the epicycle would have uniform angular velocity. The planes of the epicycles were slightly inclined to that of the ecliptic. 130 A SHORT HISTORY OF SCIENCE Thus in the figure, C is the centre of the circular deferent, E th( earth and E' the equant. The center A of the epicycle travel: at such a rate that the line E'A has uniforn angular velocity. The planet J travels ii an epicycle about A. These assumptions afforded the needful freedom for a fairly clos( approximation to observed planetary motions the mathematical computations involved be- coming naturally quite elaborate. Ptolemj disclaimed the power of determining the distances or even the order of the planets. That the system as a whole deserves our admiration as a readj means of constructing tables of the movements of sun, moon, anc planets, cannot be denied. Nearly in every detail (except the varia- tion of distance of the moon) it represented geometrically these move- ments almost as closely as the simple instruments then in use enablec observers to follow them, and it is a lasting monument to the grea1 mathematical minds by whom it was gradually developed. To the modern mind, accustomed to the heliocentric idea, it is difficult to understand why it did not occur to a mathematician like Ptolemy to deprive all the outer planets of their epicycles, which were nothing but reproductions of the earth's annual orbit transferrec to each of these planets, and also to deprive Mercury and Venus oi their deferents, and place the centres of their epicycles in the sun, as Heraclides had done. . . . The system of Ptolemy was a mere geo- metrical representation of celestial motions, and did not profess tc give a correct picture of the actual system of the world. . . Foi more than 1400 years it remained the Alpha and Omega of theoretica astronomy, and whatever views were held as to the constitution ol the world, Ptolemy's system was almost imiversally accepted as the foundation of astronomical science. — Dreyer. After Ptolemy we have no record of any important advance ir astronomy for nearly 1000 years. In reviewing Greek astronomy Berry says. The Greeks inherited from their predecessors a number of observa tions, many of them executed with considerable accuracy, which wen DECLINE OF ALEXANDRIAN SCIENCE 131 nearly sufficieat for the requirements of practical life, but in the matter of astronomical theory and speculation, in which their best thinkers were very much more interested than in the detailed facts, they re- ceived virtually a blank sheet on which they had to write (at first with indifferent success) their speculative ideas. A considerable interval of time was obviously necessary to bridge over the gulf separating such data as the eclipse observations of the Chaldeans from such ideas as the harmonical spheres of Pythagoras ; and the necessary theoretical structure could not be erected without the use of mathemati- cal methods which had gradually to be invented. That the Greeks, particularly in early times, paid little attention to making observations, is true enough, but it may fairly be doubted whether the collection of fresh material for observations would really have carried astronomy much beyond the point reached by the Chaldean observers. When once speculative ideas, made definite by the aid of geometry, had been sufficiently developed to be capable of comparison with observa- tion, rapid progress was made. The Greek astronomers of the scientific period, such as Aristarchus, Eratosthenes, and above all Hipparchus, appear moreover to have followed in their researches the method which has always been fruitful in physical science — namely, to frame pro- visional hypotheses, to deduce their mathematical consequences, and to compare these with the results of observation. There are few better illustrations of genuine scientific caution than the way in which Hip- parchus, having tested the planetary theories handed down to him and having discovered their insufficiency, deliberately abstained from building up a new theory on data which he knew to be insufficient, and patiently collected fresh material, never to be used by himself, that some future astronomer might thereby be able to arrive at an improved theory. Of positive additions to our astronomical knowledge made by the Greeks the most striking in some ways is the discovery of the ap- proximately spherical form of the earth, a result which later work has only slightly modified. But their explanation of the chief motions of the solar system and their resolution of them into a comparatively small number of simpler motions was, in reality, a far more important contribution, though the Greek epicyclic scheme has been so re- modelled, that at first sight it is difficult to recognize the relation be- tween it and our modern views. The subsequent history will, however, show how completely each stage in the progress of astronomical science has depended on those that preceded. 132 A SHORT HISTORY OF SCIENCE When we study the great conflict in the time of Copernicus be- tween the ancient and modern ideas, our sympathies naturally go ou1 towards those who supported the latter, which are now known to b( more accurate, and we are apt to forget that those who then spok« in the name of the ancient astronomy and quoted Ptolemy were indeed believers in the doctrines which they had derived from the Greeks, but that their methods of thought, their frequent refusal to face facts, and their appeals to authority, were all entirely foreign to the spirit oi the great men whose disciples they believed themselves to be. Other Works of Ptolemy. — In spite of his scientific attain- ments Ptolemy did not disdain to write an elaborate treatise on astrology. In a lost work on geometry, Ptolemy made the first known of the interminable series of attempts to give a formal proof of Euclid's parallel postulate, an attempt naturally fore- doomed to failure. In a great treatise on geography, hardly less important than the Almagest, Ptolemy gave a description of the known earth, locating not less than 5000 places by latitude and longitude. He even gave in addition to position the maximum length of day for 39 points in India, a land probably better known at this period than in the time of Mercator, near the end of the sixteenth century. Ptolemy reckoned longitude from the "Fortunate Isles," — the western boundary of the known world. Various methods of projection were discussed in connection with directions for map drawing. Ptolemy also wrote on sound and on optics, dealing particularly in the latter with refraction, with what has been called " the oldest extant example of a collection of experimental measures in any other subject than astronomy." He discovered by careful exper- iment and induction the law that light-rays passing from a rarer to a denser medium are bent towards the perpendicular, and in- vented a simple apparatus for measuring angles of incidence and reflection. Pappus. — The last two of the great Greek mathematicians were Pappus and Diophantus, who lived in Alexandria about 300 a.d. The most important work of Pappus is his Collections, in eight books, of which all but the first and a part of the second are pre- DECLINE OF ALEXANDRIAN SCIENCE 133 served. In this he comments fully on the most important Greek mathematical works known to him, making his treatise of the highest historical value, particularly in its careful summaries of books which have been lost. Book I and most of Book II are missing, the third reviews the various solutions of the duplication of the cube, adding Pappus' own, and discusses the regular in- scribed polyhedrons; the fourth deals with several less simple geometrical matters, including the higher curves, spirals, con- choid, quadratrix, etc., the problem of describing a circle tan- gent to three given circles which touch each other; the fifth is also geometrical. In Book VI Pappus gives the mathematical basis for the Ptolemaic astronomy, — i.e. trigonometry and optics. Book VII contains his well-known theorems, some- times mistakenly attributed to Gulden, that the volume of a solid of revolution is equal to the product of the area of the re- volving figure and the length of the path of its centre of gravity, and that the surface generated is equal to the product of the perim- eter and the length of the circular path described by its centre of gravity. In this final book he undertakes to deal with certain mechanical problems "more clearly and truly" than his prede- cessors have done. These include, for example, centre of gravity, inclined planes, the moving of a given weight by a given power with the help of cog-wheels, the determination of the diameter of a broken cylinder. The whole is somewhat weak on the arith- metical side. With the political decline of Greece and the awakening to in- tellectual activity of great Semitic and Egyptian populations, mathematical science changed radically from the traditional de- ductive geometry, to an arithmetical and algebraic science in harmony with the aptitudes which have characterized these races. Thus Nicomachus as we have seen was of Jewish antecedents. Hero an Egyptian in his point of view and his scientific tendencies. Beginnings of Algebra. Diophantus. — Diophantus was active in Alexandria in the first half of the fourth century a.d., though we know so little about him that even his precise name is doubtful. His chief work is his Arithmetic, which is extant 134 A SHORT HISTORY OF SCIENCE however only in somewhat mutilated form. It is the first known treatise on algebra, and is devoted to the solution of equations, employing algebraic symbols and analytical methods. Euclid had given the geometrical equivalent of the solution of a quadratic equation, and Hero could solve the same problem alge- braically but lacked a satisfactory symbolism. The algebra of Diophantus was therefore not a sudden invention, but the result of gradual evolution during several centuries of increasing interest in arithmetical problems, and declining vogue of the abstract Euclidean geometry. Writers on the history of algebra distinguish three classes or methods of algebraic expression : — (a) the rhetorical, where no symbols are used, but every term and operation is described in full. This was the only method known before Diophantus, and was later in vogue in western Europe until the fifteenth century ; (&) the syncopated, which replaces common words and operations by abbreviations, but conforms to the ordinary rules of syntax. This was the style of Diophantus ; (c) the symbolical or modern, using symbols only, without words. The syncopated method may be illustrated by the following passage from Heath's Diophantus : — Let it be proposed then to divide 16 into two squares. And let the first be supposed to be 1; yz + y -\- z = \b; zx + z+ X = 24:. Hence, by subtraction, x{z — y) +z — y =16, a; + 1 = -^ . z{x-y) +x -y = 9,z + I = — ^ , etc. z - y X - y He, on the other hand, takes a — 1 for one of the numbers and readily obtains 1 and 1 for the others, and a = — a a 5 He employs tentative assumptions with great effect. For example, " To find a cube and its root such that if the same number be added to each, the sums shall also be a cube and its root." If DECLINE OF ALEXANDRIAN SCIENCE 137 2x is the original number and x the number added, (an arbitrary and presumably erroneous assumption), 8a;' + x = 21x^, giving 19a;^ = 1. The coefifieient 19 not being a square, he now seeks to find two cubes whose difference is a square. If (a; + 1)' — a;' is equated to (2a; — 1)^ the special solution a; = 7 is easily obtained. Returning to the original problem, the new assumption is made : — let X = number to be added, 7x = original number. (7a;)' + a; = (8a;)' whence x = ^ In another type to find a square between 10 and 11, he multi- plies both by successive squares of integers until between the prod- ucts (by 16) he finds a square, 169. The number required is lOA- Such processes naturally give particular, not general, solutions. His lost Porisms are believed to have " contained propositions in the theory of numbers most wonderful for the time." Sum- marizing his methods of dealing with equations we may say that : — (1) he solves completely equations of the first degree having positive roots, showing remarkable skill in reducing simultaneous equations to a single equation in one unknown ; (2) he has a general method for equations of the second degree but employs it only to find one positive root ; (3) more remarkable than his actual solutions of equations are his ingenious methods of avoiding equations which he cannot solve. How far his work was original, how far like Euclid in his Elements it was the result of compilation, cannot be definitely ascertained. As a whole it is somewhat uneven and makes rather the impression of great learning than of exceptional originality. He seems in- debted in part to predecessors unknown to us. For him the earlier Greek distinction between computation and arithmetic has lost its force. In reviewing the work of Pappus and Diophantus Gow says : — the Collections of Pappus can hardly be deemed really important. . . . But among his contemporaries. Pappus is like the peak of Teneriffe in the Atlantic. He looks back from a distance of 500 years, to find 138 A SHORT HISTORY OF SCIENCE his peer in Apollonius. . . . His work is only the last convulsive effort of Greek geometry, which was now nearly dead, and was never effec- tually revived. ... It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations, indeed, but which has thenceforth a con- tinuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the inven- tion of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life. A melancholy interest attaches to the fate of Hypatia, daughter of Theon an Alexandrian mathematician, herself a teacher of Greek philosophy and mathematics, who was torn to pieces by a Christian mob, doubtless as a representative of pagan (Greek) learning, at Alexandria in 415 a.d. Conclusion and Retrospect. — Intellectual interests in the Greek world (now really Roman) were by this time so completely alienated from mathematics, and indeed from science in general, that the brilliant work of Pappus and Diophantus aroused but slight and temporary interest. Geometry had reached within the possible range of the Euclidean method a relatively complete development. Algebra under Diophantus attained in spite of hampering notation a level not again approached for many cen- turies. Little need be said of sciences other than those already dealt with. These, even more than mathematics and astronomy, shrank under Roman autocracy and Christian hostility. Only the works of Galen, Strabo, and Pliny need be mentioned, and with them we deal in the next chapter. The torch of science now passes from the Greeks to the Indians of the far East after their conquest by Alexander, to be in turn DECLINE OF ALEXANDRIAN SCIENCE 139 surrendered to the Mohammedan conquerors of Alexandria a.d. 641. By them it is kept from extinction until in later ages it is once more fanned to ever increasing radiance in western Europe. In attempting a retrospective estimate of Greek science it is fundamentally important to judge the whole background fairly. In science the Greeks had to build from the foundations. Other peoples had extensive knowledge and highly developed arts. Only among the Greeks existed the true scientific method with its char- acteristics of free inquiry, rational interpretation, verification or rectification by systematic and repeated observation, and con- trolled deduction from accepted principles. The Assyrians, Babylonians and Egyptians had certainly made great progress in the use of mechanical devices for moving heavy loads, in the construction of scales, and of pumps. Their measuring in- struments were well developed, and acute observations were made, but of systematic, scientific investigation there is no evidence. The Greeks received many results and suggestions from Asia Minor, Meso- potamia, and Egypt, but their achievements are essentially their own. — Wiedemann. In asking ourselves why these extraordinary beginnings seemed after a time to lose their power of continued development, we must not forget the effect of external conditions. It is conceivable indeed that scientific progress should continue from age to age, through the genius of individual teachers and students, regardless of political and social conditions. Such, however, is not the historic fact. For progress in science men of genius are indis- pensable, but in no country or age have they alone been able to make science flourish under conditions so unfavorable as were those of the early centuries of the Christian era. Greek science, however, did not " fail," learned and elaborate as are the explanations that have been given of its alleged failure. Under "the chill breath of Roman autocracy" its growth was in- deed checked, its animation suspended, for a full thousand years. Then in the Renaissance it renewed its vitality and has ever since been advancing more and more magnificently. This is not to say 140 A SHORT HISTORY OF SCIENCE that criticisms as to the imperfections of the Greek scienttfii method are invalid, but rather to assert, as most critics must agree that its merits outweighed its defects, and that the latter woulc not have proved disastrous but for the development of political economic and military conditions under which the free Greek spirii could not continue its wonderful achievements. References for Reading Ball. Chapters IV, V. Beeby. Chapter II, Articles 37-54. Dreyee. Chapters VI-IX. Gow. Chapters IV, VIII, IX, X. Heath. Diophantus of Alexandria. Aristarchus of Samot. CHAPTER VII THE ROMAN WORLD. THE DARK AGES Among them [the Greeks] Geometry was held in highest honor : nothing was more glorious than Mathematics. But we have limited the usefulness of this art to measuring and calculating. — Cicero. The Romans were as arbitrary and loose in their ideas as the Greeks, without possessing their invention, acuteness and spirit of system. — Whewell. The Romans, with their limited peasant horizon and their short- sighted practical simplicity, cherished always for true science in their iiunost hearts that peculiar mixture of suspicion and contempt which is so familiar today among the half educated. The arch dilettante Cicero boasts, even, that his coimtrymen, thank God ! are not like those Greeks, but confine the study of mathematics and that sort of thing to the practically useful. — Heiberg. The Roman World-Empibe. — For several centuries, during the decline of Greek learning both in Greece itself and in Alexan- dria, two new and powerful States were developing ; one having its centre at Carthage on the northern shore of Africa, almost opposite Sicily, the other — the Roman Empire — on the western shore of Italy in the valley of the Tiber. The latter, at first comparatively insignificant, rapidly rose to a position of world-wide power, con- quering in turn Carthage, Greece, and the East and eventually extending over the greater part of the then known world, from Britain on the north to the Cataracts of the Nile on the south, from India in the east to the Pillars of Hercules in the west. The Roman Attitude towards Science. — fOne of the most striking facts in the history of science is the total lack of any evidence of real interest in science or in scientific research among the Roman people itself or any people under Roman sway. Alex- andrian science, even, though previously flourishing, languished and went steadily to its fall after the submission of that city to the Romans in the first century B.C. The truth seems to be that 141 142 A SHORT HISTORY OF SCIENCE the Roman people, while highly gifted in oratory, literature, an history (as witness, for example, the works of Cicero, Virgil an Tacitus), were not interested and therefore not successful in scier tific work. ; This is the more impressive when we reflect upon thei marvellous military genius, and their preeminence in world-wid power, dominion and influence. In vain do we look for any Roma: scientist or philosopher of such originality or range as Aristotle o Plato ; for any Roman astronomer, like Aristarchus or Hipparchu or Ptolemy; for any Roman mathematician or inventor, lik Archimedes; for any Roman natural philosopher, like Democ ritus ; for any Roman pioneer in medicine, like Hippocrates, — for Galen was Roman neither by birth nor education, but onl; by adoption late in life. Roman Engint;ering and Architecture. — There is how ever one marked feature of Roman civilization in which extraor dinary ability was displayed and peculiar excellence achievec and in which the Romans were unquestionably far superior to al their predecessors and, until very recent times, to all their sue cessors. This feature, which is one of the most characteristic, i: the Roman genius for both military and civil engineering. It ii only necessary to mention the surviving remains of Roman walls fortresses, roads, aqueducts, theatres, baths, and bridges. Nevei before and never since has any empire built so many, so splendid and so enduring monuments for the service of its peoples in peace and in war. The surface of southern Europe, western Asia anc northern Africa is still covered after the lapse of twenty centuries with Roman remains which bid fair to resist decay and destruc- tion for another two thousand years. Roman engineering is almosi as distinguished as is Roman law. The Emperor Constantint in the fourth century wrote : " We need as many engineers as pos- sible. As there is lack of them, invite to this study persons of abou1 18 years, who have already studied the necessary sciences. Re- lieve the parents of taxes and grant the scholars sufficient means.'' The land surveyors formed a well-organized gild, but they were merely practitioners of a traditional art, perpetuating the error; of their ancient Egyptian predecessors, not dreaming of new dis- THE ROMAN WORLD 143 coveries, nor even of imparting such knowledge as they had, — outside the ranks of their own gild. Slave Labor in Antiquity. — It must never be forgotten that throughout antiquity, and to a great extent even until very recent times, the labor question was wholly different from what it is to-day. Instead of the labor-saving machinery which is so extraordinary a feature of our time, but which was practically non- existent before the end of the eighteenth century, the slave was the machine for all heavy labor. It is not likely that he was ever a particularly cheap machine, but in the mass he was powerful, and it was probably largely by his labor that the fields were cultivated and irrigated, and that dams and ditches, walls and towers, roads and bridges and pyramids and temples, were built and fortified. It is notorious that the so-called " ships " of war, the galleys, were manned by slaves, even down to modern times. It is difficult to determine the efficiency of labor of this kind because we are generally ignorant as to the time factor, but whether from our modern point of view inefficient or not, the results were often re- markable and sometimes, as in the case of the Pyramids, stupendous. Julius C^sar and the Julian Calendar. — Julius Csesar himself undertook two great problems of practical mathematical science : — the rectification of the highly confused calendar, and a survey of the whole Roman empire. In the year 47 B.C. the accumulated calendar error amounted to not less than 85 days. Reform was accomplished by a decree making the year con- sist of 365 days with an additional day in February once in four years. The survey, of which the results were to be shown in a great fresco map, was not carried out until the reign of Augustus. Vitruvius on Architecture. — The most famous ancient work on building and kindred topics, including building materials, is that entitled Be Architedura, by Vitruvius, a Roman architect and engineer living (about 14 B.C.) in the age of Augustus. This celebrated work was the only one of importance on architecture known to the Middle Ages, and was the guide and text-book of the builders of that period as well as of those of the Renaissance. The book (now easily accessible in translation) is in part a 144 A SHORT HISTORY OF SCIENCE compilation from earlier, and especially Greek, authors, and ii part original. Vitruvius uses for ir the value 3|, — less ao curate than that of Archimedes, but displaced later by the crude approximation 3. Of Vitruvius's life and work almost nothing i; known, but no other ancient treatise of a similar technical nature has had in its own field so much influence on posterity. Frontinus on the Waterworks of Rome (c. 40-103 a.d.) At about the end of the first century of our era, Sextus Julius Frontinus, a Roman soldier and engineer, wrote a highly interest- ing and valuable account of the waterworks of Rome. Frontinus served as proetor under Vespasian ; was afterwards sent to Britair as Roman governor of that island ; was superseded by Agricola ir 78 A.D. and was appointed in 97 a.d. Curator Aquarum, "an office never conferred except upon persons of very high standing." Roman Natural Science and Medicine. — Among the Romar workers and authors of importance in the history of natural science and medicine only a few require more than passing notice. This is the more remarkable when we reflect upon the vast extension o1 the Roman empire and the novel and hitherto unequalled op- portunities afforded for observation and collection in natura history, and for the study of anthropology, geography, geology meteorology, climatology, zoology, botany and the like, — not t( mention military surgery, and the hygiene and sanitation of camp' life. Lucretius (98-55 b.c.) is to-day regarded not only as a greai Roman poet but also as the most perfect exponent in his time o the natural philosophy of the Greeks who preceded him. He wa: a contemporary and a few years the junior of Cicero and Juliu Caesar. The first two books and the fifth of his De Rerun Natura (On the Nature of Things) are of interest to the moderi scientific student, because of their dealing with problems of per manent importance to mankind. He was a disciple of Epicurus and apparently also well acquainted with the works of Empedocles Democritus, Anaxagoras, and many other of the great Greek writer such as Homer, Hippocrates, Thucydides, and especially Euripides The title of his famous poem shows his interest in natural philos THE ROMAN WORLD 145 ophy, and there is evidence that he was also a teacher and re- former. He is antagonistic to superstition and a strong advocate of rationalism, but he is neither irreverent nor revolutionary. The following passages are typical : — Water in summer time flows cool in wells, Because the Earth then rarefied by heat. Its proper stores most radiate to the air. Hence more the Earth is drained of its heat. And colder grow the currents under ground. But when by cold in winter 'tis compressed. Its heat escaping passes into wells. . . . And now to tell by which of Nature's laws. The stone called Magnet by the Greeks, — since first 'Mong the Magnesians found, — can iron draw. Men gaze with wonder on the marvellous stone. With pendent chain of rings, oft five or more. Light hanging in the air suspensive, while One from another feels the influence of the stone That sends through all its wonder-working power. Here many principles we must flrst lay down And slow approach by long preparative. Rightly to solve the rare phenomenon. The more exact I then attentive ears. . . . How different is fire from piercing frost ! Yet both composed of atoms toothed and sharp. As proved by touch. Touch, O ye sacred powers — Touch is the organ whence all knowledge flows ; Touch is the body's sense of things extern. And of sensations that deep spring within ; Whether delightsome, as in genial act. Or rude collision torturing from without ; How difi'erent, then, must forms of atoms be Which such sensation varied can produce ! Strabo, — a Roman traveller, historian and geographer, lived somewhere between 63 B.C. and 24 a.d. His Geography is the most important work on that subject surviving from antiquity and, 146 A SHORT HISTORY OF SCIENCE while apparently building on the foundation laid by Eratosthenes, is plainly an original work devoted largely to his own explorations and observations during years of travel and study in different countries, including Italy, Greece, Asia Minor, Egypt, and Ethiopia. He himself says : — Westward I have journeyed to the parts of Etruria opposite Sar- dinia ; towards the South from the Euxine to the borders of Ethiopia, and perhaps not one of those who have written geographies has visited more places than I have between those limits. His work is invaluable as a picture of the limited geographical knowledge of the time, but he had no such mathematical knowledge of geography as had his great predecessors, Eratosthenes, Hip- parchus, and Ptolemy. Pliny the Elder (23-79 a.d.), sometimes called Pliny the Naturalist, is another Roman of scientific attainments, whose great work entitled Natural History, although more an encyclopaedia of miscellaneous information than a scientific treatise, is, nevertheless, like the works of Herodotus, a landmark in the history of civil- ization. It consists of thirty-seven books and is easily accessible in English. Pliny deals with the universe, God, nature, and natural phenomena ; with earth, stars, earthquakes ; with man, beasts, shells, fishes, insects, trees, fruits, gums, perfumes, timber, the diseases of plants, metals, stones, precious stones, etc. The author met his death in that eruption of Vesuvius which over- whelmed Pompeii in 79 a.d. and because of his scientific curiosity which led him to approach too near to the volcano. Galen (Claudius Galenus) who flourished in the second century a.d. was born and partly educated at Pergamum in Asia Minor, where, after much travelling, and research, chiefly in anatomy and philosophy at Smyrna and at Alexandria, he also practised the healing art. Sent for by the Roman emperor, Lucius Verus, he was afterward physician to Marcus Aurelius and his son Commodus. His writings are voluminous, encyclopedic and anatomically important, though not especially original, and his name is often linked with that of Hippocrates, partly, no doubt, THE ROMAN WORLD 147 because after Galen we find no great name in anatomy until we come to Vesalius, some 1400 years later. Late Roman Mathematical Science. — Two periods may be distinguished in ancient mathematical science, the first beginning with Pythagoras and ending with Hero. To these four to five centuries belong all the original works in geometry, astronomy, mechanics, and music. The period closes with the extension of the Tpax Romana over the Orient. The second extends to the sixth century, when Hellenism is proscribed by the new religion, the genius of invention is extinct, and men merely study the older works, commenting and coordinating. Astronomy gradually reverts to astrology, the mathematical geography well begun under Eratosthenes and Ptolemy becomes superficial and descrip- tive, with Strabo and even with Posidonius. fWhatever the eminence of the Romans in the practical arts of war, politics and engineering, their interest in abstract science was almost nil. On the other hand, commercial arithmetic, which had been studiously neglected by Greek mathematicians, now had the place of honor. The Roman numerals, clumsy as they seem to us, were superior to the Greek, and a useful system of finger-reckoning was developed, supplementing the skilful use • of the abacus. If no abacus was at hand, the corresponding lines were quickly traced on sand or dust, small stones or calculi — whence our words calculation and calculus, — serving as counters. A complete Roman abacus — of which no example has come down to us — seems to have had eight long and eight short grooves. Of the former, one held five counters or buttons, each of the others four, each of the short grooves one, these last counting as five units each. The grooves with six counters served for computa- tions with fractions. Geometry — but of Hero rather than of Euclid — was valued for its utility in surveying and architecture. Preparation for the engineering art included mathematics, optics, astronomy, history, and law. There were also teachers of me- chanics and architecture. (See Vitruvius, above.) Capella. — Early in the fifth century Martianus Capella wrote a compendium of grammar, dialectics, rhetoric, geometry, arith- 148 A SHORT HISTORY OF SCIENCE metic, music, and astronomy, of great and lasting educations influence. His classification of these "seven liberal arts" main tained itself throughout the Middle Ages and is not yet whollj extinct. Gregory of Tours for example says : — "If thou wil be a priest of God, then let our Martianus instruct thee first ii the seven sciences." BoETHius (480-524) born at Rome on the eve of its fall in 47f is the author not only of the famous Consolations of Philosophj but also of works on Music and on Arithmetic which long servec to represent Greek mathematics to the medieval world. In th( course of his public-spirited career, Boethius interested himsel in the reform of the coinage and in the introduction of water- clocks and sun-dials. His geometry consists merely of some o: the simpler propositions of Euclid, with proofs of the first thret only, and with applications to mensuration. Yet the intellectua poverty of the age was such that this remained long the standard for mathematical teaching. Boethius' Arithmetic begins : — By all men of old reputation who following Pythagoras' reputatior have distinguished themselves by pure intellect it has always beer considered settled that no one can reach the highest perfection ol philosophical doctrines, who does not seek the height of learning a1 a certain crossway — the quadrivium. For him the things of the world are either discrete (multitudes) or continuous (magnitudes). Multitudes are represented bj numbers, or in their ratios by music; magnitudes at rest are treated by geometry, those in motion by astronomy. These foui of the seven liberal arts form the quadrivium; grammar, dialec tics and rhetoric, the trivium. A Christian in faith, a pagan ir culture, Boethius has been called the " bridge from antiquity tc modern times." (See page 50.) The scholars of the time were almost without exception mer whose first interests were theological. Mathematics, having nc direct moral significance, seemed to them in itself unworthy o: attention. On the other hand, they attached exaggerated im- portance to all sorts of mystical attributes of numbers and to th( THE DARK AGES 149 interpretation of scriptural numbers. Thus Augustine says the science of numbers is not created by men, but merely discovered, residing in the natiu-e of things. Whether numbers are regarded by themselves or their laws applied to figures, lines or other motions, they have always fixed rules, which have not been made by men at aU, but only recognized by the keen- ness of shrewd people. Science and the Early Christian Church. — In the earlier centuries of our era the history of science gradually enters upon a new phase. The more highly developed civilization of Greece and Rome, weakened by corruption, has finally yielded to the attacks on the one hand of barbarous or semicivilized races, — Goths, Vandals, Huns, and Arabs, — and on the other hand to a moral revolution of humble Jewish origin. These changes were adverse to the development, or even the survival, of Greek science. The destructive relation of the northern barbarians to scientific progress may be easily imagined. The policy of official Christianity was based on antecedent antipathy for the unmoral intellectual attitude and the degenerate character which the early Christians found in close association with Greek learning, and on a too literal interpretation of the Jewish scriptures, with their primitive Chaldean theories of cosmogony and the world. Justin Martyr, in the second century, says that what is true in the Greek philosophy can be learned much better from the Prophets. Clement of Alexandria (d. 227) calls the Greek philoso- phers robbers and thieves who have given out as their own what they have taken from the Hebrew prophets. Tertullian (160-220) insists that since Jesus Christ and his gospel, scientific research has become superfluous. Isidore of Seville in the seventh century declares it wrong for a Christian to occupy himself with heathen books, since the more one devotes himself to secular learning, the more is pride developed in his soul. Lactantius early in the fourth century includes in his "Divine Institutions" a section, 'On the false wisdom of the philosophers,' of which the 24th chap- ter is devoted to heaping ridicule on the doctrine of the spherical 150 A SHORT HISTORY OF SCIENCE figure of the earth and the existence of antipodes. It is unnecessar, to enter into particulars as to his remarks about the absurdity of be heving that there are people whose feet are above their heads, am places where rain and hail and snow fall upwards, while the wonde of the hanging gardens dwindles into nothing when compared wit] the fields, seas, towns, and mountains, supposed by philosophers t^ be hanging without support. He brushes aside the argument o philosophers that heavy bodies seek the centre of the earth, as un worthy of serious notice ; and he adds that he could easily prove b; many arguments that it is impossible for the heavens to be lower tha] the earth, but he refrains because he has nearly come to the end of hi book, and it is sufficient to have counted up some errors, from whicl the quality of the rest may be imagined. It was natural that Augustine (354-430), . . . should express him self with . . . moderation, as befitted a man who had been i student of Plato as well as of St. Paul in his younger days. Witl regard to antipodes, he says that there is no historical evidence o their existence, but people merely conclude that the opposite side o the earth, which is suspended in the convexity of heaven, cannot b( devoid of inhabitants. But even if the earth is a sphere, it does no follow that that part is above water, or, even if this be the case, tha' it is inhabited ; and it is too absurd to imagine that people from ou) parts could have navigated over the immense ocean to the othe] side, or that people over there could have sprung from Adam. Witl regard to the heavens, Augustine was, like his predecessors, bounc hand and foot by the unfortunate water above the firmament. H< says that those who defend the existence of this water point tc Saturn being the coolest planet, though we might expect it to b( much hotter than the sun, because it travels every day through i much greater orbit ; but it is kept cool by the water above it. Th( water may be in a state of vapor, but in any case we must noi doubt that it is there, for the authority of Scripture is greater thai the capacity of the human mind. He devotes a special chapter t( the figure of the heaven, but does not commit himself in any waj though he seems to think that the allusions in Scripture to the heaver above us cannot be explained away by those who believe the work to be spherical. But anyhow Augustine did not, like Lactantius treat Greek science with ignorant contempt ; he appears to hav< had a wish to yield to it whenever Scripture did not pull him th( THE DARK AGES 151 other way, and in times of bigotry and ignorance this is deserving of credit. — Dreyer. Arguing elsewhere that the soul perceives what the bodily eye cannot, Augustine avails himself of the geometrical analogy of the ideal straight line which shall have length without breadth or thickness, but he lapses into mysticism when he passes to the circle. The biographer of St. Eligius (writing in 760 under Pepin) says 'What do we want with the so-called philosophies of Pythagoras, Socrates, Plato and Aristotle, or with the rubbish and nonsense of such shameless poets as Homer, Virgil and Menander ? What serv- ice can be rendered to the servants of God by the writings of the heathen Sallust, Herodotus, Livy, Demosthenes or Cicero?' Frede- gar . . . complains (about 600) that ' The world is in its decrepitude, intellectual activity is dead, and the ancient writers have no suc- cessors.' . . . — G. H. Putnam, Books of the Middle Ages. The following is a broad survey of the whole period : — The soft autumnal calm . . . which lingered up to the Antonines over that wide expanse of empire from the Persian Gulf to the Pillars of Hercules and from the Nile to the Clyde . . . was only a misleading transition to that bitter winter which filled the half of the second and the whole of the third century, to be soon followed by the abiding dark and cold of the Middle Ages. The Empire was moribund when Christianity arose. Rome had practically slain the ancient world before the Empire replaced the Republic. The barbarous Roman soldier who killed Archimedes absorbed in a problem, is but an in- stance and a type of what Rome had done always and everywhere by Greek art, civilization and science. The Empire lived upon and con- siuned the capital of preceding ages, which it did not replace. Popu- lation, production, knowledge, all declined and slowly died. . . . The sun of ancient science, which had risen in such splendour from Thales to Hipparchus, was now sinking rapidly to the horizon ; and when it at last disappeared, say, in the fifth century, the long night of the Middle Ages began. . . . The pursuit of knowledge for knowledge's sake was out of place. ... All the outlets through which modern energy is chiefly expended were then closed; a man could not serve the state as a citizen, he could not serve knowledge 152 A SHORT HISTORY OF SCIENCE as a man of science. . . . There was only one thing left for him t do, — to serve God. — J. C. Morison, The Service of Man. The Eastern Empire. Edict of Justinian. — Only half i century after the fall of Rome the Greek schools in Athens war closed, in 529 a.d., by order of the emperor Justinian, and intel lectual darkness settled down over Eastern Europe. Theolog; became more than ever the chief pursuit of the educated, and Greel learning more than ever neglected. Many Greek manuscripts however, were hidden away, and many Greek scholars, thougl scattered, kept alive the feeble spark of Greek learning. The Dark Ages. — After the mighty Roman Empire of tb West had come to its end, the peoples of Christian Europe and o the Graeco-Roman world descended into the great hollow which i; roughly called the Middle Ages, extending from the fifth to the fif teentl century, a hollow in which many great and beautiful and heroic thing were done and created, but in which knowledge, as we understand i and as Aristotle understood it, had no place. The revival of learninj and the Renaissance are memorable as the first sturdy breasting bj humanity of the hither slope of that great hollow which lies betweei us and the ancient world. The modern man, reformed and regeneratec by knowledge, looks across it and recognizes on the opposite ridge, ii the far-shining cities and stately porticoes, in the art, politics anc science of antiquity, many more ties of kinship and sympathy thai in the mighty concave between, wherein dwell his Christian ancestrj in the dim light of scholasticism and theology. — Morison. The "great hollow" here so graphically portrayed may be de- scribed as the Middle or Medieval Age (c. 450-1450 a.d.) and ol these ten centuries the first three, or thereabouts, are often callec the Dark — as they certainly were the darkest — Ages. The darkest time in the Dark Ages was from the end of the sixtl century to the revival of learning under Charles the Great (Charle- magne). Bad grammar was openly circulated and sometimes com- mended. St. Gregory the Great quoted the Bible in depreciation ol the Humanities. (Ps. Ixx. 15. 16.) The study of heathen authors was discouraged more and more. " Will the Latin grammar save ar immortal soul ? " " What profit is there in the record of pagan sageS; THE DARK AGES 153 the labors of Herciiles or of Socrates ?" Books came to be scarce. . . . But the decline of education was not universal. If studies failed in Gaul or Italy, they flourished in Ireland and afterward in Britain, and returned later from these outer borders to the old central lands of the Empire. Further, in spite of depression and discouragement, there was a continuity of learning even in the darkest ages and countries. Certain school books hold their ground . . . Capella . . . Boethius . . . Cassiodorus . . . And later Isodorus of Seville with a number of other authors are found in the ages of distress and anarchy more or less calmly gi\dng their lectures and preserving the standards of a liberal education. Much of this work was humble enough, but it was of great importance for the times that came after. . . . The darkest ages, with all their negligence, kept alive the life of the ancient world. Boethius [in the sixth century a.d., see p. 148] is the interpreter of the ancient world and its wisdom, accepted by all the tribes of Europe from one age to another, and never disqualified in his office of teacher even by the most subtle and elaborate theories of the later schools. . . . Cassiodorus (490-585) is wanting in the graces of Boethius, and he is much sooner forgotten ; but his enormous industry, his organization of literary production, his educational zeal have all left their effects indelibly in modern civilization. By his definition of the seven Liberal Arts, and by his examples of methods in teaching them, he is the spiritual author of the universities, the patron of all the available learning La the world. — Ker, Dark Ages. The Establisbmeijt of Schools by Charlemagne. — We have seen above how the schools of Athens were closed by Justinian in 529. Such schools as existed after that time were chiefly ecclesiastical and their teachings opposed to pagan or heathen {i.e. Greek) learning. At length, however, in 787 Charlemagne, moved it is said by the troublesome variety of writing as well as the general illiteracy of his people, ordered the establishment of schools in connection with every abbey of his realm, and summoned to take charge of them Peter of Pisa and Alcuin of York (735-804) (called by Guizot "the intellectual prime minister of Charlemagne"), whose names stand among the highest in a revival of learning thus begun in western Europe. 154 A SHORT HISTORY OF SCIENCE In the later part of the eighth century begins the great age ol medieval learning, the educational work of Charles the Great. . . There was some leisure and freedom and much literary ambition The Latin poets of the court of Charlemagne have an enthusiasm anc delight in classical poetry. ... In prose there was no less activity Besides the scientific treatises and the commentaries, the edifying works of Alcuin and others, there were histories. . . . The scholarlj spirit of the ninth century ... is not limited to the orthodox routine One of the chief scholars, with more Greek than most others, Erigena is famous for more than his learning, as a philosopher, who, whateve: his respect for the Church, acknowledged no authority higher thai reason. — Ker. Alcuin himself taught rhetoric, logic, mathematics and di- vinity, becoming master of the great school at St. Martin's oi Tours. Of his arithmetic the following problem is an illustra- tion : — If 100 bushels of corn are distributed among 100 people in sucl a manner that each man receives 3 bushels, each woman 2, and eacl child half a bushel ; how many men, women and children are there ' Of six possible solutions Alcuin gives but one. The mathematics taught in Charlemagne's schools woulc naturally include the use of the abacus, the multiplication table and the geometry of Boethius. Beyond this, a little Latin witl reading and writing sufficed for the needs of the church and he: servants, and was supplemented by music and theology for hei higher officers. The recognized intellectual needs of the work were indeed but slight. The civilization of Rome had beei gradually submerged by successive waves of barbaric invasioi from the north, as a similar fate was soon to be met by the stil higher culture of Alexandria. The best intellect of the times wa; perforce drawn into other forms of activity, while such scholar as remained found no favorable environment for fruitful study The Benedictine monasteries, indeed, sheltered a few studiou monks whose scientific interest scarcely extended beyond thi mathematics necessary for their simple accounts, and the com putation connected with the determination of the date of Eastei THE DARK AGES 165 Near the close of the tenth century Gerbert of Aquitaine (940- 1003), afterwards Pope Sylvester II, devoted his versatile genius in part to mathematical science. He constructed not only abaci, but terrestrial and celestial globes, and collected a valuable library. To him were also attributed a clock, and an organ worked by steam. He wrote works on the use of the abacus, on the division of numbers and on geometry. The last named contains a solution of the rela- tively difficult problem to find the sides of a right triangle whose hypotenuse and area are given. Unfortunately the latter part of his life was absorbed in political intrigue and his death in 1003 cut short his plans for attempting the recovery of the Holy Land. Out of the schools of Charlemagne gradually grew up that subtle, minute and over-refined learning of the later Middle Ages which has come to be known as Scholasticism. Based as it was upon authority instead of experiment, and magnifying, as it did, details more than principles, it sharpened rather than broad- ened the intellect, and was indifferent if not unfavorable to science. References for Reading LtJCH.ETrDS. On the Nature of Things. Strabo. Geography. Flint. Natural History. Frontinijs. The Waterworks of Rome. (Tr. by C. Herschel.) ViTRUVius. On Architecture. Ker. The Dark Ages. Gibbon. Decline and Fall of the Roman Empire. Galen. On the Natural Faculties. CHAPTER VIII HINDU AND "ARABIAN SCIENCE. THE MOORS IN SPAIN The grandest achievement of the Hindus and the one which of all mathematical investigations, has contributed most to th( general progress of intelligence, is the invention of the principle o position in writing numbers. — Cajori. Indeed, if one understands by algebra the application of arith metical operations to composite magnitudes of all kinds, whether thej be rational or irrational number or space magnitudes, then the learne( Brahmins of Hindustan are the true inventors of algebra. — Hankel. In the ninth century the School of Bagdad began to flom-ish, jus when the Schools of Christendom were falling into decay in the Wes and into decrepitude in the East. The newly-awakened Moslem in telleet busied itself at first chiefly with Mathematics and Medica Science ; afterwards Aristotle threw his spell upon it, and an immens( system of orientalized Aristotelianism was the result. From th( East, Moslem learning was carried to Spain ; and from Spain Aristotl( reentered Northern Europe once more, and revolutionized the intel lectual life of Christendom far more completely than he had revolu tionized the intellectual life of Islam. — Rashdall. Alexandria fell to the Arabs in 641 a.d. As a matter of his torical perspective it is noteworthy that the interval between it; foundation by Alexander the Great and its capture by th( Mohammedans, — during most of which period it was the in tellectual centre of the world, — is almost equal to that betweei Charlemagne's time and our own. The preservation and transmission of portions of Greek scienc( through the Dark Ages to the dawn of science in western Europe about 1200 A.D. was mainly effected through three distinct, thougl not quite independent, channels. First, there was to a limite( extent a direct inheritance of ancient learning within the Italiai peninsula, through all its political and military turmoil. Second a substantial legacy was received indirectly through the Moor in Spain ; while, third, additions of great importance came late 156 HINDU, ARABIAN AND MOORISH SCIENCE 157 through Italy from Constantinople. Before following the direct Latin-Italian line a brief sketch of Hindu and Arabic science is desirable. Hindu Mathematics. — The far-reaching conquests of Alex- ander the Great (330 B.C.) immensely stimulated communica- tion of ideas between the Mediterranean world and Asia, and the East was able to make certain great contributions to mathematical science just where the Greeks were relatively weakest, namely in arithmetic and the rudiments of algebra and trigonometry. Several centuries before our era the Pythagorean theorem and an excellent approximation for "^2 were known in India in connection with the rules for the construction of altars. The mathematicians however from whom we trace the later development of mathematics date from the sixth and following centuries. About 530 A.D. Arya-bhata wrote a book in four parts dealing with astronomy and the elements of spherical trigonometry, and enunciating numerous rules of arithmetic, algebra and plane trigo- nometry. He gives the sums of the series 1 +2 -I-... +71 1^ + 22 + ... + n^ 1^+23 + ... + n^ solves quadratic equations, gives a table of sines of successive mul- tiples of 3|° — i.e. twenty-fourths of a right angle, — and even uses the value tt = 3.1416, correct to five places. His geometry is in general inferior. Some years later, Brahmagupta composed a ' system of as- tronomy in verse, with two chapters on mathematics. In this he discusses arithmetical progression, quadratic equations, areas of triangles, quadrilaterals and circles, volume and surface of pyramids and cones. His value of tt is '^lO = 3.16 +• Typical problems and discussions are the following : — Two apes lived at the top of a cliff of height 100, whose base was distant 200 from a neighboring village. One descended the cliff, and walked to the village, the other flew up a height x and then flew in a 158 A SHORT HISTORY OF SCIEISJCJa straight line to the village. The distance traversed by each was the same. Find x. Beautiful and dear Lilavati, whose eyes are like a fawn's ! tell me what are the numbers resulting from one hundred and thirty-five, taken into twelve ? if thou be skilled in multiplication by whole or by parts, whether by subdivision or form or separation of digits. Tell me, auspicious woman, what is the quotient of the product divided by the same multiplier? The son of Pritha exasperated in combat, shot a quiver of arrows to slay Carna. With half his arrows, he parried those of his an- tagonist ; with four times the square-root of the quiver-full, he killed his horse ; with six arrows, he slew Salya ; with three he demolished the umbrella, standard and bow ; and with one, he cut off the head of the foe. How many were the arrows, which Arjuna let fly ? For the volume contains a thousand lines including precept and example. Sometimes exemplified to explain the sense and bearing of a rule ; sometimes to illustrate its scope and adaptation ; one while to show variety of inferences ; another while to manifest the principle. For there is no end of instances ; and therefore a few only are exhibited. Since the wide ocean of science is difficultly traversed by men of little understanding ; and, on the other hand, the intelligent have no occa- sion for copious instruction. A particle of tuition conveys science to a comprehensive mind ; and having reached it, expands of its own im- pulse. As oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does science infused into a wise mind spread by intrinsic force. It is apparent to men of clear understanding, that the rule of three terms constitutes arithmetic; and sagacity, algebra. Accordingly I have said in the chapter of Spherics : ' The rule of three terms is arithmetic ; spotless understanding is algebra. What is there unknown to the intelligent? Therefore, for the dull alone, it is set forth.' Five centuries later Bhaskara also wrote an astronomy contain- ing mathematical chapters, and the contents of this work soon became known through the Arabs to western Europe. While the preceding writers had no algebraic symbolism, but depended laboriously on words and sentences, Bhaskara made considerable progress in abbreviated notation. A partial list of subjects, treated HINDU, ARABIAN AND MOORISH SCIENCE 159 in his first book, includes weights and measures, decimal numera- tion, fundamental operations, addition etc., square and cube root, fractions, equations of the first and second degrees, rule of three, progressions, approximate value of tt, volumes. Applications are made to interest, discount, partnership, and the time of filling a cistern by several fountains. While there is reason to believe that the decimal system was known as early as the time of Brahma- gupta, this work contains the first systematic discussion of it, including the so-called Arabic numerals and zero. As an intermediate stage between the earlier use of entire words and our modern employment of single letters, he employs abbre- viations, but multiplication, equality and inequality have still to be written out. The divisor is written under the dividend without a line, one member of an equation under the other with verbal context to insure clearness. Polynomials are arranged in powers, though without our exponents, coefficients follow the un- known quantities. In his "rides of cipher" he even gives the equivalent of a ± = a, 0^ = 0, Vo = 0, a -=- = oo. In comparison with Greek mathematics, power and freedom are gained at the cost of some sacrifice of logical rigor. Among the Greeks, only the greatest appreciated the possibility and the importance of an unending series of numbers; but the Hindu imagination tended naturally in this direction. A notable achieve- ment of the Hindus was the introduction of the idea of negative numbers and the illustration of positive and negative by assets and debts, etc. On the whole, the Hindus, having received a part of their mathematics originally from the Greeks, made great contributions on the arithmetical and algebraic side, their influence on Euro- pean science with which they had little or no direct contact being exerted mainly through the Arabs. The Hindu mathematicians had no interest in what is termed mathematical method. They gave no definitions; preserved little logical order; they did not care whether the rules they used were properly established or not and were generally indifferent to funda- mental principles. They never exalted mathematics as a subject IbU A SHOKT HISTUKY UH »UiJl.ixn^Jl> of Study and indeed their attitude to learning may be described a; decidedly unmathematical. — G. R. Kaye. Hindu Astronomy. — In astronomy a parallel developmeni took place. It seems probable that Greek planetary theorj was introduced into India between the times of Hipparchus anc Ptolemy, but Hindu astronomy is characterized as "a curious mixture of old fantastic ideas and sober geometrical methods oi calculation." Aryabhata says indeed "The sphere of the stars is stationary, and the earth, making a revolution, produces the daily rising and setting of stars and planets," an opinion rejected by the later Brahmagupta. Mohammed and the Hegira. — During the sixth and following centuries great events were happening in Arabia, an ancientlj settled country, but up to that time a blank in the history oi civilization and of science. In 569 a.d. or thereabouts was born probably in Mecca, — an insignificant commercial town 45 miles from the middle eastern shore of the Red Sea, — that extraordinarj man Mohammed, whom millions of his fellow men still regard as the Prophet of the Almighty (Allah) . In 622 Mohammed fled witl a small company of his disciples to Medina, an agricultural towr 250 miles to the north of Mecca, where he prosecuted his prop' aganda, and completed his Koran, — the Mohammedan Bible Here also he died in 632 a.d. In Mecca, Mohammed was "the despised preacher of a smal congregation," but after his flight (hejira) to Medina, he became the leader of a powerful party and ultimately the autocrati( ruler of Arabia. Even before his death his followers numberec thousands, while the religious zeal with which they were firec has never been surpassed. Taught by Mohammed to conver or kill, they threw themselves upon their neighbors with i fanatical fury which overcame all obstacles, so that within on( short century their religion and its adherents swept like a tida wave from the barren valleys of western Arabia northward am eastward through Syria over Asia Minor and Mesopotamia, am northwestward along the African shores of the Mediterranean t( HINDU, ARABIAN AND MOORISH SCIENCE 161 the straits of Gibraltar. Egypt, Alexandria, and Carthage fell before the Mohammedans, and the Arabian or Moslem empire soon rivalled in extent its great predecessor, the Roman. In 711 Moslems crossed the straits of Gibraltar and entered Spain, soon pushing northward into western France as far as Poitiers, where their great western and northern movement was finally checked by Charles Martel, in 732. This extraordinary onrush, occurring almost within a single century, natiu-ally left the Moslems little time for the development of learning or for the arts and sciences. But after it was over, the Mohanmiedan invaders settled down in their various conquered countries and in some of them cultivated the arts of peace. The successive Arabian rulers (beginning with Al-Mansur, in 754) patronized learning, and to this end collected Greek manuscripts, which, after the closing of the Greek schools by Justinian in 529, had become scattered abroad. In particular, certain Nestorian Jews were brought to Bagdad and by them translations into Arabic were made of some of the works of Aristotle, Euclid, Ptolemy, and other Greek authors. The learning of India was also drawn upon, especially for the so-called Arabic numerals. Thus began a kind of Arabian science, chiefly imported at the outset, but destined within the next three centuries to take on characteristics of its own. It was, however, under the Caliph Al-]\Iamun (813-833), who has been called, as regards schools and learning, the Charlemagne of his people, that Aristotle was first translated into Arabic. Al-Mamun caused works on mathematics, astronomy, medicine, and philosophy to be translated from the Greek, and founded in Bagdad a kind of academy called the "House of Science," with a library and an observatory. Arabian Mathematical Science. — While the Arabs them- selves were not in general much addicted to scientific pursuits, then- relations to the Greeks and Hindus, and subsequently to the nations of western Europe are of very great importance in the history of science. Even if we accept as typical the traditional dictum attributed to the Caliph Omar, that whatever in the library of Alexandria agreed with the Koran was superfluous, whatever disagreed was worse, and all should therefore be destroyed, it was 162 A SHORT HISTORY OF SCIENCE inevitable that individuals in this active-minded race should fal under the spell of Greek mathematical science. Their religioi was in fact more tolerant towards science than was contemporar;; Christianity. It would appear that by 900 a.d. the Arabs were familiar on th( one hand with Brahmagupta's arithmetic and algebra, including the decimal system, and on the other hand with the chief works o: the great Greek mathematicians, some of which have come dowr to us only through Arabic translations. The Algebra of Alkarisml written about 830 was based on th( work of Brahmagupta, and served in turn as the foundation foi many later treatises. From its title is derived our word " algebra,' from the author's name our " algorism." The book begins : — The love of the sciences with which God has distinguished Al- Mamun, ruler of the faithful, and his benevolence to scholars, have encouraged me to write a short work on computation by completior and reduction. Herein I limited myself to the simplest matters, anc those which are most needed in problems of distribution, inheritance partnership, land measurement, etc. The first book contains a discussion of five types of quadratic equations : ax^ = bx, ax^ = c, ax^ + bx = c, ax^ + c = bx, ax^ = bx -\- c; only real positive roots are accepted, but, unlike the Greeks, hf recognizes the existence of two roots. He gives a geometri- cal solution of the quadratic equatior analogous to those of Euclid. Supposf x^ + 10.x = 39 and let AB = BC = x AH = CF = 5; then the areas ar( B The sum of these is x^ + lOx. Com- plete the square HF by adding KE = 25 G F HF = (x + 5y = 64, whence x = 3. The series 1" + 2" + 3" + ... + m" was summed for n = 1,2,3,4 and about 1000 a.d. Alkayami is said to have asserted the im^ possibility of finding two cubes whose sum should be a cube. Evei HINDU, ARABIAN AND MOORISH SCIENCE 163 a cubic equation was solved by the aid of intersecting conic sec- tions. There is no definite separation of algebra and arithmetic, and the former, in spite of relatively rapid development, remains entirely rhetorical. The division line of fractions is introduced and the check of computation by "casting out nines." In Physics, Al-Hazen (965 ? — 1038) wrote a work on optics enimciating the law of reflection and making a study of spherical and parabolic mirrors. He also devised an apparatus for studying refraction, being probably the first physicist to note the magnify- ing power of spherical segments of glass — i.e. lenses. He gave a detailed account of the htmaan eye, and attempted to explain the change of apparent shape of the sun and the moon when ap- proaching the horizon. The Arabs employed the pendulum for time measurement, and tabulated specific gravities of metals, etc. In the words of a modern physicist : — The Arabs have always reproduced what came down to them from the Greeks in thoroughly intelligible form, and applied it to new prob- lems, and thus built up the theorems, at first only obtained for par- ticular cases, into a greater system, adding many of their own. They have thus rendered an extraordinarily great service, such as would correspond in modern times to the investigations which have grown out of the pioneer work of such men as Newton, Faraday and Rontgen. — Wiedemann. Arabian Astronomy. — In connection with astronomy the Arabs, following Gjeek precedents, developed trigonometry, in- troducing sines and other functions since current. They used masonry quadrants of large size, and even a combination of a horizontal circle with two revolving quadrants mounted upon it, foreshadowing the modern theodolite. Better and more com- plete observational data facilitated, and at the same time de- manded, improved mathematical methods, while the necessary computations were accomplished much more economically, through the use of the decimal number system. Haroun Al-Raschid sent to Charlemagne an ingenious water clock, while under his suc- cessor, Al-Mamun, two learned mathematicians were commis- sioned to measure a degree of the earth's circumference. 164 A SHORT HISTORY OF SCIENCE ' Choose a place in a level desert and determine its latitude. Thei draw the meridian line and travel along it towards the pole-star Measure the distance in yards. Then measure the latitude of th( second place. Subtract the latitude of the first and divide the differ ence into the distance of the places in parasangs. The result multi plied by 360 gives the circumference of the earth in parasangs.' — Wiedemann. The writer just quoted describes a second method involving the measurement of the angle of depression of the horizon ai seen from the top of a high mountain. It is not improbable that western Europe acquired fron eastern Asia, through Arab channels, the mariner's compass anc gunpowder. ' When the night is so dark that the captains can perceive no star t( orient themselves, they fill a vessel with water and place it in the in terior of the ship, protected from wind ; then they take a needle anc stick it into a straw, forming a cross. They throw this upon the wate: in the vessel mentioned and let it swim on the surface. Hereupon thei take a magnet, put it near the surface of the water, and turn thei: hands. The needle turns upon the water; then they draw thei: hands suddenly and rapidly back, whereupon the needle points ii two directions, namely north and south.' 1232 a. d. — Wiedemann. The astronomical theory of the Arabs was merely that o: Ptolemy. But they "were not content to consider the Ptolemai( system merely as a geometrical aid to computation ; they requirec a real and physically true system of the world, and had therefor( to assume solid crystal spheres after the manner of Aristotle.' The various attempts to devise a better system all miscarried their authors having no new guiding principle, nor superior mathe matical power, and being more or less hampered by Aristoteliai traditions, though Greek theories of the rotation of the eartl seem not to have been unknown. Asiatic Observatories. — Besides the work of the Arabiai astronomers themselves, it is an interesting fact that their bar barian Mongol conquerors in the East acquired a temporaril; HINDU, ARABIAN AND MOORISH SCIENCE 165 active scientific interest, founding a fine observatory at Meraga near the northwest frontier of modern Persia. The instruments used here are said to have been superior to any used in Europe until the time of Tycho Brahe in the sixteenth century. The principal achievement of this observatory was the issue of a revised set of astronomical tables for computing the motions of the planets, together with a new star catalogue. The excellence of their work may be inferred from a determination of the precession of the equinoxes within 1". This development lasted only a few years in the latter half of the thirteenth century. A similar brief out- burst of astronomical activity occurred among the Tartars at Samarcand (Russian Turkestan) nearly 200 years later, that is, a little before the time of Copernicus, and here the first new star catalogue since that of Ptolemy was compiled. It is noteworthy that there was no hostility between science and the Moham- medan church. One of the uses of astronomy indeed was to de- termine the direction of Mecca. No great original idea can be attributed to any of the Arab and other astronomers here discussed. They had, however, a remark- able aptitude for absorbing foreign ideas, and carrying them slightly further. They were patient and acciu-ate observers, and skilful calculators. We owe to them a long series of observations, and the invention or introduction of several important improvements in mathematical methods. . . . More important than the actual contributions of the Arabs to astronomy was the service that they performed in keeping alive interest in the science and preserving the discoveries of their Greek predecessors. — Berry. The Moors in Spain.— We have aheady touched above upon the rapid spread of Mohammedanism westward from its home in Arabia, and the remarkable conquests of its followers in Spain and western France. These western Mohammedans included not only some of pure Arabian stock, but more of mixed descent, especially from that part of northern Africa once known as Maure- tania, — whence the term Moors, generally applied to the con- quering Mohammedans of the west. The Moors entered Spain 166 A SHORT HISTORY OF SCIENCE early in the eighth century, bringing after them the learning of th Arabs, so that Hindu and Arabian science, and to some exten Greek science, were making their way into southwestern Europ even before the schools of Charlemagne were established towar( the end of the same century in central (Christian) Europe. In th ninth and tenth centuries a remarkable civilization arose in Spain, - the highest that the Arabian race has ever reached. The develop ments of science in Mohammedan Spain are more or less typica of what occurred throughout the whole Arabian empire, and i] such cities as Cordova, Toledo, and Seville a type of civilizatioi and a stage of learning were reached higher in many respects thai existed at the same time and even for centxu-ies afterward any where in Christian Europe. Scarcely had the Arabs become firmly settled in Spain when the^ commenced a brilliant career. Adopting what had now become thi estabhshed policy of the Commanders of the Faithful in Asia, tH( Emirs of Cordova distinguished themselves as patrons of learning and set an example of refinement strongly contrasting with the con dition of the native European princes. Cordova, under their ad ministration, at its highest point of prosperity, boasted of more thai two hundred thousand houses, and more than a million of inhabitants After sunset, a man might walk in a straight line for ten miles by th( light of the public lamps. Seven hundred years after this time then was not so much as one public lamp in London. Its streets wen solidly paved. In Paris, centuries subsequently, whoever stepped ovei his threshold on a rainy day stepped up to his ankles in mud. — Draper. The Mohammedans made some additions to medical science and yet theu- medicine hardly goes beyond that of Galen, whon they specially revered. In alchemy they are notable, thougl more by their attempts than their achievements. Too oftei it was simply a search for "potable gold" or other " elixirs o: life," "the philosopher's stone," and the like. In the arts anc industries, however, the Moors deserve special mention. Cordovai and Morocco leather are well known. Toledo and Damascui blades (swords) were long famous. Arabian horses fur HINDU, ARABIAN AND MOORISH SCIENCE 167 nished Europe with one of the most serviceable strains of that useful animal, and many Arabian words have been adopted into our language, e.g. alcohol, elixir, algebra, alembic, zenith, nadir, etc. "... Under the caliphs, Moslem Spain became the richest, most populous, and most enlightened country in Europe. The palaces, the mosques, bridges, aqueducts, and private dwellings reached a luxiu-y and beauty of which a shadow still remains in the great mosque of Cordova. New industries, particularly silk weaving, flourished exceedingly, 13,000 looms existing in Cordova alone. Agriculture, aided by perfect systems of irrigation for the first time in Europe, was carried to a high degree of perfection, many fruits, trees and vegetables hitherto unknown being introduced from the East. Mining and metallurgy, glass making, enamelling, and damascening kept whole populations busy and prosperous. From Malaga, Seville, and Almeria went ships to all parts of the Mediterranean loaded with the rich produce of Spanish Moslem taste and industry, and of the natural and cultivated wealth of the land. Caravans bore to farthest India and darkest Africa the precious tissues, the marvels of metal work, the enamels, and precious stones of Spain. All the luxury, culture, and beauty that the Orient could provide in return, found its way to the Moslem cities of the Peninsula. The schools and libraries of Spain were famous throughout the world ; science and learning were cultivated and taught as they never had been before. Jew and Moslem, in the friendly rivalry of letters, made their country illustrious for all time by the productions of their study. . . . The schools of Cordova, Toledo, Seville, and Saragossa attained a celebrity which subsequently attracted to them students from all parts of the world. At first the principal subjects of study were literary, such as rhetoric, poetry, history, philosophy, and the like, for the fatalism of the faith of Islam to some extent retarded the adoption of scientific studies. To these, however, the Spanish Jews opened the way, and when the barriers were broken down, the Arabs themselves entered with avidity into the domain of science. Cordova then became the centre of scientific investigation. Medicine and surgery especially were piu-sued with intense diligence and success, and veterinary surgery may be said to have there first crystallized into a science. Botany and pharmacy also had their famous professors, and astronomy was studied and 168 A SHORT HISTORY OF SCIENCE taught as it had never been before; algebra and arithmetic wert applied to practical uses, the mariner's compass was invented, anc science as applied to the arts and manufactures made the products ol Moslem Spain — the fine leather, the arms, the fabrics, and the metaj work — esteemed throughout the world. . . . Canals and watei wheels for irrigation carried marvellous fertility throughout the soutl: of Spain, where the one thing previously wanting to make the land s paradise was water. Rice, sugar, cotton, and the silkworm were all introduced and cultivated with prodigious success ; the silks, brocades, velvets, and pottery of Valencia, the beautiful damascened steel o\ Seville, Toledo, Murcia, and Granada, the stamped embossed leathei of Cordova, and the fine cloths of Seville brought prosperity tc Moslem and Mozarab alike under the rule of the Omeyyad caliphs^ while the systematic working of the silver mines of Jaen, the corals or the Andalusian coasts, and the pearls of Catalonia supplied the ma- terial for the lavish splendor which the rich Arabs affected in then attire and adornment. The Moors of Andalusia and Valencia acclimatized and cultivated a large number of semitropical fruits and plants hitherto little knowr in Europe, and studied arboriculture and horticulture not only practi- cally but scientifically. The famous work on the subject by Abu Zacaria Al-Awan was the foundation of such books, and of the applica- tion of science to gardening. It was mainly derived from Chaldean Greek, and Carthaginian manuscripts now lost. Curiously, Spain hac produced under the Romans a famous book on agriculture by Colu- mella : but for scientific knowledge it cannot be compared to th< Treatise on Agriculture by Abu Zacaria. . . . From the earliesi times the wool of Spain had been the finest in the world. . . . Vasi herds of stunted, ill-looking, but splendidly fleeced sheep belonged t( the nobles and ecclesiastical lords, and quite early in the period of re conquest, when these classes were all-powerful, a confederacy of sheej owners was formed, which by the fourteenth and fifteenth centurie: had developed into a corporation of immense wealth. This was calle( the Mesta. . The fleeces were extremely fine, often weighini 12 pounds per animal, and the wool was sought after throughout th^ world, especially by Flemish and French cloth workers. Even in th ninth century Spanish wool was famous in Persia and in the East and as early as the time of the Phoenicians it was considered th finest in the world. — Hume. HINDU, ARABIAN AND MOORISH SCIENCE 169 The tenth century was the golden age of Moorish science in Spain. Another hundred years and it had gone down forever. Its permanent importance, even in conserving the work of the ancients, has been questioned, and a recent writer cleverly com- pares the whole western movement of the Arabians to the sands of their deserts, — now fierce and pitiless when driven by some force such as the wind, — now sinking into inert, helpless, infertile heaps when left to themselves. An Arab renaissance as early as the eighth century had revived something of classic knowledge. The poetry and philosophy of Greece were studied, and the taste for learning was cultivated with the en- thusiasm which the Arabs infused into all their undertakings. Every one knows the fascinating account of these things in the pages of Gibbon. How the tide of progress flowed from Samarcand and Bo- khara to Fez and Cordova. How a vizir consecrated 200,000 pieces of gold to the foundation of a college. How the transport of a doctor's books required four hundred camels. How a single library in Spain contained 600,000 volumes, while seventy public libraries were opened in Andalusia alone. How the Arabian schools of Spain and Italy were resorted to by scholars from every country in Europe. . . . Here was a state of luxury and learning which contrasted strongly enough with the barbarism of the age. But under all this show where was the substantial basis? How much of all this was real? Arab architecture, in so far as it was Arab, and not built for them by the Greeks, was a concoction of whim and fantasy. In those nervous hands every strong and simple feature was distorted into endless com- plications, and, as always happens, lost in stability what it gained in eccentricity. Their learning was of the same character. Though they disputed interminably on the rival merits of the Greek philos- ophers, they were content to receive all their knowledge of them through indifferent translations. When the real revival of learning came, and a genuine Renaissance set in, the six or seven centuries of Arab civilization were simply ignored and passed over. . . The Arab mind seems to turn by a sort of instinct to the occult, the mystical, the fantastic. It is always sighing for new worlds to conquer before it has made good the ground it stands on. It has the curious gift of turning everything it touches from substance to shadow. 170 A SHORT HISTORY OF SCIENCE Astronomy changes into astrology, and the main business of the scieno becomes the casting of horoscopes. The study of medicine changei into the composition of philtres and talismans and the reciting of incan tations. Chemistry changes into a search for the secret of the trans mutation of metals and the elixir of immortal health. In short, th( tendency always was to shift the appeal from the intellect and reasoi to the fancy and imagination ; and their zeal, instead of being devotee to laying firm foundations, evaporated in vague aspirations after th( unintelligible or the unobtainable. . . . And the consequence is that not only has the Arab left us litth or nothing, but his whole history seems already more legendary thai real. Other civilizations abide our question. Not the Greek anc Roman only, but the remote Assyrian and Egyptian, are definite anc real in comparison with the Arabian. This seems of another texture It is such stufl^ as dreams are made of. Those so-called conquests of his [the Arab's] were really the taking advantage of a unique opportunity for destroying and pulling down The collapse of the Western Empire, and weakness and paralysis oi the Eastern, afi'orded the Arab a fine field for the display of his peculiai prowess. He took to the lumber and debris of these crumbling empires as fire takes to rotten wood. But if in the void that separates ancieni civilisation from modern the Arab appears to advantage, there nc sooner entered on the scene nations of solid character and creativ< genius than he retired before them, and yielded to their advance. — March Phillips. The Golden Age of Moorish learning in the tenth century cam( and went, leaving behind it singularly few permanent results Owing to the racial and religious hatreds of the time the Christiai conquerors of the Moslems, like their Roman prototypes in th( first few centuries after Christ, had small respect for Greek — anc less for Mohammedan — learning. Hence, doubtless, it cam( about that to-day in Cordova, for example, almost no traces re main of that Arabian learning of which it was once the celebratec seat. Even the site of its illustrious university has faded fron memory and only its great mosque (of which the heart is occupiec by a Christian church) remains to bear visible witness to Moham' medan Cordova. The same is true of other once famous centre; HINDU, ARABIAN AND MOORISH SCIENCE 171 of Spanish Mohammedan-Greek learning. Toledo still possesses some of its Arabian walls and gateways, and Seville its lovely Giralda — " the first astronomical observatory in Europe " — and its Tower of Gold ; but it is only in the Alhambra of Granada that any adequate vision can be had of Mohammedan life and influence in Spain. Here the quiet, the seclusion, the rich orna- mentation, and the music of abundant running waters, still com- municate an impression of wealth, taste, and power, and suggest possibilities of uninterrupted study and an intellectual life. Else- where, evidences of the Mohammedan love of inquiry, of libraries, of decoration, and even of fruits and gardens, have been almost wholly blotted out. References for Reading Ball. Chapter IX. Berrt. Chapter III. Draper. History of the Intellectual Development of Europe. Vol. II. Dreyer. Chapter XI. Hume, M. History of the Spanish People. March Phillips. In the Desert. Gibbon. Decline and Fall. CHAPTER IX PROGRESS OF SCIENCE TO 1450 A.D. It cannot be too emphatically stated that there is no historical evidence for the theory which connects the new birth of Eiu-ope with the passing away of the fateful millennial year and with it, the awful dread of a coming end of all things. Yet, although there was no breach of historical continuity at the year 1000, the date will serve as well as any other that could be assigned to represent the turning-point of European history, separating an age of religious terror and theological pessimism from an age of hope and vigor and active religious en- thusiasm. . . . The change which began to pass over the schools of France in the eleventh century and culminated in the great intellectual Renaissance of the following age, was but one effect of that general revivification of the human spirit which should be recognized as con- stituting an epoch in the history of European civilization not less momentous than the Reformation or the French Revolution. . . . The schools of Christendom became thronged as they were never thronged before. A passion for inquiry took the place of the old routine. The Crusades brought different parts of Europe into con- tact with one another and into contact with the new world of the East, — with a new Religion and a new Philosophy, with the Arabic Aris- totle, with the Arabic commentators on Aristotle, and eventually even with Aristotle in the original Greek. . . . Whatever the causes of the change, the beginning of the eleventh century represents, as nearly as it is possible to fix it, the turning-point in the intellectual history of Europe. — Rashdall. The Crusades. — From the time of Mohammed's hegira from Mecca to Medina in 622 a.d. to the siege of Vienna by his followers in 1683 — a period of more than 1000 years — Europe stood in constant dread of Mohammedan conquest. Fifteen years after the hegira, Jerusalem was captured by Omar, and remainec under Mohammedan control till the end of the first Crusade, sinc( which time it has been sometimes in Christian, sometimes ir Mohammedan, possession. Toleration of Christians in the Holj 172 PROGRESS OF SCIENCE TO 1450 A.D. 173 Land was, however, the rule until the eleventh century, and between 700 and 1000 a.d. pilgrimages to Jerusalem were fre- quently undertaken by Christians in the West. But after 1010 such pilgrimages began to be seriously interfered with, and matters steadily grew worse, until in 1071 Seljukian Turks displaced Arabian Mohammedans as rulers of Jerusalem. These Turks, though more rough than intolerant, eventually interfered with both trade and pilgrimages, until for this and other reasons the conquest of the Holy Land became a passion with the Christian nations. In the spring of 1097, after several years of widespread prepara- tion, a great host of western Christians, variously estimated at 150,000 to 600,000, gathered at Constantinople charged with war- like and religious zeal and bent on wresting Jerusalem and the Holy Land from the possession of the Mohammedan "Infidel." This was the beginning of those expeditions under the banner of the Cross, — hence known as the Crusades, — which may be regarded as intermittent reactions of the Christian West against the pressure of the Mohammedan East. The spirit of Christian Europe in the Middle Ages being essentially religious and ec- clesiastical, it was natural that its more bold and adventurous youth should regard with jealousy and indignation the wide extent of the Mohammedan empire and especially its possession of Jerusalem and other holy places. In all, eight such Crusades are recognized by historians, and of these the influence upon Christian Europe must have been immense. In the first place, the expansion of the intellectual outlook due to the mere experiences of travel, for men born and bred under the parochial limitations of feudalism and monasticism, must have been great. Then, too, the arts and appliances observed abroad, the different stand- ards of all sorts, the wealth and luxury of the distant East, doubt- less had a powerful effect upon Europe when reported or intro- duced by the Crusaders upon their return. When we reflect upon the ages of darkness which had rested upon Christian Europe from the fall of Rome into the hands of the barbarians to the fall of Jerusalem into the hands of the Turks — a period of almost exactly six hundred years — we may agree with those who are 174 A SHORT HISTORY OF SCIENCE disposed to look upon the Crusades as an age of discovery com parable with that of the new world by Columbus and his follow, ers, — but a discovery of the East instead of the West. The period of the Crusades extends over about two centuries viz : — from 1090 to 1290, and thus immediately precedes th( Renaissance, of which it was apparently one of the most im- portant factors. Trivium : QuADRiviuM. Scholasticism. — Meantime, follow' ing the mandate of Charlemagne establishing schools in connectior with all the abbeys and monasteries of his vast domain of centra Europe, a characteristic technical and essentially verbal scholarshif gradually arose which, although chiefly ecclesiastical in substance, and so narrow in its range as almost completely to neglect natura science, was often thorough and sometimes profound. This learning in its later development is known as "Scholasticism," oi which the foundation and essence was the famous curriculum oi "the seven liberal arts," founded upon the educational doctrine; of Plato, but adapted to the fashion of the Middle Ages. Thest consisted of a quadrivium — geometry, astronomy, music and arith- metic — and a trivium — grammar, logic, and rhetoric, (p. 148.^ In the introduction to the Logic of Aristotle which was in th( hands of every student even in the Dark Ages, the Isagoge of Por- phyry, the question was expHcitly raised in a very distinct anc emphatic manner. The words in which this writer states, withou' resolving, the problem of the Scholastic Philosophy, have played per haps a more momentous part in the history of thought, than anj other passage of equal length in all literature outside the canonica Scriptures. They are worth quoting at length : ' Next, concerning genera and species, the question indeed whethe they have a substantial existence, or whether they consist in ban intellectual concepts only, or whether, if they have a substantial ex istence, they are corporeal or incorporeal, and whether they are sep arable from the sensible properties of the things (or particulars o sense), or are only in those properties and subsisting about them, shall forbear to determine. For a question of this kind is a very dee] one and one that requires a longer investigation.' — Rashdall. PROGRESS OF SCIENCE TO 1450 A.D. 175 To show the low state of natural history it suffices to refer to an extraordinary work, the so-called Physiologus or Bestiary, a kind of scriptural allegory of animal life, originally Alexandrian, but surviving in mutilated forms and widely used in medieval times. The childish and grotesque character of this curious compendium shows how ill-adapted were the centuries of crusad- ing to the calm pursuits of science ; they were indeed almost barren in this direction. Scholasticism, nevertheless, lingered long after the Crusades were ended, and abundant survivals of it exist even today. Medieval Universities. — The origin of the universities which play so great a part in the cultivation and dissemination of learn- ing in the later middle ages is involved in obscurity. The medical school at Salerno in southern Italy seems to have become known in the ninth century, so that the University of Salerno is some- times called the oldest in Europe. It was still famous in the thirteenth century. The law school at Bologna, in northern Italy, became well known about 1000 a.d., though the date of the University of Bologna is usually given as near the end of the twelfth century. The University of Paris is often dated from the early part of the same century. None of these early univer- sities was much more than an association or gild of masters and pupils. Laboratories for instruction were of course unknown. In the eleventh and twelfth centuries there was a gradual de- velopment from the previous monastic schools to the beginnings of modern universities at Paris, Bologna, Salerno, Oxford, and Cambridge, the schools themselves however continuing along their previous lines ; and from that time onward to our own, the universities have played the chief part in the advancement of learning in general and of science in particular. In their develop- ment theological influences were naturally dominant, and it is interesting to observe that the use of Aristotle's Natural Philoso- phy, which became later the stronghold of orthodox conserva- tism, was prohibited in the thirteenth century. Medieval academic standards were naturally low. The univer- sity was a voluntary and privileged society of scholars. Not until 176 A SHORT HISTORY OF SCIENCE 1426 is there a record of the refusal of a degree for poor scholarship and the victim then sought redress by legal proceedings, thougl in vain. In most of the early universities logic, philosophy, an( theology were cultivated rather than even mathematical science. Transmission of Science through Moorish Spain. — Th( meagre rivulet of classical science derived directly from Greel and Roman sources is now mingled with the current which founc its way through northern Africa and Spain under the Moors Boethius' rudimentary work was supplanted, and before 1400 the first five books of Euclid were taught at many universities Ptolemy's Almagest was also translated from the Arabic int( Latin early in the twelfth century, probably with the use o: Arabic numerals. Near the close of the Moorish domination o: Spain, King Alfonso X of Castile (1223-1284) collected at Toledc a body of Christian and Jewish scholars who under his direc' tion prepared the celebrated Alfonsine Tables, using the nevi Arabic numerals. These enjoyed a high reputation for three centuries, though first printed in 1483. While we thus owe to the Arabs a considerable debt for pre serving for the use of later ages the precious heritage of Greel learning, the revival of learning in the fourteenth century came chiefly from other quarters and would probably have come in du( time even if Arabic influences had not been at work. Yet it ii noteworthy that early in the twelfth century re-translations of th( Greek classics began to be made from the Arabic, and these maj well have supplied the very limited demand for them tolerated bj the church for the next hundred years. In spite of jealous ex- clusiveness the learning of the great schools of Granada, Cordova and Seville gradually found its way to Paris, Oxford, and Cam- bridge. During the course of the twelfth century a struggle had been goinj on in the bosom of Islam between the Philosophers and the Theo- logians. It was just at the moment when, through the favor of th( Caliph Al-Mansur, the Theologians had succeeded in crushing th( Philosophers, that the torch of Aristotelian thought was handed on t( Christendom. . . . PROGRESS OF SCIENCE TO 1450 A.D. 177 It was from this time and from this time only (though the change had been prepared in the region of pure Theology by Peter the Lom- bard) that the Scholastic Philosophy became distinguished by that servile deference to authority with which it has been in modern times too indiscriminately reproached. And the discovery of the new Aris- totle was by itself calculated to check the originality and speculative freedom which, in the paucity of books, had characterized the active minds of the twelfth century. The tendency of the sceptics was to transfer to Aristotle or Averroes the authority which the orthodox had attributed to the Bible and the Fathers of the Church. — Rashdall. Dawn of the Renaissance. — In the thirteenth century it becomes plain that a new spirit is arising in Europe. We cannot faU to detect at this time the existence, even at places as far apart as Oxford and Bologna — infinitely further apart then than now, — of a widespread desire for knowledge and a zeal for learning such as had not been known for centuries. Arabic mathematical science is introduced from northern Africa by Leonardo Pisano. A fresh and notable philosopher — Albertus Magnus — appears. Thomas Aquinas writes his famous Imi- tatio Christi. Great Gothic cathedrals arise, more universities are founded, and, most noteworthy of all for the history of science, an original student of nature appears, in Roger Bacon. By the beginning of the thirteenth century, in consequence of the opening up of communications with the East — through inter- course with the Moors in Spain, through the conquest of Constanti- nople, through the Crusades, through the travels of enterprising scholars — the whole of the works of Aristotle were gradually making their way into the Western world. Some became known in translations direct from the Greek; more in Latin versions of older Syriac or Arabic translations. And now the authority which Aristotle had long enjoyed as a logician — nay, it may almost be said the authority of logic itself — communicated itself in a manner to all that he wrote. Aristotle was accepted as a well-nigh final authority upon Meta- physics, upon Moral Philosophy, and with far more disastrous results upon Natural Science. The awakened intellect of Europe busied 178 A SHORT HISTORY OF SCIENCE itself with expounding, analysing and debating the new treasure unfolded before its eyes. . . . And of the scientific side of this revival Italy was the centre This branch of the movement began, indeed, before the twelft] century. It was in Italy that the Latin world first came into con tact with the half-forgotten treasures of Greek wisdom, with th^ wisdom which the Arabs had borrowed from the Greeks and witl original products of the remoter East. Of the Medical School of Sa lerno we have already spoken. It was probably in Italy and througl the Arabic that the Englishman Adelard of Bath translated Euclic into Latin during the first half of the eleventh century. At abou the same time modern musical notation originated with the discoverie: of the Camaldulensian monk, Guido of Arezzo. In the first years o: the following century the Algebra and the Arithmetic which th( Arabs had borrowed from the Hindus were introduced into Italj by the Pisan merchant, Leonardo Fibonacci. . . It was to thii Arabo-Greek influence that Bologna owed its very important SchoO; of Medicine and Mathematics — two subjects more closely connected then than now through their common relationship to Astrology. — Rashdall. Mathematical Science in the Thirteenth Centuky. — Increasing activity in mathematical science was due largely tc Leonardo Pisano of Italy, Jordanus Nemorarius of Saxony, and Roger Bacon of England. Leonardo Pisano or Fibonacci (born 1175) was educated in Barbary, where his father was in charge of the custom-house, and thus became familiar with Alkarismi's algebra, and the Arabic decimal system. He appreciated their advantages and on his return to Italy published in his Liber Abaci an account which gave them currency in Europe "in order that the Latin race might no longer be deficient in that knowledge." As the mathe- matical masterpiece of the Middle Ages, it remained a standard for more than two centuries. His algebra is rhetorical, but gains by the employment of geometrical methods. He discusses the fundamental operations with whole numbers and fractions, using the present line for division. Fractions are decomposed into parts with unit numerators as in early Egypt. Through the Arab; PROGRESS OF SCIENCE TO 1450 A.D. 179 Leonardo inherits Egyptian as well as Greek traditions, for ex- ample, the type of fraction just mentioned, square and cube root, progressions, the method of false assumption. It would appear that when the Arabs conquered Alexandria some of the old Egyptian culture was preserved. The rule of three, partner- ship, powers and roots, and the solution of equations are also included. In 1225 the emperor, impressed by the accounts of Pisano's mathematical power, arranged a mathematical tournament of which the challenge questions are preserved : ' To find a number of which the square, when either increased or diminished by 5, would remain a square. 'To find by the methods used in the tenth book of Euclid a line whose length x should satisfy the equation a^ + 2a;^ + lO.r = 20. ' Three men. A, B, C, possess a sum of money u, their shares being in the ratio 3:2:1. A takes away x, keeps half of it, and deposits the remainder with D ; B takes away y, keeps f of it, and deposits the remainder with D ; C takes away all that is left, namely z, keeps f of it, and deposits the remainder with D. This deposit is found to belong to A, B, and C in equal proportions. Find u, x, y and z.' Leonardo gave a correct solution of the first and third, also a root of the cubic equation correct to nine decimals. — Ball. Jordanus Nemorarius wrote important Latin works on arith- metic, geometry, and astronomy. His De Triangulis — the most important of these — -consists of four books dealing not only with triangles, but with polygons and circles. He generally uses Arabic numerals, and denotes quantities known or unknown by letters. He solves the problem of finding two numbers having a given simi and product, by a method equivalent to our elemen- tary algebra. This is practically the first European syncopated algebra, but seems to have become too little known to have far- reaching results in a tune not yet ripe for this invention. A book on Weights contains elements of mechanics. Albertus Magnus, born near the end of the twelfth century, became an ardent champion of the newly discovered but pro- scribed works of Aristotle. In particular he interpreted the Milky 180 A SHORT HISTORY OF SCIENCE Way as an accumulation of small stars, and ridiculed the curren objections to antipodes, striving, however, always to harmonizi the ancient science with the theology of his church. Two Oxford scholars, John of Holywood (Sacrobosco) and Roge Bacon, have next to be mentioned. Sacrobosco lectured at Pari: on arithmetic and algebra, and wrote standard books on the forme with rules but no proofs, and an astronomy of which more thai sixty editions were afterwards printed. Roger Bacon (1214-1294?). — In the history of natural sciena one thirteenth century name stands out before all others, viz. that of Roger or "Friar" Bacon, a member of the Franciscar order, born at Ilchester, England, in 1214. He was a pupil o: Robert Grosseteste "who had especially devoted himself t( mathematics and experimental science," and had studied th( works of the Arabian authors. Bacon also travelled abroad anc studied at the University of Paris, — at that time the centrt of European learning. Here he took the degree of Doctor oJ Theology and probably also here became a Franciscan friar. Hf taught at Oxford, where he had a kind of laboratory for alchemica experiments. Doubtless it was for this that he became reputed as a worker in "magic" and the "black arts," for in 1257 he was forbidden by the head of his order to teach, and was sent to Paris where he underwent great privations. In 1266 he was invitee by Pope Clement IV to prepare and send to him a treatise on th( sciences, and within 18 months he had written and sent three important works — his Opus Majtis, Optis Minus, and Optu Tertium. In 1268 he returned to Oxford and there composed several more works, but under a later Pope his books were con- demned and he was thrown into prison where he remained unti about a year before his death. In Paris, Bacon devoted himself particularly to physical scienc( and mathematics. His 0pm Majus (1267) contains both i summary of ancient and current physical science, and a philosophy of learning based on Greek, Roman, and Arabic authorities. H( insisted that natural science must have an experimental basis and that astronomy and the physical sciences must be founded oi PROGRESS 6F science TO 1450 A.D. 181 mathematics, "the alphabet of all philosophy." On the other hand he says : — We must consider that words exercise the greatest influence. Al- most all wonders are accomplished through speech. In words the highest enthusiasm expresses itself. Therefore words, deeply thought . . . keenly realized, well calculated, and spoken with emphasis, have notable power. Bacon enunciated the essential principles of calendar reform, recognizing that the current plan of 365| days led to an error of one day in 130 years. He made an acute criticism of the arbi- trary assumptions and the artificial complexity of the Ptolemaic astronomy; he discussed reflection and refraction, spherical aberration, rainbows, magnifying glasses, and shooting stars; he attributed the tides to the action of the lunar rays. In a chapter on geography he " comes to the conclusion that the ocean between the east coast of Asia and Europe is not very broad. This . . . was quoted by Columbus in 1498. ... It is pleasant to think that the persecuted English monk, then two hundred years in his grave, was able to lend a powerful hand in widening the horizon of mankind." (See Appendix.) Most of this remarkable work — not printed for nearly 500 years — was so far in advance of the age that it not only failed of appreciation, but exposed the author to accusations of magic, and even to imprisonment. In spite of his many attainments he believed in astrology, in the doctrine of " signatures " and in the " philosopher's stone," and " knew " that the circle had been squared. He prophesied ships propelled swiftly by mechanical means and carriages without horses. He repudiated belief in witch- craft,' and paid the penalty for his corn-age by many years in prison. Dante Alighieei (1265-1321). — Another notable scholar of the thirteenth century is Dante, the greatest poetical genius of the Middle Ages, who requires our notice not only because of his influence in awakening and stimulating the minds of his own and later times, but also as the author of a treatise On Water • Not merely astrology and alchemy but even magic and necromancy were at this time the subjects of university lecture courses. 182 A SHORT fflSTORY OF SCIENCE and the Earth {De Aqua et Terra) which, according to himself was delivered at Mantua in 1320 as a contribution to the question then much discussed, " whether on any part of the earth's surfaci water is higher than the earth." In his cosmology, Dante seem: to derive from Aristotle and Pliny, without having attainec familiarity with the Ptolemaic system. Computation in the Middle Ages. — During the fourteentl century there was continued activity in the gradual disseminatioi of Arabic learning, largely through the medium of almanacs anc calendars, so that Arabic computations, Euclidean geometry, anc Ptolemaic astronomy became widely known. Some of these calendars emphasized the religious side and gave dates of churcl festivals for a series of years, others specialized in astrology medicine, or astronomy. For ecclesiastical purposes Romar numerals were preferred, but at least an explanation of the ne^ Arabic characters and their use was generally given. The arithmetic of Boethius, based on Roman numerals, retainec its vogue in northern Europe as late as about 1600. Arabi( arithmetic, or algorism, based on the Liher Abaci of Leonardc Pisano, employing the decimal scale and including the element; of algebra, came into general use among the Italian merchant; in the thirteenth and fourteenth centuries, though not withoui meeting serious opposition. Outside of Italy, however, accounti were kept in Roman numerals till about 1550, and in the mor( conservative religious and educational institutions, for a hundrec years longer. The Florentines at the same time considerably simplified the classification of arithmetical operations, in accord ance with our modern list : — numeration, addition, subtraction multiplication, division, involution and evolution. Addition and subtraction were begun at the left. The multi plication table, at first little known, ended with 5x5. Fo: further products up to 10 X 10, a system of finger reckoning wa widely used, the rule running : — Let the number five be represented by the open hand ; the numbe six by the hand with one finger closed ; the number seven by the han( with two fingers closed; the number eight by the hand with thre PROGRESS OF SCIENCE TO 1450 A.D. 183 fingers closed ; and the number nine by the hand with four fingers closed. To multiply one number by another let the multiplier be represented by one hand, and the number multiplied by the other, according to the above convention. Then the required answer is the product of the number of fingers (counting the thumb as a finger) open in the one hand by the number of fingers open in the other to- gether with ten times the total number of fingers closed.' Long division naturally required the skill of a mathematical expert. For example, if it is necessary to divide 1330 by 84 {Ball, p. 191) the Arabic or Persian method may be represented as follows, the right hand figure summing up the whole process: 1 3 3 8 4 1 3 3 8 X 5 3 4 4 8 X 9 4 0. 1 1 3 3 8 5 3 4 4 9 4 9 2 7 8 4 8 4 8 4 1 5 The galley or " scratch " method generally employed in Italy would take for the same problem, the successive forms (Ball, p. 192): 5 ;^30(1 ^4 4 ^9 ;^^o(i 4 jg9 ;^^0(i5 m ^9 ;^^0(i5 ;^^o(i5 m ' In modern notation : If a; is the number of fingers closed in one hand, y the Dumber closed in the other, then (5 +x) (5 +y) = (5 - x) (5 - 2/) + 10 (x + y). 184 A SHORT HISTORY OF SCIENCE This method was considered simpler than our modern Ion] division, and remained in use till the seventeenth century. The signs +, — , H-, and the use of decimal fractions belong t( a somewhat later period. The characteristics of the algoristic arithmetic are: (1) the us- of the Hindu-Arabic system of notation ; (2) the system of local value (3) the use of the zero ; (4) the entire discarding of the abacus ; (5 the combined use of symbols and numbers (in reality a combinatioi of algebra and arithmetic, as these terms are understood to-day) and (6) the introduction into Western Europe of a vast amount o arithmetical material from the East by means of Latin translation; from Arabian sources. While the general tendency of this perioc was to approach the study of arithmetic from its practical anc scientific sides, the mystical aspects of the subject — so popula: in the earlier periods — are by no means neglected. The fantas tic treatment of the properties of numbers is still common in thii age. . . . Thus the beginning of the thirteenth century marks the introduc tion of the Arabian system of notation and its adoption in place o: both the Roman notation and the abacus. This fundamental revolu tion was brought about only gradually, and that of the algorism car be traced in the translated literature of the Hindu-Arabian arith- metic. — Ahelson. Mathematics in the Medieval Universities. — The state oi mathematics in the universities toward the close of the fourteentl century may be inferred from the requirements for the master'; degree at Prague (1384) and Vienna (1389). The former includec Sacrobosco's Sphere, Euclid Books I-VI, optics, hydrostatics theory of the lever, and astronomy. Lectures were given or arithmetic, finger-reckoning, almanacs, and Ptolemy's Almagest At Vienna, Euclid I-V, perspective, proportional parts, mensu- ration, and a recent version of Ptolemy were required. In Leipsic however, in 1437 and 1438 mathematical ( ?) lectures were confinec to astrology, and conditions seem to have been much the same a1 the Italian universities, while Oxford and Paris probably occupiec an intermediate level. PROGRESS OF SCIENCE TO 1460 A.D. 185 There can be no doubt that at all times medieval schools taught all that their respective generations knew of arithmetic; that the teachers of arithmetic in the schools were often the famous mathe- maticians of their day ; that this teaching, since it kept pace with the increase in the knowledge of the subject, was progressive in character, and that at no time, not even in the barren generations at the close of the Middle Ages, when the scholastic education had outlived its use- fulness, did arithmetic cease to be a subject of study in the arts facul- ties of the medieval universities. — Ahelson. The Renaissance. — With the fourteenth century we enter upon one of the most interesting and noteworthy periods of human history; viz. the Renaissance. Neither the term nor the period is, however, sharply defined, the former signifying an awakening or "new birth," the latter covering loosely the fourteenth to the sixteenth centuries. It is only necessary to recapitulate briefly some of the phenomena touched upon in the present chapter, to realize that the civilization of the later Middle Ages has been under- going great changes. The Crusades marked the first and perhaps most important of these, while the rediscovery or recovery of the classics from Arabian and other sources in the eleventh to the thir- teenth centuries, followed by the revival of (classical) learning in the fourteenth must have been powerful ferments of the medieval scholastic mind, expanded and uplifted as it was by the poetical philosophy of Dante and challenged by the naturalism and ration- alism of Roger Bacon. The great events of the fourteenth century were in part new, and in part the natiiral extension and development of those of the thirteenth. A strange and appalling natural phenomenon was the famous epidemic known as the " black death," a quickly fatal disease which carried ofl from one quarter to one half of all the inhabitants of Europe, producing social changes — such as the rise of wages — which are still felt. Humanism. — The development of better education begun in the thirteenth century was marked in the fourteenth by the found- ing of many now famous universities and colleges and by that revival of ancient learning which is associated especially with the 186 A SHORT fflSTORY OF SCIENCE name of Petrarch (1304-1374). This revival, while at first chieflj literary and philosophical, brought with it translations into Latir — the current language of scholars at that time — of Aristotle anc other classical writers of scientific importance, and thus aided ir bringing on a new birth or renaissance in science as well as ir other branches. Precisely as there is one great name in thirteenth centurj literature, viz. that of Dante, which must be regarded wit! attention by all students of history, so in the fourteenth the name and work of Petrarch require careful consideration. Francesco Petrarca, commonly called Petrarch, a gifted Italian poet and scholar, greatly promoted the revival of ancient learning by insisting on the importance and merits of the Greek and Roman authors, Petrarch was less eminent as an Italian poet than as the foundei of Humanism, the inaugurator of the Renaissance in Italy. . . , Standing within the kingdom of the Middle Ages, he surveyed the kingdom of the modern spirit and, by his own inexhaustible indus- try in the field of scholarship and study, he determined what we call the revival of learning. By bringing the men of his own generatior into sympathetic contact with antiquity, he gave a decisive impulse to that European movement which restored freedom, self-conscious- ness and the faculty of progress to the human intellect. ... He was the first man to collect libraries, to accumulate coins, to advocate the preservation of antique monuments, and to collate manuscripts Though he knew no Greek, he was the first to appreciate its vast im- portance ; and through his influence, Boccaccio laid the earliest founda- tions of its study. . . . For him the authors of the Greek and Latir world were Hving men, — more real in fact than those with whom he corresponded; and the rhetorical epistles he addressed to Cicero Seneca and Varro prove that he dwelt with them on terms o: sympathetic intimacy. — Symonds. Rich as the fourteenth and fifteenth centuries are in mathe matical science and geographical discovery, and in art and inven tion, they are almost destitute of positive achievement in natura science. Doubtless the scientific spirit of curiosity and inquin was alive and active, but thus far it had taken other directions. PROGEESS OF SCIENCE TO 1450 A.D. 187 Alchemy. — What astrology was to astronomy, alchemy was to chemistry ; viz. the crude and often magic-working predecessor. The search for such will o' the wisps as the "philosopher's stone," the "elixir of life," "potable gold" and the "transmutation of elements," is probably as old as human history. The ancients seem to have dabbled in it, the Arabs to have been devoted to it, and the men of the Middle Ages, and even of the fourteenth and fifteenth centuries, to have spent much time upon it. Alembics and receivers, "Moors' Heads" and "Moors' Noses," calci- fication, distillation, and the like typify interesting and by no means fruitless gropings after the real composition of things. The names of Albertus Magnus, Bernard of Treviso, Eck of Salz- burg, and Basil Valentine are some which have come down to us as most important at this time, and as we read of the prepara- tion of the "spirits of salt" (hydrochloric acid), the calcification (oxidation) of mercury, etc., we realize that their labors, though often misdirected, were the prelude to better things. The Mariner's Compass. — The loadstone was certainly known to antiquity as a stone having the power of attracting and carrying a load of iron, but its directive property seems to have been first recognized and used for guidance on land or sea by the Chinese, since according to Humboldt, Chinese ships navigated the Indian Ocean with the magnetic needle in the third century of our era. The Arabs are also credited with its invention and use, as stated in the preceding chapter. The first reference to it in Christian Europe is said to be in a poem by Guyot of Provence, dated 1190, while references are also made to the compass in works of the thirteenth century. One of these runs : — No master mariner dares to use it lest he should be suspected of being a magician ; nor would the sailors venture to go to sea under the command of a man using an instrument which so much appeared to be under the influence of the powers below. It is probable, however, that the compass was first made commonly useful to western Europe early in the fourteenth century, by Flavio Gioja, a native of Amalfi, a small port 188 A SHORT HISTORY OF SCIENCE near Naples in Italy, who first poised the needle on a pivo1 instead of a card floating on water, as had been the custon before his time. (See page 164.) Clocks. — Clocks with wheels seem to have come into occa- sional use from the twelfth to the fourteenth centuries, and one oJ the first is said to have been sent by the Sultan of Egypt ir 1232 to the Emperor Frederick II. It resembled a celestial globe, in which the sun, moon and planetj moved, being impelled by weights and wheels so that they pointed out the hour, day and night, with certainty. Another is mentioned as in Canterbury cathedral, while still an- other at St. Albans, made by R. Wallingford who was abbot there in 1326, is said to have been so notable "that all Europe could not produce such another." It remained for Huygens in the seven- teenth century to apply pendulums to clocks. Wool and Silk. Textiles in the Middle Ages. — As an example of the industrial history of the times the following account of conditions in Spain is given : — The cloth manufactures in Spain continued to be of the coarsesl character until after the marriage of Catharine of Lancaster to tin heir of Castile (1388) when finer cloths were manufactured anc improved methods adopted. Up to that time the cloths used bj people of the higher class came from Bruges, from London, and fron Montpellier. James II of Aragon — the sovereign of Barcelona where there were at the time hundreds of looms at work making £ coarse woolen — wished to send a present to the Sultan of Egypi (1314 and 1322), and chose green cloths from Chalons and red cloths from Rheims and Douai, but sent no Spanish stuff ; while the stew- ard's accounts of Fernando V show that all his household were dressec in garments of imported stuffs. The great centre for the sale of woo was at Medina del Campo, and the cloth factories of Segovia anc Toledo were the most active and celebrated in Castile, while thos( of Barcelona were the principal in the east of Spain. It is asserte( that the improvement in the qualities of the Spanish cloth afte the coming of the Plantagenet princess to Spain was partly owinj to the fact that some herds of Enghsh sheep formed part of he PROGRESS OF SCIENCE TO 1450 A.D. 189 dowry, and the blending of staples enabled a better cloth to be made. The Flemish weavers mixed Spanish with English wool for their best textures. During the Arab domination of the south, Jaen, Granada, Valencia, and Seville had been great centres of silk culture and manufacture. Edrisi says that in the kingdom of Jaen in the thirteenth century there were 3000 villages where the cultivation of the silkworm was carried on, while in Seville there were 6000 silk looms, and Almeria had 800 looms for the manufacture of fancy brocades, etc. We are also told that a minister of Pedro the Cruel owned 125 chests of silk and gold tissue. In the twelfth century, a very flourishing trade in silks, velvets, and brocades was carried on with Constantinople and the East generally. Even in the fourteenth and early fifteenth cen- turies, the silks of Valencia and the bullion embroideries and gold and silver tissues of Cordova and Toledo were unsurpassed in Christen- dom, though heavily handicapped by the growing burdens placed upon craftsmen by labor laws and racial prejudice, and the dis- couragement of luxury by sumptuary regulations. — Hume. The Invention of Printing. — Before the middle of the fifteenth century, printing was done chiefly from fixed blocks of wood, metal, or stone, as is the case to-day in the printing of en- gravings, wood cuts and the like. The introduction of movable types, capable of an almost infinite variety of combination was therefore a forward step of fundamental importance, since the same letter or picture could be used over and over in new com- binations where previously it could be used but once. Until quite recently, it was generally held that the invention of the art of printing from movable types was the work of Johann Gutenberg (1397-1468) of Mainz on the Rhine, aided by Johann Faust or Fust, a rich citizen of Mainz. Of late, however, the claim of Gutenberg has been much disputed. The controversy about the person and nationahty of the inventor [of the art of printing] and the place of invention resembles the rival claims of seven cities to be the birthplace of Homer. . . . The best authorities agree on Gutenberg. Jacob Wimpheling wrote in 1507 . . . 'Of no art can we Germans be more proud than of the art of printing. 190 A SHORT HISTORY OF SCIENCE which made us the intellectual bearers of the doctrines of Christianity of all divine and earthly sciences, and thus benefactors of the whoK race.' — Schaff. AsELSON. The Seven Liberal Arts. Ball. History of Mathematics, Chapters VI, VIII, X. Cajori. History of Mathematics. _ Cajori. History of Physics. Kefekences ])i{^pj;r. Intellectual Development of Europe. FOB MuiR. Alchemy and the Beginnings of Chemistry. Reading Rashdall. Universities of the Middle Ages. Schaff. The Renaissance. Symonds. The Renaissance. White. Warfare of Science with Theology. A Map of the Globe in the Time of Columbus (After J. H. Robinson. Courtesy of Messrs. Ginn & Co.) In 1492 a German mariner, Behaim, made a globe which is still preserved in Nuremberg. H did not know of the existence of the American continents or of the vast Pacific Ocean. . . He places Japan (Cipango) where Mexico lies. In the reproduction many names are omittec and the outlines of North and South America are sketched in. — J. H. Robinson, Medicsval and Modem Times. CHAPTER X A NEW ASTRONOMY AND THE BEGINNINGS OF MODERN NATURAL SCIENCE The breeze from the shores of Hellas cleared the heavy scholastic atmosphere. Scholasticism was succeeded by Humanism, by the acceptance of this world as a fair and goodly place given to man to enjoy and to make the best of. In Italy the reaction became so great that it seemed destined to put paganism once more in the place of Christianity ; and though it produced lasting monuments in art and poetry, the earnestness was wanting which in Germany brought about the revival of science, and later on the rebellion against spiritual tyranny. . . Astronomy profited more than any other science by this revival of learning, and about the middle of the fifteenth century the first of the long series of German astronomers arose who paved the way for Copernicus and Kepler, though not one of them deserves to be called a preciirsor of these heroes. — Dreyer. The silent work of the great Regiomontanus in his chamber at Nuremberg computed the ephemerides which made possible the discovery of America by Columbus. — Rudio. The extension of the geographical field of view over the whole earth and the release of thought and feeling from the restrictions of the Middle Ages mark a division of equal importance with the fall of the ancient world a thousand years earlier. — Dannemann. Science begins to dawn, but only to dawn, when a Copernicus, and after him a Kepler or a Galileo, sets to work on these raw materials, and sifts from them their essence. She bursts into full daylight only when a Newton extracts the quintessence. There has been as yet but one Newton ; there have not been very many Keplers. — Tait. The Age of Discovert. — With the end of the fifteenth century and the beginning of the sixteenth opens one of the most marvellous chapters in all history ; viz. the Discovery of the New World. At about the same time further explorations of the old world attained equal extent and interest. We have referred above (p. 174) to the Discovery of the East by the Crusaders, and now 191 192 A SHORT HISTORY OF SCIENCE with Columbus, Magellan, and their successors, we have an evei more pregnant Discovery of the West. Meanwhile, Diaz and dj Gama pushed the explorations of Prince Henry of Portugal, " th( Navigator," to the south, and in rounding the Cape of Gooc Hope completed the Discovery of the South. To the north, ex plorers had already advanced to regions of perpetual snow anc ice, so that in all directions there were new problems of intense interest profoundly moving the imagination of mankind. The Reformation. — Another potent element was added tc the already complex fermentation of medieval ideas when in 151< a widespread insurrection began in the Christian Church, the most conservative and most powerful institution of the Middle Ages This revolution, — for such it proved to be, — with which the name of Luther will always be chiefly associated, soon aroused a wave of determined opposition, naturally strongly conservative known to-day as the "counter-reformation," of which the In- quisition was one instrument. The increased importance of the art of navigation reacted powerfully on the underlying sciences of mathematics and astron- omy, particularly through the demand for improved astronomi- cal tables. The Church, even, had a strong, if restricted, interesi in astronomy on account of the necessity of more accurate data for its calendar. Pioneers of the New Astronomy. — Nicholas of Cusa (1401-1464), later Bishop of Brixen, wrote on Learned Ignorance arguing that the universe, being infinite in extent, could have nc centre, and that the earth has diurnal rotation. " It is now cleai that the earth really moves, if we do not at once observe it since we perceive motion only through comparison with some- thing immovable." In mathematics he follows Euclid anc Archimedes, co5perating in a translation of the latter frorc Greek into Latin, and dealing with the squaring of the circle. He makes a map of the known world, using central projection He is said to have determined areas of irregular boundary by th( then novel method of cutting them out and weighing, and is one of the first to emphasize the importance of measurement in al A NEW ASTRONOMY 193 investigations. He showed independence of thinking, but his astronomical theories were too Httle developed — and too specula- tive — to constitute real progress in an age not yet quite ripe for their reception. Peurbach (1423-1461), who had as a youth met Nicholas of Cusa in Rome, became professor of astronomy and mathematics in Vienna and has been called " the founder of observational and mathematical astronomy in the West." Recognizing the imper- fections of the Alfonsine tables he published a new edition of the Almagest with tables of natural sines — instead of chords — computed for every ten minutes. He depended mainly, however, on imperfect Arabic translations. His more eminent pupil and successor, Johann Miiller, of Konigsberg, better known as Regiomontanus (1436-1476), was the most distinguished scientific man of his time. After the fall of Constantinople he was among the first to avail himself of the opportunities for more direct acquaintance with the works of Archimedes, Apollonius, and Diophantus. For the defective version of the Almagest which had come through Arabic channels he substituted the Greek original, while his tables, pub- lished in 1475, were important both for astronomy and for the voyages of discovery of Vasco da Gama, Vespucci, and Colum- bus. These tables covered the period 1473 to 1560, giving sines for each minute of arc, longitudes for sun and moon, latitude for the moon, and a list of predicted eclipses from 1475 to 1530. An- other work on astrology includes a table of natural tangents for each degree. A wealthy merchant of Nuremberg erected an elaborately equipped observatory for Regiomontanus, and the printing-press recently established there became the most impor- tant in Germany. Accepting, however, a summons to Rome to reform the calendar, he was murdered at the age of 40. His De Triangulis (1464) is the earliest modern trigonometry. Four of its five books are devoted to plane trigonometry, the other to spherical. He determines triangles from three given conditions, using sines and cosines, and employs quadratic equa- tions successfully in some of his solutions. One of his problems 194 A SHORT HISTORY OF SCIENCE is "to determine a triangle when the difference of two sides, the perpendicular on the base, and the difference between the seg- ments into which the base is divided are given : i.e. a — b, a sin B, acos B - b cos A are known; to find, a, b, c, A, B, C." An- other is to construct from four given lines a quadrilateral which can be inscribed in a circle. Conditions Necessary for Progress. — The genius oi Hipparchus and Ptolemy had brought Greek astronomy to its culmination. Higher it could not rise until tliree conditions should be fulfilled, even though here and there the heliocentric hypothesis might be adopted through an unsupported inspiration of individuals. First, there must be better astronomical instru- ments and more accurate observations, extended over long periods. Second, there must be improved methods of mathematical com- putation for the reduction and interpretation of these observations. Third, there must be substantial progress towards clear thinking as to the fundamental facts and laws of motion. These conditions were met one after another during the sixteenth and seventeenth centuries by an extraordinary series of men of genius, among whom the chief were Copernicus, Tycho Brahe, Kepler, Galileo, and Newton. Their work constitutes a great part of the history oJ science during these two centuries — and one of the most won^ derful chapters of all time. Of these five, Copernicus and Kepler were predominantly inter- ested on the mathematical and theoretical side, Tycho Brahe was a great observer, Galileo combined experimental and observa tional skill with a new appreciation of physical laws, while Newton building on the foundation laid by all the others, made a magnifi cent synthesis of their results into a rational and consistent mathe matical theory of the solar system. These five represent Poland South Germany, Denmark, Italy, and England. Scientific progres: is no longer localized or dependent on princely patronage. It ha now become international. NicoLATJS Copernicus (1473-1543) was born in the remot little city of Thorn on the Vistula, and having relatives in th' Church, prepared himself for an ecclesiastical career. This le( A NEW ASTRONOMY 195 him, after medical study at Cracow, first to the university of Vienna, then to the chief Italian universities, Bologna, Padua, Ferrara, and Rome, where he found opportunity to cultivate his mathematical talents and to master what was then known of astronomy. He became canon at Frauenburg in his native land in 1497, and from 1512 until his death thirty years later, was settled there, rendering varied public services, and practising gratuitously, as needful, the medical art he had also learned. At the same time he found it possible to devote much attention to astronomical studies. In his study of the classical writers he came upon a statement that certain Pythagorean philosophers explained the phenomena of the daily and yearly motions of the heavenly bodies by sup- posing the earth itself to rotate on its axis and to have also an orbital motion. ' Occasioned by this, I also began to think of a motion of the earth, and although the idea seemed absurd, still, as others before me had been permitted to assume certain circles in order to explain the motions of the stars, I beheved it would readily be permitted me to try whether on the assumption of some motion of the earth better explanations of the revolutions of the heavenly spheres might not be found. And thus I have, assuming the motions which I in the following work at- tribute to the earth, after long and careful investigation, finally found that when the motions of the other planets are referred to the circu- lation of the earth and are computed for the revolution of each star, not only do the phenomena necessarily follow therefrom, but the order and magnitude of the stars and all their orbs and the heaven itself are so connected that in no part can anything be transposed without confusion to the rest and to the whole universe.' — Dreyer. ' I made every effort to read anew all the books of philosophers I could obtain, in order to ascertain if there were not some one of them of the opinion that other motions of the heavenly bodies existed than are assumed by those who teach mathematical sciences in the schools. So I found first in Cicero that Hicetas of Syracuse believed the earth moved. Afterwards I found also in Plutarch that others were likewise of this opinion. . . . Starting thence I began to re- flect on the mobihty of the earth.' — Timerding. 196 A SHORT HISTORY OF SCIENCE Copernicus was not a great observational astronomer. Hi instruments were poor, his eyesight not keen, his location un favorable for clear skies. His recorded observations are few chiefly of eclipses or oppositions of planets, and of no high degrei of accuracy. His interest and genius lay rather in the directioi of profound analysis and careful mathematical revision of thi current geocentric theory, practically unchanged since its formu lation by Ptolemy thirteen centuries earlier. Unfortunately th( conditions of the time were adverse to the publication of so radica an innovation as a heliocentric theory of the solar system; no; was Copernicus ever greatly interested in any publication of hii results, being both indifferent to reputation and averse to con troversy. ' The scorn, ' he says, ' which I had to fear in consequence of th< novelty and seeming unreasonableness of my ideas, almost moved m( to lay the completed work aside.' Moreover, he realized the futility of publishing his revolu tionary theories until he should have buttressed them with i planetary system so completely worked out that its superiority to the long-intrenched Ptolemaic system should be unquestionabh — a herculean, if congenial labor. Nevertheless, he gradually formulated his astronomical system in manuscript, and aboui 1529 issued a Covimentariolus giving an outline of his theory which thus became gradually but vaguely known to scholars Ten years later George Joachim — Rheticiis — a young professoi of mathematics from the Lutheran university of Wittenberg visited Copernicus, eager to learn more of the new doctrine The Lutheran church was not more hospitable than the Romai Catholic to scientific novelty and Luther himself called Copernieu: a fool. De Revolutionibus. — In 1,540 appeared the Prima Narratii by Rheticus containing a considerable admixture of astrology, am in 1543 the immortal De Revolutionibus Orbium Celestium, a cop; reaching Copernicus, it is said, on his death-bed. He begins wit) certain postulates : first, that the universe is spherical ; second, tha A NEW ASTRONOMY 197 the earth is spherical ; third, that the motions of the heavenly bodies are uniform circular motions or compounded of such motions. The slender basis for the first and third of these may be inferred from his statement in regard to certain hypothetical causes of want of uniformity : — Both of which things the intellect shrinks from with horror, it being unworthy to hold such a view about bodies which are con- stituted in the most perfect order. He makes the relative character of the motions involved of fundamental importance. In his own words : — For all change in position which is seen is due to a motion either of the observer or of the thing looked at, or to changes in the position of both, provided that these are different. For when things are moved equally relatively to the same things, no motion is perceived, as between the object seen and the observer. Thus the daily revolution of sun, moon, and stars about a station- ary earth would have the same apparent effect as rotation of the earth in the opposite direction about its own axis, and the ap- parent yearly motion of the sun about the earth is equivalent to an orbital motion of the latter. 'It is,' he says, 'more probable that the earth tm-ns about its axis than that the planets at their various distances, the comets sweep- ing through space, and the endless multitude of the fixed stars, describe the same regular daily motion about the earth.' The apparent irregularities in the motions of the five known planets had been a perpetual stumbling-block to the ancient astronomers, requiring more and more complicated hypotheses for their explanations as accuracy of observations increased. The heliocentric theory of Copernicus, inaccurate as it was in some respects, afforded a simple explanation of the fact that Mercury and Venus seem merely to oscillate east and west of the sun, while Mars, Jupiter, and Saturn recede indefinitely from it, ex- hibiting also periodic reversals of the direction of their motion. 198 A SHORT HISTORY OF SCIENCE The new explanation obviously accounted also for the variation; in the brightness of these planets. ' It is certain, ' he says, ' that Saturn, Jupiter and Mars are always nearest the earth when they rise in the evening, that is when they are in opposition to the sun, as the earth is situated between them anc The Copernican System the sun. On the contrary, Mars and Jupiter are farthest from th earth when they set in the evening, the sun lying between them anc us. This proves sufficiently that the sun is the centre of their orbits as of those of Venus and Mercury. Since thus all planets mov about one centre it is necessary that the space which remains betwee] the circles of Venus and Mars, contain the earth and its accompanyin moon.' He is, therefore, not afraid to maintain that the earth with th moon encircling it, traverses a great circle in its annual motio] among the planets about the sun. The universe, however, is si vast, that the distances of the planets from the sun are insignifican A NEW ASTRONOMY 199 in comparison with that of the sphere of the stars. He holds all this easier of comprehension, than if the mind is confused by an almost endless mass of circles, as is necessary foi^ those who put the earth in the centre of the universe. 'So in fact the sun seated on the royal throne guides the family of planets encircling it. We find thus in this arrangement a har- monious connection not otherwise realized. For here one can see why the forward and backward motions of Jupiter seem greater than those of Saturn and smaller than those of Mars.' His adherence to the Greek assumption of uniform circular motion leaves him still under the necessity of retaining an elabo- rate system of epicycles, but he rejects Ptolemy's equant. . . . He his fabric of the heavens Hath left to their disputes, perhaps to move His laughter at their quaint opinions wide ; Hereafter when they come to model heaven And calculate the stars, how will they wield The mighty frame ! how build, unbuild, contrive To save appearances ! how gird the sphere With centric and eccentric scribbled o'er. Cycle in epicycle, orb in orb ! — Milton, Paradise Lost, VIII. The epicycles of Copernicus numbered however but 34, — sufficing " to explain the whole construction of the world and the whole dance of the planets " — against the 79 to which the Ptole- maic theory had gradually attained. The completeness of mathe- matical detail with which the whole theory is worked out can not here be adequately described. He includes so much trigonometry as his astronomical work requires, also a revision of Ptolemy's star catalogue. He computes a very accurate value of the equinoctial precession, and interprets this correctly as due to a slow conical motion of the earth's axis, like that of a top coming to rest. Copernicus estimates the relative sizes of moon, earth and 200 A SHORT HISTORY OF SCIENCE sun as 1 : 43 : 6937, and the distance from earth to sun — according to the method of Aristarchus — at about 1200 earth-radii, that is about ^ of the actual. Revolutionary as were the theories expounded by Copernicus they were not clothed in such popular form as to occasion imme- diate or general controversy. In dedicating his work to the Pope, Copernicus says in substance : — It seems to me that the church can derive some advantage from my labors. Under Leo X indeed the rectification of the calendai was not possible, since the length of the year and the motions ol the sun and moon were not exactly determined. I have sought tc determine these more closely. What I have accomplished, I leave to the judgment of your Holiness, and of the learned mathemati- cians. (See Appendix.) Moreover criticism was in considerable measure disarmed by a fraudulent preface inserted by Osiander, a Lutheran theologian of Nuremberg, to whom the care of publication had been par- tially intrusted by Rheticus. In this preface, ostensibly bj Copernicus himself, it is stated, — that though many will take offence at the doctrine of the earth's motion, it will be found on further consideration that the authoi does not deserve blame. For the object of an astronomer is t( put together the history of the celestial motions from careful ob servations, and then to set forth their causes or hypotheses abou them, if he cannot find the real causes, so that those motions can bi computed on geometrical principles. But it is not necessary tha his hypotheses should be true, they need not even be probable; i is sufficient if the calculations founded on them agree with the obser vations. Nobody would consider the epicycle of Venus probable as the diameter of the planet in its perigee ought to be four times a great as in the apogee, which is contradicted by the experience of al times. Science simply does not know the cause of the apparent! irregular motions, and an astronomer will prefer the hypothesis whic is most easily understood. Let us therefore add the following nei hypotheses to the old ones, as they are admirable and simple, bu A NEW ASTRONOMY 201 nobody must expect certainty about astronomy, for it cannot give it ; and whoever takes for truth what has been designed for a different purpose, will leave this science as a greater fool than he was when he approached it. Influence of Copernicus. — The publication of Be Revolu- tionibiis was naturally a powerful stimulus to astronomical and mathematical studies. Thus Rheticus, whose relations to Coper- nicus had been so fruitful, calculated a new and extensive set of mathematical tables, while Reinhold, who had hailed Coper- nicus as a new Ptolemy, published astronomical tables — the Prutenic or Prussian — on the basis of Copernicus' work, superior to the Alfonsine, previously current. Before the new doctrine should be completely justified or the reverse, it was necessary that certain mechanical notions should be clarified, and that more accurate observational data should be systematically collected. Copernicus had based his imposing structiu-e on a very slender foundation of actual fact, and had professed his complete satisfaction if his theoretical results should come within ten minutes of the observed positions of the planets, — a degree of accuracy which he did not, in fact, attain. On the other hand, he could indeed answer, but not rise entirely above, the traditional notions that the four elements of the ancients must have rectilinear, the heavenly bodies circular, motion ; also, that if the earth rotated in twenty-four hours, loose bodies would long since have been thrown off, falling bodies would not fall, and clouds would always be left behind in the west. As suggested by Dreyer : — It is interesting, though useless, to speculate on what would have been the chances of immediate success of the work of Copernicus if it had appeared fifty years earlier. Among the humanists there certainly was considerable freedom of thought, and they would not have been prejudiced against the new conception of the world because it upset the medieval notion of a set of planetary spheres inside the empyrean sphere, with places allotted for the hierarchy of angels. If one of the leaders of the Church (at least in Italy) at the beginning 202 A SHORT HISTORY OF SCIENCE of the sixteenth century had been asked whether the idea of the eart moving through space was not clearly heretical, he would probabl merely have smiled at the innocence of the enquirer and have answere in the words of Pomponazzi that a thing might be true in philosoph and yet false in theology. But the times had changed. The sun c the Renaissance had set when, in 1527, the hordes of the Constable o Bourbon sacked and desecrated Rome ; the Reformation had put a: end to the religious and intellectual solidarity of the nations, and th contest between Rome and the Protestants absorbed the menta energy of Europe. During the second half of the sixteenth centur; science was therefore very little cultivated, and though astronomy anc astrology attracted a fair number of students (among whom was oni of the first rank), still theology was thought of first and last. Ant theology had come to mean the most literal acceptance of every won of Scripture ; to the Protestants of necessity, since they denied thi authority of Popes and Councils, to the Roman Catholics from i desire to define their doctrines more narrowly and to prove how un justified had been the revolt against the Church of Rome. Then was an end of all talk of Christian Renaissance and of all hope of rec onciling faith and reason ; a new spirit had arisen which claimec absolute control for Church authority. Neither side could therefon be expected to be very cordial to the new doctrine. Robert Recorde, in his Pathway to Knowledge (1551), has his "Master" state to a "scholar " : ' Eraclides Ponticus, a great philosopher, and two great clerkes ol Pythagoras schole, Philolaus and Ecphantus, were of the contrarj opinion, but also Nicias Syracusius and Aristarchus Samius seem wit! strong arguments to approve it.' After saying that the matter is toe difficult and must be deferred till another time, the Master state; that ' Copernicus, a man of great learning, of muche experience anc of wondrefuU diligence in obseruation, hathe renewed the opinion oi Aristarchus Samius, and affirmeth that the earthe not only mouetl circularlye about his own centre, but also may be, yea and is con- tinually out of the precise centre 38 hundredth thousand miles ; bul bicause the vnderstanding of that controuersy dependeth of profoundei knowledge than in this introduction may be vttered conueniently, I wil let it passe tyll some other time.' A NEW ASTRONOMY 203 A little later Francis Bacon writes : — ' In the system of Copernicus there are many and grave difficulties ; for the threefold motion with which he encumbers the earth is a serious inconvenience, and the separation of the sun from the planets, with which he has so many affections in common, is likewise a harsh step ; and the introduction of so many immovable bodies into nature, as when he makes the sun and the stars immovable, the bodies which are pecul- iarly lucid and radiant, and his making the moon adhere to the earth in a sort of epicycle, and some other things which he assumes, are proceedings which mark a man who thinks nothing of introducing fic- tions of any kind into nature, provided his calculations turn out well.' Bacon himself was very ignorant of all that had been done by mathematics ; and, strange to say, he especially objected to astronomy being handed over to the mathematicians. Leverrier and Adams, calculating an unknown planet into a visible existence by enormous heaps of algebra, fiu'nish the last comment of note on this specimen of the goodness of Bacon's view. . . . Mathematics was beginning to be the great instrument of exact inquiry ; Bacon threw the science aside, from ignorance, just at the time when his enormous sagacity, applied to knowledge, would have made him see the part it was to play. If Newton had taken Bacon for his master, not he, but somebody else, would have been Newton. — De Morgan. Copernicus cannot be said to have flooded with light the dark places of nature — in the way that one stupendous mind subsequently did — but still, as we look back through the long vista of the history of science, the dim Titanic figure of the old monk seems to rear itself out of the dull flats around it, pierces with its head the mists that over- shadow them, and catches the first gleam of the rising sun, . . . Like some iron peak, by the Creator Fired with the red glow of the rushing morn. — E. J. C. Morton. Ttcho Brahe (1546-1601). — The first great need of the new Copernican astronomy — adequate and accurate data — was soon to be supplied by Tycho Brahe, born in 1546 of a noble Danish family. While a student at the University of Copenhagen his interest in astronomy was enlisted by an eclipse, and later, at Leipsic, he persisted in devoting to his new avocation the time 204 A SHORT HISTORY OF SCIENCE and attention he was expected to give to subjects more highb esteemed for a man of birth and fortune. From a lunar eclipse which took place while he was at Leipsic Tycho foretold wet weather, which also turned out to be correct. Here, too, he began his life work of procuring and improving the best instruments for astronomical observations, at the same tim< testing and correcting their errors. Returning to Denmark fron travels in Germany, his predilection for astronomy was powerfullj stimulated by the appearance in the constellation Cassiopeia, ir November, 1572, of a brilliant new star, which remained visible fo: 16 months. The great importance attached to this occurrence bj Tycho and his contemporaries was due to the evidence it afforded against the truth of the Aristotelian conviction that the heavens were immutable, since Tycho's careful observations showed thai the star must certainly be more distant than the moon, and that i1 had no share in the planetary motions. He reluctantly published an account of the new star, expressing still his adherence tc the current pre-Copernican notions of crystalline spheres for tht different heavenly bodies and of atmospheric comets, all com- bined with astrological reflections and inferences, as illustrated bj the following passages from Dreyer's biography : — The star was at first like Venus and Jupiter, and its effects will therefore first be pleasant ; but as it then became like Mars, there will next come a period of wars, seditions, captivity, and death of princes and destruction of cities, together with dryness and fiery meteors ir the air, pestilence, and venomous snakes. Lastly, the star becamt like Saturn, and there will therefore, finally, come a time of want death, imprisonment, and all kinds of sad things. As the star seen by the wise men foretold the birth of Christ the new one was generally supposed to announce His last coming and the end of the world. That an unusual celestial phenomenon occurring at that particulai moment should have been considered as indicating troublous times is extremely natural when we consider the state of Europe in 1573 The tremendous rebellion against the Papal supremacy, which for s long time had seemed destined to end in the complete overthrow o Tycho Brake's Quadrant. 'A NEW ASTRONOMY 205 the latter, appeared now to have reached its limit, and many people thought that the tide had already commenced to turn. Tycho considered that the new star was formed of ' celestial matter,' not differing from that of which the other stars are composed, except that it was not of such perfection or solid composition as in the stars of permanent duration. It was therefore gradually dissolved and dwindled away. It became visible to us because it was illuminated by the sun, and the matter of which it was formed was taken from the Milky Way, close to the edge of which the star was situated, and in which Tycho believed he could now see a gap or hole which had not been there before. But the star had a truer mission than that of announcing the arrival of an impossible golden age. It roused to unwearied exertions a great astronomer, it caused him to renew astronomy in all its branches by showing the world how little it knew about the heavens ; his work became the foundation on which Kepler and Newton built their glorious edifice, and the star of Cassiopeia started astronomical science on the brilliant career which it has pursued ever since, and swept away the mist that obscured the true system of the world. As Kepler truly said, 'If that star did nothing else, at least it announced and produced a great astronomer.' At the same time the book bears witness to the soberness of mind which distinguishes him from most of the other writers on the subject of the star. His account of it is very short, but it says all there coidd be said about it — that it had no parallax, that it remained immovable in the same place, that it looked like an ordinary star — and it describes the star's place in the heavens accurately, and its variations in light and color. Even though Tycho made some re- marks about the astrological significance of the star, he did so in a way which shows that he did not himself consider this the most valu- able portion of his work. To appreciate his little book perfectly, it is desirable to glance at some of the other numerous books and pam- phlets which were written about the star, and of most of which Tycho himself has in his later work given a very detailed analysis. In 1575 Tycho obtained while travelling a copy of Copernicus' Commentariohis, and in the following year received from King Frederick II the island of Hveen, with funds for the maintenance of an observatory upon it. As to the former his opinion is that 206 A SHORT fflSTORY OF SCIENCE ' The Ptolemean system was too complicated, and the new one which that great man Copernicus had proposed, following in th« footsteps of Aristarchus of Samos, though there was nothing in i1 contrary to mathematical principles, was in opposition to those oi physics, as the heavy and sluggish earth is unfit to move, and the system is even opposed to the authority of Scripture.' — Dreyer, Tycho Brahe. Uraniborg. — The observatory of Uraniborg — the castle ol the heavens — at Hveen was an extraordinary establishment. In a large square inclosure oriented according to the points ol the compass, were several observatories, a library, laboratory, living-rooms and, later, workshops, a paper-mill and printing- press, and even underground observatories. The whole estab- lishment was administered with lavish extravagance, while Tychc was neither careful of his obligations nor free from arbitrary ar- rogance in his personal and administrative relations. In spite oi these difficulties " a magnificent series of observations, far transcend- ing in accuracy and extent anything that had been accomplished by his predecessors" was carried on for not less than 21 years At the same time medicine and alchemy were also cultivated. Concerned as he was to secure the greatest possible accuracy Tycho constructed instruments of great size; for example, e wooden quadrant for outdoor use with a brass scale of some ter feet radius, permitting readings to fractions of a minute. The best artists in Augsburg, clockmakers, jewellers, smiths, anc carpenters, were engaged to execute the work, and from the zea which so noble an instrument inspired, the quadrant was completec in less than a month. Its size was so great that twenty men coulc with difficulty transport it to its place of fixture. The two principa rectangular radii were beams of oak; the arch which lay betweei their extremities was made of solid wood of a particular kind, and thi whole was bound together by twelve beams. It received additiona strength from several iron bands, and the arch was covered witl plates of brass, for the purpose of receiving the 5400 divisions inti which it was to be subdivided. A large and strong pillar of oak, shoe with iron, was driven into the ground, and kept in its place by soli( Ukaniborg. A NEW ASTRONOMY 207 mason work. To this pillar the quadrant was fixed in a vertical plane, and steps were prepared to elevate the observer, when stars of a low altitude required his attention. As the instrument could not be conveniently covered with a roof, it was protected from the weather by a covering made of skins; but notwithstanding this and other precautions, it was broken to pieces by a violent storm, after having remained uninjured for the space of five years. — Brewster. A smaller but more serviceable azimuth quadrant of brass gave angles to the nearest minute. He had a copper globe constructed at great expense with the positions of some 1000 stars carefully marked upon it. The very precision of his observations tended to confirm his scepticism of the Copernican hypothesis, as it seemed incredible that the earth's supposed orbital motion should cause no change which he could detect in the position and brightness of the stars. He was also misled by supposing that the stars had measurable angular magnitude. He was not successful in making any funda- mental improvement in the relatively crude methods of time measurement, depending himself on wheel-mechanism without the regulating pendulum, and an apparatus of the sand-glass or clepsydra type. In 1577 Tycho made observations on a brilliant comet, and drew from them important theoretical inferences; namely, that instead of being an atmospheric phenomenon, the comet was at least three times as remote as the moon, and that it was revolving about the sun at a greater distance than Venus — unimpeded by the familiar crystalline spheres. He was even led, in discussing apparent irregularities of its motion, to suggest that its orbit might be oval — foreshadowing one of Kepler's great discoveries. According to the current view of his time, comets were formed by the ascending from the earth of human sins and wickedness, formed into a kind of gas, and ignited by the anger of God. This poisonous stuff falls down again on people's heads, and causes all kinds of mischief, such as pestilence, Frenchmen ( I), sudden death, bad weather, etc. — Dreyer, Tycho Brahe. 208 A SHORT HISTORY OF SCIENCE Eleven years later Tycho published a volume on the comet a; a part of a comprehensive astronomical treatise which was, how ever, never completed. About the same time his royal patroi died, and the new administration proved less sympathetic witl the great astronomer's work and less indulgent with his extrava gance and personal eccentricities. After a series of disagreements, Tycho withdrew from his ob servatory in 1597, spent the winter in Hamburg, and after negO' tiations with different sovereigns, accepted the invitation of the Emperor Rudolph to settle in Prague in 1599. Here he agau organized a staff of assistants, including, to the great advantage of himself and of his science, the young Kepler, but his furthei progress was prematurely terminated by death in 1601, at the age of 55. Tycho's chief services to the progress of astronomy consistec first, in the superior accuracy of his instruments and observations heightened by repetition and systematic correction of errors second, in the extension of these observations over a long series o: years. In both respects he departed from current practice, anc anticipated the modern. In point of accuracy his errors of star places seem rarely to have exceeded 1' to 2', and he even de termined the length of the year within one second. While h( recomputed almost every important astronomical constant, h( accepted the traditional distance of the sun. Kepler gave striking evidence later of his confidence in Tycho': accuracy by writing : — ' Since the divine goodness has given to us in Tycho Brahe a mos careful observer, from whose observations the error of 8' is shewn ii this calculation, . . it is right that we should with gratitude recog nize and make use of this gift of God. . . . For if I could hav( treated 8' of longitude as negligible I should have already correcte( sufRciently the hypothesis . . . discovered in chapter xvi. But a they could not be neglected, these 8' alone have led the way toward the complete reformation of astronomy, and have made the subject matter of a great part of this work.' — Berry. A NEW ASTRONOMY 209 On the other hand, Tycho was not strong on the theoretical side. He was never willing to accept the Copernican hypothesis of rotation and orbital motion of the earth — maintaining, for ex- ample, that if the earth moved, a stone dropped from the top of a tower must fall at a distance from the foot. Again with refer- ence to the apparent displacement of the stars which would be expected to result from orbital motion of the earth, he says : — A yearly motion would relegate the sphere of the fixed stars to such a distance that the path described by the earth must be insig- nificant in comparison. Dost thou hold it possible that the space between the sun, the alleged centre of the universe, and Saturn amoimts to not even -j^^ of that distance ? At the same time this space must be void of stars. Sensible, however, of the weakness of the Ptolemaic theory, he devised an ingenious compromise in which the planets revolved about the Sun in their respective periods, and the entire heavens about the earth daily — all of which is not mathematically dif- ferent from the Copernican theory. We see in him at the same time a perfect son of the sixteenth century, believing the universe to be woven together by mysterious connecting threads which the contemplation of the stars or of the elements of nature might unravel, and thereby lift the veil of the future; we see that he is still, like most of his contemporaries, a believer in the solid spheres and the atmospherical origin of comets, to which errors of the Aristotelean physics he was destined a few years later to give the death-blow by his researches on comets; we see him also thoroughly discontented with his surroundings, and looking abroad in the hope of finding somewhere else the place and the means for carrying out his plans. As a practical astronomer Tycho has not been surpassed by any observer of ancient or modern times. The splendor and number of his instruments, the ingenuity which he exhibited in inventing new ones and in improving and adding to those which were formerly known, and his skill and assiduity as an observer, have given a character to his labors and a value to his observations which will be appreciated to the latest posterity. — Brewster. 210 , A SHORT HISTORY OF SCIENCE Kepler. — Pierre de la Ramee, or Petrus Ramus, a Frend mathematician and philosopher, impatient with the cumbroui astronomical hypotheses of the ancients, and unsatisfied witl Copernicus' proposed simplification, published a work in 1565 expressing the hope ' that some distinguished German philosopher would arise and founc a new astronomy on careful observations by means of logic and mathe- matics, discarding all the notions of the ancients.' Within a few months he discussed the matter at length witl Tycho Brahe at Augsburg. Without accepting Ramus' views the young astronomer did make it his life work to lay the neces- sary foundation for such a new astronomy. Thirty years later Mastlin, professor at Tiibingen, wrote his former student Keplei — then aged 28 — that Tycho 'had hardly left a shadow of what had hitherto beer taken for astronomical science, and that only one thing was certain which was that mankind knew nothing of astronomical matters.' Born late in 1571 in Wiirtemberg, of Protestant parents ir very straitened circumstances, Johann Kepler's whole life was i struggle against poverty, ill-health, and adverse conditions. Ir 1594, abandoning with some hesitation theological studies, foi which his acceptance of the new Copernican hypothesis dis qualified him, he was appointed lecturer on mathematics at Gratz Students were few, and his duties included the preparation of i yearly almanac, containing, besides what its name implies, £ variety of weather predictions and astrological information "Mother Astronomy," he says, "would surely have to suffe: hunger if the daughter Astrology did not earn their bread." Becoming thus more interested in astronomy, "there were," hi says, "three things in particular: viz., the number, the size, am the motion of the heavenly bodies, as to which I searched zealousb for reasons why they were as they were and not otherwise." Th first result which seemed to him important, though somewha fantastic from our standpoint, was a crude correspondence be Kepler {Opera omnia). A NEW ASTRONOMY 211 tween the planetary orbits and the five regular solids, published in 1596 under a title which may be abridged to Cosmographic Mystery. The Earth is the circle, the measure of all. Round it describe a dodecahedron, the circle including this will be Mars. Round Mars de- scribe a tetrahedron, the circle including this will be Jupiter. De- scribe a cube round Jupiter, the circle including this will be Satiun. Then inscribe in the Earth an icosahedron, the circle inscribed in it will be Venus. Inscribe an octahedron in Venus, the circle inscribed in it will be Mercury. Kepler declared that he would not renounce the glory of this discovery "for the whole Electorate of Saxony." The corre- spondence of the dimensions of this fantastic geometrical con- struction with the distances of members of our solar system is in reality far from close, but both Tycho Brahe and Galileo seem to have been favorably impressed by the book. The difficulties of Kepler's position as a Protestant in Gratz led him, after a preliminary visit, to accept an engagement as Tycho's assistant at Prague. The powers of original genius were then for the first time as- sociated with inventive skill and patient observation, and though the astronomical data provided by Tycho were sure of finding their ap- plication in some future age, yet without them, Kepler's speculations would have been vain and the laws which they enabled him to deter- mine would have adorned the history of another century. — Brewster. In 1602 Kepler succeeded Tycho as imperial mathematician. Most fortunately, also, he secured possession of his chief's great collection of observations, though not of the instruments, — a matter of less consequence, since Kepler like Copernicus was a mathematician rather than an observer. To the study of these records he devoted the next 25 years. Among all the planetary observations of Tycho Brahe those of Mars presented the irregu- larities most difficult of explanation, and it was these which, having been originally assigned to Kepler, engrossed his attention for many years, and in the end led to some of his finest discoveries. 212 A SHORT HISTORY OF SCIENCE The Copernican theory like the Ptolemaic involved the resolu- tion of the motion of each planet into a main circular motion, modified by superimposing other circular motions — epicycles — successively upon it, each circle being the path of the centre of the next. Even after disentangling the essential irregularities of Mars' orbit from those merely due to irregular motion of the earth, he could still obtain no satisfactory agreement with Tycho's records, of which, as has been said, he refused to doubt the ac- curacy. Taking advantage of his own failure — as happens to men of true genius — he abandoned the restriction of circular motions, and experimented with other closed curves, of which the ellipse is simplest. Taking the sun at a focus, the problem was at last solved, theory and observation reconciled within due limits of error. At the same time uniform motion was naturally abandoned, for with a non-circular orbit, it was evident that the planet could not describe both equal distances and equal areas in equal times. Here, again, Kepler's scientific imagination led him to the great discovery that the planet traverses its orbit in such a manner that a line joining it to the sun would describe sectors of equal area in equal times, the planet thus moving fastest when nearest the sun. Of Kepler's celebrated three laws, the first two are: The planet describes an ellipse, the sun being in one focus. The straight line joining the planet to the sun sweeps out equal areas in equal intervals of time. These residts were published in 1609 as part of extended Com- mentaries on the Motions of Mars. The great problem was solved at last, the problem which had baffled the genius of Eudoxus and had been a stumbling-block to the Alexandrian astronomers, to such an extent that Pliny had called Mars the inobservabile sidus. The numerous observations made by Tycho Brahe, with a degree of accuracy never before attained, had in the skilful hand of Kepler revealed the unexpected fact that Mars describes an ellipse, in one of the foci of which the sun is situated, and that the radius vector of the planet sweeps over equal areas in equal times. And the genius and astounding patience of Kepler had A NEW ASTRONOMY 213 proved that not only did this new theory satisfy the observations, but that no other hypothesis could be made to agree with the obser- vations, as every proposed alternative left outstanding errors, such as it was impossible to ascribe to errors of observation. Kepler had therefore, unlike all his predecessors, not merely put forward a new hypothesis which might do as well as another to enable a computer to construct tables of the planet's motion; he had found the actual orbit in which the planet travels through space. In the history of astronomy there are only two other works of equal importance, the book De Revolutionibus of Copernicus and the Principia of Newton. The 'astronomy without hypothesis' demanded by Ramus had at last been produced, and well might Kepler proclaim : ' It is well. Ramus, that you have run from this pledge, by quitting life and your professorship ; if you held it still, I should, with justice, claim it.' Resuming later the tendency of his Cosmographic Mystery, he published in 1619 his Harmony of the World, containing his third law : — The squares of the times of revolution of any two planets (in- cluding the earth) about the sun are proportional to the cubes of their mean distances from the sun. In his delight he exclaims 'Nothing holds me, I will indulge in my sacred fury ; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my God, far away from the confines of Egypt.' — ' What sixteen years ago, I urged as a thing to be sought, that for which I joined Tycho Brahe, for which I settled m Prague, for which I have devoted the best part of my life to astronomical contempla- tions — at length I have brought to light, and recognized its truth beyond my most sanguine expectations. It is not eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the imveiled sun, most admirable to gaze on, burst out upon me.' . . . Archimedes of old had said " Give me a place to stand on, and I shall move the world." Tycho Brahe had given Kepler the place to stand on, and Kepler did move the world. 214 A SHORT HISTORY OF SCIENCE It should be borne in mind that Kepler's results depend no on a priori theory for their confirmation, but upon actual ob servations supporting them and interpreted by them. The grea further step of showing that the three laws are not independen and empirical, but mathematical consequences of a single me chanical law still awaited the genius of Newton. Kepler's notions in regard to force and motion are still crude Thus, for example, having in mind an analogy with magnetism Kepler says in his Epitome of the Copernican Astronomy (1618-1621): — 'There is therefore a conflict between the carrying power of th( sun and the impotence or material sluggishness (inertia) of the planet each enjoys some measure of victory, for the former moves the planei from its position and the latter frees the planet's body to some extern from the bonds in which it is thus held . . . but only to be capturec again by another portion of this rotatory virtue.' Elsewhere he says : — ' We must suppose one of two things : either that the moving spirits in proportion as they are more removed from the sun, are more feeble or that there is one moving spirit in the centre of all the orbits namely, in the sun, which urges each body the more vehemently ir proportion as it is nearer; but in more distant spaces languishes ii consequence of the remoteness and attenuation of its virtue.' — Whewell. He recognized the nfecessity of a force exercised by the sun, bul believed it inversely proportional to the distance instead of tc the square of the distance. His notions of gravity are expressec in his book on Mars : — ' Every bodily substance will rest in any place in which it is placer isolated, outside the reach of the power of a body of the same kind Gravity is the mutual tendency of cognate bodies to join each othei (of which kind the magnetic force is), so that the earth draws a stone much more than the stone draws the earth. Supposing that the earth were in the centre of the world, heavy bodies would not seel the centre of the world as such, but the centre of a round, cognate A NEW ASTRONOMY 215 body, the earth ; and wherever the earth is transported heavy bodies will always seek it; but if the earth were not round they would not from all sides seek the middle of it, but would from different sides be carried to different points. If two stones were situated anywhere in space near each other, but outside the reach of a third cognate body, they would after the manner of two magnetic bodies come together at an intermediate point, each approaching the other in proportion to the attracting mass. And if the earth and the moon were not kept in their orbits by their animal force, the earth would ascend towards the moon one fifty-fourth part of the distance, while the moon would descend the rest of the way and join the earth, provided that the two bodies are of the same density. If the earth ceased to attract the water all the seas would rise and flow over the moon. — Dreyer. Kepler's last important published work was his Rudolphine Tables (1627), embodying the accumulated results of Tycho's work and his own, and remaining a standard for a century. It is noteworthy that during Kepler's work on these tables, mathe- matical computation was peacefully revolutionized by the intro- duction of logarithms, newly discovered by Napier and Biirgi. In 1628, after vain attempts to collect arrears of his salary as imperial mathematician, he even joined Wallenstein as astrologer, but died soon after at Regensburg in 1630. Kepler also wrote an important work on Dioptrics with a mathe- matical discussion of refraction and the different forms of the newly invented telescope, the whole constituting the foundation of modern optics. In it he develops the first correct theory of vision, "Seeing amounts to feeling the stimulation of the retina, which is painted with the colored rays of the visible world. The picture must then be transmitted to the brain by a mental cur- rent, and delivered at the seat of the visual faculty." He sup- poses that color depends on density and transparency, and that refraction is due to greater resistance of a dense medium. He enunciates the law that intensity of light varies inversely as the square of the distance. " In proportion as the spherical surface from whose centre the light proceeds is greater or smaller, so is the strength or density of the light-rays which fall on the smaller 216 A SHORT HISTORY OF SCIENCE sphere to the strength of those rays which fall on the largei sphere." He explains the estimation of distance by binoculai vision. He supposes the velocity of light to be infinite. Hij more purely mathematical work will be mentioned in a latei chapter. Kepler added Plato's boldness of fancy to his own patient and candid habit of testing his fancies by a rigorous and laborious com- parison with the phenomena ; and thus his discoveries led to those oi Newton. — Whewell. If Kepler had burnt three-quarters of what he printed, we should in all probability have formed a higher opinion of his intellectual grasp and sobriety of judgment, but we should have lost to a greal extent the impression of extraordinary enthusiasm and industry, and of almost unequalled intellectual honesty, which we now get from a study of his works. — Berry. Kepler says : ' If Christopher Columbus, if Magellan, if tht Portuguese, when they narrate their wanderings, are not only ex- cused, but if we do not wish these passages omitted, and should lose much pleasure if they were, let no one blame me for doing the same. Kepler's talents were a kindly and fertile soil, which he cultivated with abundant toil and vigor, but with great scantiness of agricultura skill and implements. Weeds and the grain throve and flourishec side by side almost undistinguished ; and he gave a peculiar appear- ance to his harvest, by gathering and preserving the one class of plant; with as much care and diligence as the other. — Whewell. Endowed with two qualities, which seemed incompatible witl each other, a volcanic imagination and a pertinacity of intellect whicl the most tedious numerical calculations could not daunt, Kepla conjectured that the movements of the celestial bodies must be con nected together by simple laws, or, to use his own expression, bj harmonic laws. These laws he undertook to discover. A thousanc fruitless attempts, errors of calculation inseparable from a colossa undertaking, did not prevent him a single instant from advancinj resolutely toward the goal of which he imagined he had obtained i glimpse. Twenty-two years were employed by him in this investiga tion, and still he was not weary of it ! What, in reality, are twenty two years of labor to him who is about to become the legislator o worlds; who shall inscribe his name in ineffaceable characters upoi Galileo [Opere, 1744). A NEW ASTRONOMY 217 the frontispiece of an immortal code; who shall be able to exclaim in dithyrambic language, and without incurring the reproach of any- one, ' The die is cast ; I have written my book ; it will be read either in the present age or by posterity, it matters not which ; it may well await a reader, since God has waited six thousand years for an inter- preter of his words.' — Arago. The philosophical significance of Kepler's discoveries was not recognized by the ecclesiastical party at first. It is chiefly this, that they constitute a most important step to the establishment of the doctrine of the government of the world by law. But it was im- possible to receive these laws without seeking for their cause. The result to which that search eventually conducted not only explained their origin, but also showed that, as laws, they must, in the necessity of nature, exist. It may be truly said that the mathematical exposi- tion of their origin constitutes the most splendid monument of the intellectual power of man. — Draper. Galileo. — Columbus discovered America when Copernicus was but 19, and before the birth of Tycho Brahe, Magellan had completed the proof of the earth's rotundity by actually sailing around it, while Luther had stirred up the great religious revolt of Protestantism. The later years of Kepler and Galileo fell within the period of the Thirty Years' War, of which neither was to witness the close. Permanent English settlements in America had just begun. Galileo (1564-1642), born on the day of Michael Angelo's death, " nature seeming to signify thereby the passing of the sceptre from art to science, " and in the same year with Shakespeare, exerted a mighty influence on the development of science in many fields, and in particular laid the foundations of modern dynamics. It is a remarkable circumstance in the history of science that astronomy should have been cultivated at the same time by three such distinguished men as Tycho, Kepler and Galileo. While Tycho in the 54th year of his age was observing the heavens at Prague, Kepler, only 30 years old, was applying his wild genius to the determination of the orbit of Mars, and Galileo, at the age of 36, was about to direct the telescope to the unexplored regions of space. The diversity of gifts which Providence assigned to these three philosophers was no 218 A SHORT HISTORY OF SCIENCE less remarkable. Tycho was destined to lay the foundation of moder astronomy by a vast series of accurate observations made with th largest and the finest instruments ; it was the proud lot of Kepler t deduce the laws of the planetary orbits from the observations of hi predecessors; while Galileo enjoyed the more dazzling honor c discovering by the telescope new celestial bodies and new system of worlds. — Brewster. Coming into a world still dominated by the Aristotelian tradi tion, Galileo is puzzled by the conflict between his own obser vations and the accepted theories, but firm and fearless in hi convictions, he eagerly and powerfully controverts the older notions incidentally gaining enemies as well as disciples. What thos( accepted theories were may be exemplified by the following pas sages from a work of Daniel Schwenter (1585-1636), professor o mathematics at Altdorf : — 'When a body falls it moves faster the nearer it approaches th( earth. The farther it falls the more power it possesses. For every thing which is heavy, hastens according to the opinion of philosopheri towards its natural place, that is the centre of the earth, just as mai returning to his fatherland becomes the more eager the nearer h( comes, and therefore hastens so much the more. Still another natura cause contributes to this. The air which is parted by the falling ball hastens together again behind the ball and drives it always harder.' If the Copernican theory were true, the bullet remaining twc minutes in the air would be left many miles behind by the revolv- ing earth, — a distance which the moving atmosphere could noi possibly carry it. The rainbow is "a mirror in which the humar understanding can behold its ignorance in broad day." Th( powder drives the bullet in an oblique line to the highest poin1 of its path, then follows motion in an arc, finally, the natura motion vertically downward. In his whole point of view and habit of mind Galileo embodiec the attitude and spirit of modern science. He was keenly aleri in observing, analyzing, and reflecting on natural phenomena eager and convincing in his expositions, sceptical and intolerani A NEW ASTRONOMY 219 of mere authority, whether in science, philosophy, or theology. It was a true instinct of the conservatives to recognize in him the champion of a principle fatally hostile to their own. Between these antagonistic principles no permanent peace was possible. While still a mere youth, he discovered the regularity of pendulum vibrations by observing the slow swinging of the cathedral lamp of Pisa (1582). Before he was 25 he published work on the hydrostatic balance (1586), and on the centre of gravity of solids. Only a little later he conducted at the leaning tower simple ex- periments in falling bodies, which upset world-old notions on this everyday matter, showing that the velocity of descent is not, as was commonly supposed, proportional to weight. And "yet the Aristotelians, who with their own eyes saw the unequal weights strike the ground at the same instant, ascribed the effect to some unknown cause, and preferred the decision of their master to that of nature herself." He further showed that the hypothesis of uniform acceleration accounted correctly for the observed relations between space, time and velocity, and that the path of a projectile is a parabola. In the words of a recent authority, when Galileo deduced by experiment, and described with mathematical pre- cision, the acceleration of a falling body, he probably contributed more to the physical sciences than aU the philosophers who had preceded him. Hearing of the telescope newly invented in Holland, he con- structed one for himself, by means of which he discovered sun spots, the mountains of the moon, the satellites of Jupiter, the rings of Saturn, and the phases of Venus. The sensation created by these discoveries is described in the following passages from Fahie's Life of Galileo and Brewster's Martyrs of Science. 'As the news had reached Venice that I had made such an in- strument, six days ago I was summoned before their Highnesses, the Signoria, and exhibited it to them, to the astonishment of the whole senate. Many of the nobles and senators, although of a great age, mounted more than once to the top of the highest church tower in 220 A SHORT HISTORY OF SCIENCE Venice, in order to see sails and shipping that were so far off that il was two hours before they were seen, without my spy-glass, steering full sail into the harbour ; for the effect of my instrument is such thai it makes an object 50 miles off appear as large as if it were onlj five.' ' But the greatest marvel of all is the discovery of four new planets, I have observed their motions proper to themselves and in relation to each other, and wherein they differ from the motions of the othei planets. These new bodies move round another very great star, ir the same way as Mercury and Venus, and, peradventure, the othei known planets, move round the sun. As soon as my tract is printed, which I intend sending as an advertisement to all philosophers and mathematicians, I shall send a copy to his Highness, the Grand Duke, together with an excellent spy-glass, which will enable him to judge for himself of the truth of these novelties.' • — Fahie. Galileo's discoveries on the surface of the moon were ill received by the followers of Aristotle. According to their preconceived opin- ions, the moon was perfectly spherical and absolutely smooth ; and tc cover it with mountains and scoop it out into valleys was an act ol impiety which defaced the regular forms which Nature herself had imprinted. It was in vain that Galileo appealed to the evidence of observation and to the actual surface of our own globe. The verj irregularities on the moon were, in his opinion, a proof of divine wisdom : and had its surface been absolutely smooth, it would have been ' bul a vast unblessed desert, void of animals, of plants, of cities, and mer — the abode of silence and inaction — senseless, lifeless, soulless, anc stripped of all those ornaments which now render it so varied and sc beautiful.' In examining the fixed stars and comparing them with the planets Galileo observed a remarkable' difference in the appearance of theii discs. All the planets appeared with round globular discs like th( moon; whereas the fixed stars never exhibited any disc at all bui resembled lucid points sending forth twinkling rays. Stars of al magnitudes he found to have the same appearance ; those of th( fifth and sixth magnitude having the same character, when seei through a telescope, as Sirius, the largest of the stars, when seen bi the naked eye. Important and interesting as these discoveries were, they wer thrown into the shade by those to which he was led during a carefu A NEW ASTRONOMY 221 examination of the planets with a more powerful telescope. On the 7th of January, 1610, at one o'clock in the morning, when he directed his telescope to Jupiter, he observed three stars near the body of the planet, two being to the east and one to the west of him. They were all in a straight line, and parallel to the ecliptic and appeared brighter than other stars of the same magnitude. Believing them to be fixed stars, he paid no great attention to their distances from Jupiter and from one another. On the 8th of January, however, when, from some cause or other, he had been led to observe the stars again, he found a very different arrangement of them ; all the three were on the west side of Jupiter, nearer one another than before and almost at equal distances. Though he had not turned his attention to the extraordi- nary fact of the mutual approach of the stars, yet he began to con- sider how Jupiter could be found to the east of the three stars, when but the day before he had been to the west of two of them. The only explanation which he could give of this fact was that the motion of Jupiter was direct, contrary to the astronomical calculations and that he had got before these two stars by his own motion. In this dilemma between the testimony of his senses and the results of calculation, he waited for the following night with the utmost anxiety, but his hopes were disappointed, for the heavens were wholly veiled in clouds. On the 10th, two only of the stars appeared, and both on the east side of the planet. As it was obviously impossible that Jupiter could have advanced from west to east on the 8th of January, and from east to west on the 10th, Galileo was forced to conclude that the phenomenon which he had observed arose from the motion of the stars, and he set himself to observe diligently their change of place. On the 11th there were still only two stars, and both to the east of Jupiter, but the more eastern star was now twice as large as the other one, though on the preceding night they had been per- fectly equal. This fact threw a new light upon Galileo's difficulties, and he immediately drew the conclusion, which he considered to be indubitable, ' that there were in the heavens three stars which revolve around Jupiter, in the same manner as Venus and Mercury revolve around the sun.' On the 12th of January he again observed them in new positions, and of different magnitudes; and on the 13th he discovered a fourth star, which completed the four secondary planets with which Jupiter is surrounded. — Brewster. His results were published in 'The Sidereal Messenger,' announc- 222 A SHORT fflSTORY OF SCIENCE ing ' great and very wonderful spectacles, and offering them to th« consideration of every one, but especially of philosophers and as- tronomers; which have been observed by Galileo Galilei . . . bj the assistance of a perspective glass lately invented by him ; namelj in the face of the moon, in innumerable fixed stars in the Milky Way, in nebulous stars, but especially in four planets which revolve around Jupiter at different intervals and periods with a wonderful celerity which, hitherto not known to any one, the author has recently beer the first to detect, and has decreed to call the Medicean stars.' — Whewell. The reception which these discoveries met with from Kepler is highly interesting, and characteristic of the genius of that great man He was one day sitting idle and thinking of Galileo, when his friend Wachenfels stopped his carriage at his door to communicate to him some intelligence. 'Such a fit of wonder,' says he, 'seized me at a report which seemed to be so very absurd, and I was thrown into sucl: agitation at seeing an old dispute between us decided in this way that between his joy, my coloring, and the laughter of both, confounded as we were by such a novelty, we were hardly capable, he of speaking or I of listening. On our parting, I immediately began to think ho\^ there could be any addition to the number of the planets withoul overturning my Cosmographic Mystery, according to which Euclid's five regular solids do not allow more than six planets round the sun. . I am so far from disbelieving the existence of the four circumjovia planets, that I long for a telescope, to anticipate you, if possible, ii discovering two round Mars, as the proportion seems to require, si! or eight round Saturn, and perhaps one each round Mercury anc Venus.' In a very different spirit did the Aristotelians receive the Siderea Messenger of Galileo. The principal professor of philosophy a Padua resisted Galileo's repeated and urgent entreaties to look a the moon and planets through his telescope ; and he even labored t( convince the Grand Duke that the satellites of Jupiter could not pos sibly exist.* ' There are seven windows given to animals in the domicile of th head, through which the air is admitted to the tabernacle of the body ' ' As I wished to show the satellites of Jupiter to the professors in Florence, the would neither see them nor the telescope. These people believe there is no trut to seek in nature, but only in the comparison of texts.' A NEW ASTRONOMY 223 to enlighten, to warm, and to nourish it. What are these parts of the microcosmos ? Two nostrils, two eyes, two ears, and a mouth. So in the heavens, as in a macroeosmos, there are two favorable stars, two unpropitious, two luminaries, and Mercury undecided and indifferent. From this and many other similarities in nature, such as the seven metals, etc. which it were tedious to enumerate, we gather that the number of planets is necessarily seven. Moreover, these satellites of Jupiter are invisible to the naked eye, and therefore can exercise no influence on the earth, and therefore would be useless, and therefore do not exist. Besides, the Jews and other ancient nations, as well as modern Europeans, have adopted the division of the week into seven days, and have named them after the seven planets. Now, if we increase the number of the planets, this whole and beautiful system falls to the ground.' — Fahie. It was inevitable that such a man as Galileo should accept the Copernican hypothesis. He writes to Kepler in 1597 : — 'I esteem myself fortunate to have found so great an ally in the search for truth. It is truly lamentable, that there are so few who strive for the true and are ready to turn away from wrong ways of philosophizing. But here is no place for bewailing the pitifulness of our times, instead of wishing you success in your splendid investiga- tions. I do this the more gladly, since I have been for many years an adherent of the Copernican theory. It explains to me the cause of many phenomena which under the generally accepted theory are quite unintelligible. I have collected many arguments for refuting the latter, but I do not venture to bring them to publication. 'That the moon is a body like the earth I have long been assured. I have also discovered a multitude of previously invisible fixed stars, outnumbering more than ten times those which can be seen by the naked eye, — forming the Milky Way. Further I have discovered that Saturn consists of three spheres which almost touch each other.' While none of Galileo's astronomical discoveries were either necessary or sufficient to confirm the Copernican theory, their support was exceedingly important. Thus the slow motion of the sun spots across the disc and their subsequent reappearance 224 A SHORT HISTORY OF SCIENCE showed rotation of that body, the satellites of Jupiter and par- ticularly the phases of Venus, analogous to those shown by the moon, obviously harmonized with the Copernican theory. This implied at least that the planets shone by reflected sunlight, and it had indeed been insisted against that theory that Venus and Mercury under it must show phases till then undiscovered. In 1632 Galileo published his celebrated Dialogue on the Twc Chief Systems of the World, the Ptolemaic and the Copernican, a work comparable in magnitude and importance with Copernicus Revolutions. In the curious preface he says : — ' Judicious reader, there was published some years since in Rome a salutiferous Edict, that, for the obviating of the dangerous Scandals of the present Age, imposed a reasonable Silence upon the Pythag- orean Opinion of the Mobility of the Earth. There want not such as unadvisedly affirm, that the Decree was not the production of a sobei Scrutiny, but of an illformed passion ; and one may hear some muttei that Consultors altogether ignorant of Astronomical observation! ought not to clipp the wings of speculative wits with rash prohibitions My zeale cannot keep silence when I hear these inconsiderate com' plaints. I thought fit, as being thoroughly acquainted with thai prudent Determination, to appear openly upon the Theatre of th( World as a Witness of the naked Truth. ... I hope that by thes( considerations the world will know that if other Nations have Navi gated more than we, we have not studied less than they; and tha our returning to assert the Earth's stability, and to take the contrary only for a Mathematical Capriccio, proceeds not from inadvertenci of what others have thought thereof, but (had one no other induce ments), from these reasons that Piety, Religion, the Knowledge of thi Divine Omnipotency, and a consciousness of the incapacity of man'; understanding dictate unto us.' In the first of the four conversations into which the work i divided, the Aristotelian theory of the peculiar character of th heavenly bodies is subjected to destructive criticism, with em phasis on such phenomena as the appearance of new stars, o comets and of sun spots, the irregularities of the moon's surface the phases of Venus, the satellites of Jupiter, etc. Galileo's Dialogue. A NEW ASTRONOMY 225 'When we consider merely the vast dimensions of the celestial sphere in comparison with the littleness of our earth . . . and then think of the speed of the motion by which a whole revolution of the heavens must be accomplished in one day, I cannot persuade myself that the heavens turn while the earth stands fast.' Adducing not merely the sun spots themselves, but their rapid variation, he insists that the universe is not rigid and permanent, but constantly changing or, as science has more and more em- phasized since his day, passing through consecutive, related phases or evolving. 'I can listen only with the greatest repugnance when the quality of unchangeability is held up as something preeminent and complete in contrast to variability. I hold the earth for most distinguished exactly on account of the transformations which take place upon it.' He begins to see the fallacy of the objections that if the earth rotated, a body dropped from a masthead would be left behind by the ship and that movable objects could be thrown off centrif- ugally at the equator. As positive arguments in support of the Copernican system, he urges particularly the retrogressions and other irregularities of the planets, and also the tides. Of the famous controversy of Galileo with the Inquisition, it may here suffice to quote the judgment of the court (see Appen- dix) : — 'The proposition that the sun is in the centre of the world and immovable from its place is absm'd, philosophically false and formally heretical ; because it is expressly contrary to the Holy Scriptures,' etc. and a passage from the biographer already cited at so much length : — For over fifty years he was the knight militant of science, and almost alone did successful battle with the hosts of Churchmen and Aristotelians who attacked him on all sides — one man against a world of bigotry and ignorance. If then, . . . once, and only once, when face to face with the terrors of the Inquisition, he, like Peter, denied his Master, no honest man, knowing all the circumstances, will be in a hurry to blame him. Q 226 A SHORT HISTORY OF SCIENCE Of Galileo's still more remarkable services to physics anc dynamics, something will be added in a later chapter. Medical and Chemical Sciences. — These were still at th( low medieval level. There was as yet no scientific medicine, anc no chemistry but alchemy, which was now in its final stage, iatro (medical) chemistry. Here one great name is that oi Paracelsus (1493-1541), erratic and radical Swiss physician and alchemist, whose chief merit is his courage in opposing mere authority in science, and whose influence long after caused "salt, sulphur, and mercury" to be highly regarded and carefully studied. He also introduced and insisted upon the importance of antimony as a remedy, and is said to have been the first to use that tincture of opium which is still known by his name foi it ; viz. laudanum. Paracelsus, on the other hand, in spite oi the fact that he was a popular surgeon, rejected the study oi anatomy, taught medical knowledge through scanning of the heavens, and considered diseases as spiritual in origin. "The true use of chemistry," he said, "is not to 1 make gold but tc prepare medicines." Another name worthy of remembrance in the chemistry of the sixteenth century is that of Landmann (Latin, Agricola) whose great work on Metallurgy (De Re Metallica, 1546) is the most im- portant of this period, and who must also be regarded as the first mineralogist of modern times. Anatomy. Vesalius. — Hardly less important, meantime, thai the studies of Copernicus, Tycho Brahe, Galileo and Kepler upor the heavenly bodies were those of the Belgian anatomist, Andrea: Vesalius, upon the human body. For more than 1000 years ther( had been almost no progress in anatomy or medicine, Hippocratei and Galen being still regarded as the final authorities in these matters up to the middle of the sixteenth century. Vesaliui (1514-1564), born in Brussels and educated in Paris, was the firs in modern times to dissect the human body, and to publish excel lent drawings of his dissections. It was said that he opened thi body of a nobleman before the heart had entirely ceased beating and thereby incurring the displeasure of the Inquisition, was sen BEGINNINGS OF MODERN NATURAL SCIENCE 227 tenced to perform a penitential journey to Jerusalem. At all events, he went to Jerusalem and was shipwrecked and lost while returning. After Vesalius the study of human anatomy was vigorously and successfully prosecuted in Italy as was natural, since it was in Italy that Humanism and the revival of learning first took firm hold of Christian Europe. One of Vesalius' Italian contemporaries, Eustachius, whose name is still familiarly associated with the pas- sage or "tube" connecting the throat and the middle ear, is hardly less famous in the history of anatomy than is Vesalius himself. The name of Fallopius, professor at Pisa in 1548 and at Padua in 1551, is also similarly associated with the human oviducts, — the so-called Fallopian tubes. His disciple Fabricius of Acquapendente discovered the valves in the veins, and was the teacher of William Harvey. A Spanish anatomist of note, Michael Servetus, — born 1509, — perished as a martyr at the stake in 1553 because of heretical writings abhorrent alike to the In- quisition and to Calvin. Of physiology we have as yet little or no account. Doubtless aU the anatomists just mentioned and many other "philosophers" had pondered, as did Aristotle and his predecessors, on the workings of the animal, and especially the human, mechanism. But from Aristotle (b.c. 322) to William Harvey (1578-1657) no real progress was made. It is a melancholy commentary on superstition and human prejudice that long after the brilliant work of Vesalius and the Italian anatomists, no proper "anatomy acts" existed to make lawful dissection either possible or easy, so that for several cen- turies afterward anatomists, surgeons, and medical students felt themselves at times obliged to resort to "body-snatching." Natubal History and Natural Philosophy. — No great progress was made in this field after the observations of Aristotle, Theophrastus, Xenophanes, and Pythagoras until the sixteenth century. Fossils mostly remained unexplained or were regarded as " freaks " of nature. Animals and plants were comparatively neg- lected and, if studied, considered either as the raw material for sup- posed remedies or medicines, or else as treated by Aristotle. The 228 A SHORT fflSTORY OF SCIENCE twenty-six books De Animalihus, of Albertus Magnus {d. 1282 were not printed until 1478, but were apparently well known ii manuscript copies. No great worker appears in this almost neg lected field until we come to Conrad Gesner (or Gessner) (1516- 1565), the first famous naturalist of modern times, on accoun of his vast erudition surnamed "the German Pliny." Professor o Greek at Lausanne and later of Natural History at Basel, he wa; almost as prolific an author as was della Porta fifty years later for he wrote extensively upon plants, animals, milk, medicine and theology, as well as various classical subjects. Yet he rank; high in the history of biology, both for the extent and the qualitj of his work in zoology and botany. It is significant that Gesne: was a Swiss, and as such probably safe from persecution at a time when William Tiu-ner, an English ornithologist, worked and published in Cologne. At the end of the fifteenth and beginning of the sixteenth centuries Leonardo da Vinci (1452-1519) turned his attention in part from art to science, engineering, and inventions, making interesting studies in architecture, hydraulics, geology, etc. He is regarded as the first engineer of modern times, and has been called "the world's most universal genius." Palissy, "the Potter," later examined minutely various fossils and took the ther advanced ground (as Xenophanes and Pythagoras had done, however, some two thousand years earlier) that these are ir reality what they appear to be, i.e. petrified remains of plant and animal life, and not "freaks of nature." Palissy's bold stand oi this subject marks one of the first steps in modern times toward rational geology. It was not until the end of the sixteenth century, when Wil- liam Gilbert, an eminent practising physician of Colchester England (1540-1603), published his now famous work on th( magnet {Be Magnete) that further progress was made througl the first rational treatment of electrical and magnetic phenomena To him is due the name electricity {vis electrica). He regarded the earth as a great magnet and, accepting the Copernicar theory, attributed the earth's rotation to its magnetic character BEGINNINGS OF MODERN NATURAL SCIENCE 229 He even extended this idea to the heavenly bodies, with an ani- mistic tendency. Gilbert is also reputed to have done important work in chemistry, but none of this has survived. His work is one of the finest examples of inductive philosophy that has ever been presented to the world. It is the more remarkable because it preceded the Novum Organum of Bacon, in which the inductive method of philosophizing was first explained. — Thomas Thomson. The most prolific \\Titer on natural philosophy and physical science of the sixteenth century was G. della Porta (1543-1615), a native of Naples and a resident of Rome, founder of an early scientific academy there, and afterwards of the famous Accademia del Lincei of Rome. His writings are voluminous and in many books, of which we need mention here only his Magia Naturalis, (1569), De Refractione (1593), Pneumatica (1691), De Distilla- tione (1604), De Munitione (1608) and De Aeris Transmutationi- bus (1609). In his Natural Magic, della Porta is the first to describe a camera obscura, besides touching on many interesting properties of lenses, and referring to spectacles, some forms of which had long been known. His work On Refraction deals largely with binocular vision, and is a criticism of the work of Euclid and Galen on that subject. The author hints also at a crude tele- scope, and may have known some form of stereoscope. Della Porta's compositions range all the way from natural magic to Italian comedies, and entitle him to high rank as a tireless and original, if not especially fruitful, thinker and worker. References for Reading Berry. History of Astronomy. Chapters IV-VII. Brewster. Martyrs of Science. Dretee. Tycho Brake; Planetary Systems. Chapters XII-XVT. FAmE. Life of Galileo. Gilbert. On the Magnet. LocT. Biology and its Makers. Lodge. Pioneers of Science. CHAPTER XI PROGRESS OF MATHEMATICS AND MECHANICS IN THE SIXTEENTH CENTURY It was not alone the striving for universal culture which attracted the great masters of the Renaissance, such as Brunelleschi, Leonardo da Vinci, Raphael, Michael Angelo and especially Albrecht Diirer, with irresistible power to the mathematical sciences. They were conscious that, with all the freedom of the individual phantasy, art is subject to necessary laws and, conversely, with all its rigor of logical structure, mathematics follows esthetic laws. — Rudio. The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms. — Cajori. The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, witt the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's Descriptio. — Glaisher. It is Italy, which is the fatherland of Archimedes, whose creative power embraces all domains of the mechanical science, the land oi the Renaissance, from out of which those mighty waves of new ideas and new impulses in science and art have come forth into the world — the fatherland of Galileo the creator of experimental physics, ol Leonardo da Vinci the engineer, of Lagrange who has given its forn: to modern analytical mechanics. — W. v. Dyck. Dynamics is really a product of modern times, and affords the rare example of a development fulfilled in a single great personage — Galileo. Nothing is finer than how he, beginning in the Aristoteliai spirit, gradually frees himself from its bondage and, instead of emptj metaphysics, introduces well-directed methodical investigations o nature. — • Timerding. The period from the invention of printing about 1450 to that o analytic geometry in 1637 was one of very great importance fo mathematics and mechanics as well as for astronomy. At th 230 PROGRESS OF MATHEMATICS AND MECHANICS 231 beginning, Arabic numerals were known, but the mathematics even of the universities hardly extended beyond the early books of Euclid and the solution of simple cases of quadratic equations, in rhetorical form. At the end of the period the foundations of modern mathematics and mechanics were securely laid. Aims and Tendencies of Mathematical Progress. — In the centuries just preceding, the chief applications of mathematics had connected themselves with the relatively simple needs of trade, accounts and the calendar, with the graphical constructions of the architect and the military engineer, and with the sines and tangents of the astronomer and the navigator. During the period in ques- tion some of these applications became increasingly important, and at the same time mathematics was more and more cultivated for its own sake. Mathematicians became gradually a more and more distinctly differentiated class of scholars ; mathematical text- books took shape. The beginnings of this evolution have been dealt with abeady ; its further progress is now to be traced. The larger achievements and tendencies of the period in mathe- matical science were the following : — In Arithmetic, decimal fractions and logarithms were introduced, regulating and immensely simplifying computation; a general theory of numbers was developed ; in Algebra, a compact and ade- quate symbolism was worked out, including the use of the signs +, -H, X, — , =, 0, V , and of exponents; equations of the third and fourth degree were solved, negative and imaginary roots accepted, and many theorems of our modern theory of equations discovered. In Geometry, the computation of tt was carried to many deci- mals, the beginnings of projective geometry were made, and a so-called method of indivisibles developed, foreshadowing the integral calculus; in plane and spherical Trigonometry, the theorems and processes now in use were worked out, and extensive tables computed. In Mechanics, ideas about force and motion, equilibrium and centre of gravity, were gradually clarified. Underlying some of these new developments are the dawning 232 A SHORT HISTORY OF SCIENCE fundamental concepts : function, continuity, limit, derivative, in finitesimal, on which our modern mathematics has been built up Descartes, Newton and Leibnitz are soon to make their revolu tionary discoveries in analytic geometry and the calculus. We have seen that up to about 1500 the chief stages in the de velopment of mathematics have been the introduction and im provement of Arabic arithmetic for commercial purposes (thougl accounts were kept in Roman numerals until 1550 to 1650), th( rediscovery of Greek geometry, and the improvement of trigo nometry in connection with its increasing use in astronomy, navi- gation and military engineering. The development of science hai been powerfully promoted by the general intellectual emancipatioi of the Renaissance, while mathematical progress, beginning earlier has been both a cause and a consequence of the general advance The diffusion and the preservation of scientific knowledge hav( derived immense advantage from the new art of printing and fron expanding commercial intercourse. Algebra, almost helpless ii Greek times because, for lack of proper symbolism, expressed onlj in geometrical or rhetorical form, has been converted by i process of abbreviation, at first into a syncopated form, inter- mediate between the rhetorical and our modern purely symbolic notation. Pacioli. — The earliest printed book on arithmetic and algebrs was published at Venice in 1494 by Lucas Pacioli, a Franciscai monk born in Tuscany about 1450. Rules are here given for th( fundamental operations of arithmetic, and for extracting square roots. Commercial arithmetic is treated at considerable lengtl by the newer algoristic or Arabic methods. The method of arbi trary assumption corrected by proportion is used effectively, foi example : — To find the original capital of a merchant who spent a quarte: of it in Pisa and a fifth of it in Venice, who received on these transac tions 180 ducats, and who has in hand 224 ducats. Assume that his original capital was 100 ducats ; then the surplu: would be 100 - 25 - 20 = 55, but this is f of his actual surplu; 224 - 180, therefore his original capital was f of 100 = 80 ducats. PROGRESS OF MATHEMATICS AND MECHANICS 233 Some of Pacioli's commercial problems are exceedingly compli- cated. He solves numerical equations of the first and second degree, but admits only positive roots and considers the solution of cubic equations, as well as the squaring of the circle, impossible. Addition is denoted by p or p, equality sometimes by ae, a begin- ning of syncopated algebra. The introduction of the radical sign with indices V2. V3 and of the signs -|- and — date from about this time. In geometry Pacioli, like Regiomontanus, employs algebraic methods. Among other problems he determines a triangle from the radius of the inscribed circle and the segments into which it divides one of the sides. His solution, though highly esteemed at the time, is much less simple than he might have obtained by the formulas at his command. In the spirit of the Renaissance he brings the feeble mathematics of the universities into fruitful relations with the practical mathe- matics of the artist and the architect. The inscribed hexagon and the equilateral triangle play their part as gild secrets in the develop- ment of Gothic architecture. The question is not " How to prove," but "How to do." On the other hand, the current tendency to drift into mysti- cal interpretation is exemplified by the following extract from Pacioli : — There are three principal sins, avarice, luxury, and pride ; three sorts of satisfaction for sin, fasting, almsgiving, and prayer; three persons ofPended by sin, God, the sinner himself, and his neighbom' ; three witnesses in heaven, Paier, verbum, and spiritus sanctus; three degrees of penitence, contrition, confession, and satisfaction, which Dante has represented as the three steps of the ladder that leads to purgatory, the first marble, the second black and rugged stone, and the third red porphyry. There are three sacred orders in the church militant, subdiaconati, diaconati, and presbyterati ; there are three parts not without mystery, of the most sacred body made by the priest in the mass; and three times he says Agnus Dei, and three times, Sanctus; and if we well consider all the devout acts of Christian wor- ship, they are found in a ternary combination ; if we wish rightly to 234 A SHORT fflSTORY OF SCIENCE partake of the holy communion, we must three times express our con- trition, Domine non sum dignus ; but who can say more of the ternary number in a shorter compass, than what the prophet says, tu signaculum sandae trinitatis. There are three Furies in the infernal regions; three Fates, Atropos, Lachesis, and Clotho. There are three theo- logical virtues ; Fides, spes, and charitas. Tria sunt pericula mundi : Equum currere; navigare, et sub tyranno vivere. There are three enemies of the soul : the Devil, the world, and the flesh. There are three things which are of no esteem : the strength of a porter, the advice of a poor man, and the beauty of a beautiful woman. There are three vows of the Minorite Friars; poverty, obedience, and chastity. There are three terms in a continued proportion. There are three ways in which we may commit sin : corde, ore, ope. Three principal things in Paradise: glory, riches, and justice. There are three things which are especially displeasing to God: an avaricious man, a proud poor man, and a luxurious old man. And all things in short, are founded in three ; that is, in number, in weight, and ii measure. Geometey in Art. — Brunelleschi (1377-1446), the famous architect of the early Renaissance, made a perspective view of th( Signoria in Florence in a sort of box with clouds. The famous doori of the Baptistery by his contemporary Ghiberti show the develop ment of perspective in the marked contrast between the earlie: and the later panels. Raphael in his School of Athens include; himself and Bramante in a group of mathematicians. Painteri were even called for a time perspectivists — prospettim. Leonardo da Vinci (1452-1519), one of the intellectual giant of the Renaissance, eminent alike in art, science and engineer ing, gave the first correct explanation of the partial illuminatio] of the darker part of the moon's disc by reflection from the eartl He calls mechanics the paradise of the mathematical sciences because through it one first gains the fruit of these sciences. H denies the possibility of perpetual motion, saying "Force is th cause of motion and motion the cause of force." He discusses th lever, the wheel and axle, bodies falling freely or on inclined plane; foreshadowing Galileo. Contrary to the Aristotelian tradition h asserts that everything tends to continue in its given state, and h PROGRESS OF MATHEMATICS AND MECHANICS 235 even enunciates the fundamental principle that for simple machines forces in equilibriimi are inversely as the virtual velocities. 'Whoever,' he says, 'appeals to authority applies not his intellect but his memory.' ' While Nature begins with the cause and ends with the experiment, we must nevertheless pursue the opposite plan, beginning with the experiment and by means of it investigating the cause.' 'No human investigation can call itself true science, unless it comes through mathematical demonstration.' 'He who scorns the certainty of mathematics will not be able to silence sophistical theories wliich end only in a war of words.' Unfortunately his work in this field remained unpublished, and therefore relatively unfruitful. Leonardo and other great artists of his time — notably Albrecht Durer of Nuremberg (1471-1528) — developed the geometrical theory of perspective. For the purpose of accurately representing the human head Diirer made both plans and elevation. "Intelli- gent painters and accurate artists," he says, " at the sight of works painted without regard to true perspective must laugh at the blind- ness of these people, because to a right understanding nothing im- presses more disagreeably than falsehood in a painting, regardless of the diligence with which it has been made. That such painters, however, are pleased with their own mistakes is due to the fact that they have not learned the art of measurement, without which no one can become a true workman." All this had importance both for modern art and modern geometry. Characteristic of this period is the so-called Margarita Philo- sophiea published in many editions from 1503 to 1600. It was the first modern encyclopaedia printed, and gives in its twelve books " a compendium of the trivium, the quadrivium, and the natural and moral sciences." A younger contemporary of Pacioli, Michael Stifel (1487-1567), a German monk converted to Lutheranism, developed a fantastic arithmetical interpretation of the Bible, identifying Pope Leo X with the beast in Revelation and predicting the immediate end of the world, — with results disastrous to his person as well as his reputation. 236 A SHORT HISTORY OF SCIENCE He relates . . . that whilst a monk at Esslingen in 1520, and whec infected by the writings of Luther, he was reading in the library of his convent the 13th Chapter of Revelations, it struck his mind that the Beast must signify the Pope, Leo X ; He then proceeded in pious hope to make the calculation of the sum of the numeral letters in Lee decimus, which he found to be M, D, C, L, V, I ; the sum which these formed was too great by M, and too little by X ; but he bethought him again, that he has seen the name written Leo X ; and that there were ten letters in Leo decimus, from either of which he could obtain the deficient number, and by interpreting the M to mean mysterium, he found the number required, a discovery which gave him such un- speakable comfort, that he believed that his interpretation must have been an immediate inspiration of God. — Peacock. Stifel's writings on arithmetic and algebra embody some improve- ments of current notation. He introduced for example the symbol; lA, \AA, lAAA for what we should denote by x, x^, x^. The low state of computation at this time is illustrated wltt startling clearness by a bulletin on the blackboard at Wittenberg, in which Melanchthon urgently invited the academic youth tc attend a course on arithmetic, adding that the beginnings of th( science are very easy, and even division can with some diligence b( comprehended. Robert Recorde (1510-1558) studied at Oxford and graduatec in medicine at Cambridge in 1545, later becoming "royal physl cian." His "Grounde of Artes" or arithmetic, one of the earliesi mathematical books printed in English (1540), ran through more than 27 editions and exerted a great influence on English education In the "Preface to the Loving Reader" he says: — Sore ofttimes have I lamented with myself the unfortunate con dition of England, seeing so many great Clerks to arise in sundr^ other parts of the World, and so few to appear in this our Nation whereas for pregnancy of natural wit (I think) few Nations do excel English-men. But I cannot impute the cause to any other thing then to the contempt or misregard of Learning. For as English-mei are inferiour to no men in mother Wit, so they pass all men in vaii Pleasures, to which they may attain with great pain and labour ; am PROGRESS OF MATHEMATICS AND MECHANICS 237 are slack to any never so great commodity, if there hang of it any pain- full study or travelsome labour. The book itself is in the form of a dialogue or catechism be- ginning : — The Scholar speaketh. ' Sir, such is your authority in mine estimation, that I am content to consent to your saying, and to receive it as truth, though I see none other reason that doth lead me thereunto ; whereas else in mine own conceit it appeareth but vain, to bestow any time privately in learning of that thing that every Child may and doth learn at all times and hours, when he doth any thing himself alone, and much more when he talketh or reasoneth with others.' He employs the symbol + "whyche betokeneth too muche, as this line — plaine without a crosse line betokeneth too little." In 1557 he published an algebra under the alluring title " Whet- stone of Witte," using the sign = for equality, which he says he selected because "noe 2 thynges can be moare equalle" than two parallel straight lines. Algebraic Equations of Higher Degree. — Two great Ital- ian mathematicians vied with each other in giving a powerful im- petus to the development of algebra in the sixteenth century. Niccolo Fontana or Tartaglia (1500-1557) a man of the hum- blest origin, lectured at Verona and Venice, and first won fame by successfully meeting a challenge to solve mathematical prob- lems, all of which proved, as he had anticipated, to involve cubic equations. His Nova Scienza (1537) discusses falling bodies, and many problems of military engineering and fortification, the range of projectiles, the raising of sunken galleys, etc. The title-page is chiefly occupied by a large plate, which represents the courts of Philosophy, to which Euclid is doorkeeper, Aristotle and Plato being masters of an inmost court, in which Philosophy sits throned, Plato declaring by a label that he will let nobody in who does not understand Geometry. In the great court there is a cannon being fired, all the sciences looking on in a crowd — such as Arithmetic, 238 A SHORT HISTORY OF SCIENCE Geometry, Music, Astronomy, Cheiromancy, Cosmography, Necro mancy. Astrology, Perspective, and Prestidigitation ! A wonderf ulb modest-looking gentleman, with his hand upon his heart, stands amonj the number, with a you-do-me-too-much-honour loot upon his coun tenance ; Arithmetic and Geometry are pointing to him, and undei his feet his name is written — Nicolo Tartalea. — Morley, Jerome Cardan. The Inventioni (1546) gives his solution of the cubic equation A treatise on Numbers and Measures (1556, 1560) gives a methoc for finding the coefficients in the expansion of (1 + a;) " for w = 2 ... 6. It contaifis also a wide range of problems from commercia arithmetic and a collection of mathematical puzzles. The follow- ing examples may illustrate these : — 'Three beautiful ladies have for husbands three men, who ar( young, handsome, and gallant, but also jealous. The party art travelling, and find on the bank of a river, over which they have tc pass, a small boat which can hold no more than two persons. Hov, can they pass, it being agreed that, in order to avoid scandal, nc woman shall be left in the society of a man imless her husband i; present ? ' ' A ship carrying as passengers 15 Turks and 15 Christians en- counters a storm, and the pilot declares that in order to save the shij and crew one half of the passengers must be thrown into the sea To choose the victims, the passengers are placed in a circle, and it i; agreed that every 9th man shall be cast overboard, reckoning from i certain point. In what manner must they be arranged so that thf lot may fall exclusively upon the Turks ? ' ' Three men robbed a gentleman of a vase containing 24 ounces o: balsam. Whilst running away they met in a wood with a glass-sellei of whom in a great hurry they purchased three vessels. On reaching i place of safety they wish to divide the booty, but they find that thei: vessels contain 5, 11, and 13 ounces respectively. How can the^ divide the balsam into equal portions ? ' — Ball. There is no other treatise that gives as much information con cernmg the arithmetic of the sixteenth century, either as tc theory or application. The life of the people, the customs of th( PROGRESS OF MATHEMATICS AND MECHANICS 239 merchants, the struggles to Improve arithmetic, are all set forth here by Tartaglia in an extended but interesting fashion. Tartaglia, anticipating Galileo, taught that falling bodies of different weight traverse equal distances in equal times, and that a body swung in a circle if released flies off tangentially. GiROLAMO Caedan (1501-1576) led a life of wild and more or less disgraceful adventure, strangely combined with various forms of scientific or semi-scientific activity, — particularly the practice of medicine. He studied at Pavia and Padua, travelled in France and England, and became professor at Milan and Pavia. His Ars Magna (1545) contains the solution of the cubic equa- tion fraudulently obtained from his rival Tartaglia. After its publi- cation the aggrieved Tartaglia challenged Cardan to meet him in a mathematical duel. This took place in Milan, August 10, 1548, but Cardan sent his pupil Ferrari in his place. Tartaglia relates that he was accompanied only by his brother, Ferrari by many friends. Cardan had left for parts unknown. As Tartaglia began to explain to the crowd the origin of the strife and to criticise Ferrari's 31 solutions, he was interrupted by a demand that judges be chosen. Knowing no one present he declines to choose; all shall be judges. Being finally allowed to proceed he convicts his opponent of an erroneous solution, but is then overwhelmed by tumultuous clamor with demands that Ferrari must have the floor to criticise his solution. In vain he insists that he be allowed to finish, after which Ferrari may talk to his heart's content. Fer- rari's friends are vehement ; he gains the floor and chatters about a problem which he claims Tartaglia has not been able to solve till the dinner hour arrives and Tartaglia, apprehending still worse treatment, withdraws in disgust. Ferrari (1522-1565), this disciple of Cardan, even succeeded in giving a general solution of the equation of the fourth degree, beyond which, as has been shown only in quite recent times, the solution can in general no longer be similarly expressed. Some idea of the difficulty of these sixteenth century achievements may be conveyed by the corresponding modernized solutions. If the given equation is a:^ + bx' + ex + d = the coefficient of the 240 A SHORT fflSTORY OF SCIENCE first term is made 1 by division and that of the second is mad( by the substitution x = 2/ - -— - The new equation having the f om o a y' + ey+f = yfe now put y = z--^, whence z* - — ^ +/ = a quadratic equation in ^. The solution of the original equation oi degree three is thus made to depend on that of an equation of degre( one less. Similarly if the given equation of the fourth degree is in ouj notation ax^ +hx^ + cx^ + dx + e = Q the coefficient of th( first term is made 1 by division and that of the second is made bj the substitution x = y 4a The new equation having the form y'+fy' + gy + h =0. We put y'+fy^ + gy + h = {y^ - ay + fi) {y^ + ay + y) whence / = (3 + 7 — a^ g =(fi - y) a h =/37. We obtain a, /3, 7 from these three equations by eliminating tw( and solving the cubic equation obtained for the other ; that is, the solu tion of the original equation of degree four is made to depend on tha of a new equation of degree one less. One of Cardan's scientific inventions was an improved suspen sion of the compass needle. He was also eminent as an astrologer Symbolic Algebra : Vieta. — Of still greater importance n the history of algebra is F. Vieta (1540-1603) a lawyer of th( French court. He won the interest of Henry IV by solving a com plicated problem proposed by an eminent mathematician, as wai the custom of the time, as a challenge to the learned world. Thii involved an equation of the 45th degree which he succeeded ii solving by a trigonometric device. Later he was employed to inter pret the cipher despatches of the hostile Spaniards. His In Artev Analyticam Isagoge is the earliest work on symbolic algebra In it known quantities are denoted by consonants, unknown b^ vowels, the use of homogeneous equations is recommended, thi PROGRESS OF MATHEMATICS AND MECHANICS 241 first six powers of a binomial given, and a special exponential notation introduced. He shows that the celebrated classical problems of trisecting a given angle and duplicating a cube involve the solution of the cubic equation, and makes important discoveries in the general theory of equations — for example resolving poly- nomials into linear factors and deriving from a given equation other equations having roots which differ from those of the first by a constant or by a given factor. He solves Apollonius' famous problem of determining the circle tangent to three given circles, and expresses ir by an infinite series. He devises systematic methods for the solution of spherical triangles. Development of Trigonometry. — Many circumstances com- bined to promote the development of trigonometry at this period. It was needed by the military engineer, the builder of roads, the astronomer, the navigator, and the mapmaker whose work was tributary to all of these. Rheticus (George Joachim, 1514^1576), — "the great computer whose work has never been superseded," — worked out a table of natural sines for every 10 seconds to fifteen places of decimals. We owe to him our familiar formulas for sin 2x and sin 3a;. The notation sin, tan, etc. and the determination of the area of a spherical triangle date from about this time. To this period belong also the very important work of Mercator on map-making and the reform of the calendar by Pope Gregory XHI. Map-making. — Mercator (Gerhard Kramer, 1512-1594) de- voted himself in his home city, Louvain, to mathematical geogra- phy, and gained his livelihood by making maps, globes and astronomical instruments, combined in later life with teaching. His great world map, completed in 1569, marks an epoch in cartography. The first "Atlas" was published by his son in 1595. He gives a mathematical analysis of the principles underlying the projection of a spherical surface on a plane. 'If,' he says, 'of the four relations subsisting between any two places in respect to their mutual position, namely difference of latitude, difference of longitude, direction and distance, only two are regarded, the others also correspond exactly, and no error can be committed as 242 A SHORT HISTORY OF SCIENCE must so often be the case with the ordinary marine charts and so muc the more the higher the latitude.' Mercator's geometrical method amounts to projecting th spherical surface of the earth on a cylinder tangent to the eart along the equator and having the same axis with the earth Under this method of projection, angles are preserved in magni tude, but areas remote from the equator are disproportionatel; expanded. A straight line on the chart corresponds with th course of a ship steering a constant course. The Gregorian Calendar. — Until 1582 the Julian calenda (p. 143) remained in force with 365 j days each year and a graduall; increasing error amounting at this time to ten days. Under thi auspices of Pope Gregory the days from October 5 to 15, 1572 were dropped and the number of leap-years in 400 reduced fron 100 to 97. Religious jealousies prevented the adoption of thi reform in Protestant Germany for a century, while Englanc postponed it until 1752. A New Intention for Computation. — The invention o logarithms would appear to have been a natural sequel of anj adequate theory and notation for exponents. Thus Stifel in hi: arithmetic (1544) had tabulated small integral powers of 2 — fron I to 64 — and shown the correspondence between multiplicatioi of these powers and addition of the indices or exponents, but hii use of exponents was too limited, he lacked the apparatus of deci' mal fractions necessary for the practical application of the methoc and probably had no conception of the vast labor-saving possi- bilities so near at hand. In 1614 John Napier published at Edinburgh his MirificiLoga- rithmorum Canonis Descriptio, for which the time was so fully rip( that an enthusiastic reception was at once assured. Napier as i devout Protestant, stimulated by fear of an impending Spanist invasion, busied himself with inventions " profRtabill & necessarj in theis dayes for the defence of this Hand & withstanding oi strangers enemies of God's truth & relegion." Among these wer( a mirror for burning distant ships, and a sort of armored cliariot Impressed by the tremendous calculations then in progress bj PROGRESS OF MATHEMATICS AND MECHANICS 243 Rheticus, Kepler, and others in connection with the development of the new astronomy, Napier made a vastly more important inven- tion. His definition of a logarithm rests on the following kinetic basis : — Q -S Tj^ ^J Qi -S^ r S is a straight line of definite length ; fi Si extends to the right indefinitely. Moving points P and Pi start from T and Ti with equal initial speeds; the latter continues at the same rate, the former is retarded so that its speed is always proportional to its distance from S. If equal intervals are taken on Ti Si the cor- responding intervals in TS will grow smaller to the right. When P is at any position Q the logarithm of QS is represented by the corresponding length TiQi on the other line. It may be shown in fact that if in our notation PS = x, TiPi = y, TS = I, — = —7. This conception involving a functional relation be- dy I tween two variables went much deeper than the comparison of discrete numbers by Stifel. Napier's conception of a logarithm involved a perfectly clear apprehension of the nature and consequences of a certain functional relationship, at a time when no general conception of such a relation- ship had been formulated, or existed in the minds of mathematicians, and before the intuitional aspect of that relationship had been clarified by means of the great invention of coordinate geometry made later in the century by Rene Descartes. A modern mathematician re- gards the logarithmic function as the inverse of an exponential func- tion; and it may seem to us, familiar as we all are with the use of operations involving indices, that the conception of a logarithm would present itself in that connection as a fairly obvious one. We must however remember that, at the time of Napier, the notion of an index, in its generality, was no part of the stock of ideas of a mathe- matician, and that the exponential notation was not yet in use. — Hohson. 244 A SHORT HISTORY OF SCIENCE Independent tables were computed by the astronomer Biirg and published at Prague in 1620. Both Napier and Burgi, basinj their work on the relation which we should express by the equiva lent equations a- = a" and 1/ = log,, x, avoid fractional values of y bj taking values of a near 1, their actual values being a = .999999^ and a = 1.0001 respectively. In choosing a base less than 1, Napiei is also influenced by his desire that sines and cosines as prope: fractions shall have positive logarithms. If we intro- duce our modern graphical interpretation oi y = logoS;, Biirgi is concerned with the determination of abscissas of points where the exponen- tial curve is met by the horizontal straight lines % = c where c takes successive integral values. Choosing a base a near 1 naturally gives values of x near each other Napier's choice of a base less than 1 would correspond with the same curve inverted. In 1615 Henry Briggs, afterwards Savilian Professor of Geom- etry at Oxford, wrote of Napier " I hope to see him this summer if it please God, for I never saw book which pleased me better, o] made me more wonder." In connection with this and later visiti it was soon discovered that great simplification in the practica use of logarithms would result from taking log 1=0 and log 10 = 1 and giving up the restriction of logarithms to integral values, thui making the decimal parts of all logarithms depend wholly on th( sequence of digits. Napier had been so predominantly interested ii trigonometric applications that his table consisted not of logarithm; of abstract numbers, but of 7-place logarithms of the trigonometrii functions for each minute. In connection with his change of tb base, Briggs developed interesting methods of interpolating am testing the accuracy of logarithms. He gives the logarithms fron 1 to 20,000 and from 90,000 to 100,000 to 14 places, computing als^ 10-place trigonometric tables with an angular interval of 10 seconds PROGRESS OF MATHEMATICS AND MECHANICS 245 Kepler recognized immediately the enormous significance of the new logarithmic method and addressed an enthusiastic panegyric to Napier in 1620, not knowing that he had died in 1617. What if logarithms had been invented in time to save Kepler his vast com- putations ? A few years ago we have been shown in a rectorial address what the telescope has meant for observational astronomy. An equally great significance attached to logarithms for the computing astronomer. — Gutzmer. Vlacq of Leyden soon after filled the gap in Briggs' table, and this is the basis for the tables since published. The first tables to base e, commonly called Napierian, were published in 1619. In more recent times methods of interpolation have been employed which are more powerful and less laborious, while ordinary com- putation has been simplified by avoiding the use of too many deci- mal places, and by the mechanical device of the slide-rule. The modern computing machine naturally tends to supersede the logarithmic method. Among the remarkable computations characteristic of the sixteenth century may be mentioned Ludolph von Ceulen's achievement in computing tt to 35 decimal places, using regular polygons of 96 and 192 sides. German writers in consequence have sometimes attached his name to this important constant. In England Thomas Harriott (1560-1621) and William Oughtred (1575-1660) rendered important services in introducing the most recent advances in arithmetic, algebra and trigonometry. The former rejected negative and imaginary roots indeed, but used the signs > and <, denotes a^ by a a, etc. Oughtred uses the symbols X and : :, also the contractions for sine, cosine, etc. " Two New Branches of Science." — Even after Galileo's condemnation by the Inquisition, though old, infirm, and nearly blind, his scientific ardor was unquenched, and in 1638 he pub- lished (at Leyden) a work on mechanics under the title. Conver- sations and Mathematical Demonstrations on two New Branches of Science, which constituted the most notable progress in mechan- ics since Archimedes. He says : 246 A SHORT HISTORY OF SCIENCE My purpose is to set forth a very new science dealing with a ver ancient subject. There is, in nature, perhaps nothing older thai motion, concerning which the books written by philosophers are neithe few nor small; nevertheless I have discovered by experiment som properties of it which are worth knowing and which have not hitherti been either observed or demonstrated. Some superficial observation have been made, as, for instance, that the free motion (naiuraleri motum) of a heavy falling body is continuously accelerated ; but t( just what extent this acceleration occurs has not yet been announced for so far as I know, no one has yet pointed out that the distance traversed, during equal intervals of time, by a body falling from rest stand to one another in the same ratio as the odd numbers beginninj with unity. It has been observed that missiles and projectiles describe i curved path of some sort ; however no one has pointed out the fac that this path is a parabola. But this and other facts, not few ii number or less worth knowing, I have succeeded in proving ; and wha I consider more important, there have been opened up to this vast anc most excellent science, of which my work is merely the beginning, way and means by which other minds more acute than mine will explore it remote corners. This discussion is divided into three parts; the first part deal with motion which is steady or uniform ; the second treats of motion a we find it accelerated in nature; the third deals with the so-calle( violent motions and with projectiles. . . . Throughout this work Galileo depends on results of experimen rather than on mere speculation. He recognizes that air ha weight and that water can be raised but a certain height b; the ordinary pump,i but he still accepts the ancient notion tha 1 ' This pump worked perfectly so long aa the water in the cistern stood abov a certain level ; but below this level the pump failed to work. When I first noticei this phenomenon I thought the machine was out of order ; but the workman whor I called in to repair it told me the defect was not in the pump but in the wate which had fallen too low to be raised through such a height ; and he added that i was not possible, either by a pump or by any other machine working on the principl of attraction, to lift water a hair's breadth above eighteen cubits; whether th pump be large or small this is the extreme hmit of the lift. Up to this time I had bee: so thoughtless that, although I knew a rope, or rod of wood, or of iron, if sufE ciently long, would break by its own weight when held by the upper end, it neve occurred to me that the same thing would happen, only much more easily, to PROGRESS OF MATHEMATICS AND MECHANICS 247 "nature abhors a vacuum" as an explanation. He shows experi- mentally that a body descends an inclined plane with uniformly accelerated motion. In a board 12 ells in length a groove half an inch wide was made. It was drawn straight and lined with very smooth parchment. The board was then raised at one end, first one ell, then two. Then Galileo let a polished brass ball roll through the groove and determined the time of descent for the whole length of the groove. If on the other hand he let the ball roll through only one quarter of the length, this required just half the time. . . . The distances were to each other as the squares of the times, a law verified by hundredfold repetitions for all sorts of distances and slopes. The time was still determined by weighing water escaping through a small orifice. He shows by ingenious experi- ments the dependence of velocity on height alone, and that a freely falling body has the necessary energy to reach its original level. The whole theory of the falling body is now easily deduced. When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner. This we readily understand when we consider the intimate relation- ship between time and motion ; for just as uniformity of motion is de- fined by and conceived through equal times and equal spaces (thus we call a motion uniform when equal distances are traversed during equal time-intervals), so also we may, in a similar manner, through equal time-intervals, conceive additions of speed as taking place without complication ; thus we may picture to our mind a motion as uniformly and continuously accelerated when, during any equal intervals of time whatever, equal increments of speed are given to it. . Hence the definition of motion which we are about to discuss may column of water. And really is not that thing which is attracted in the pump a column of water attached at the upper end and stretched more and more until finally a point is reached where it breaks, like a rope, on account of its excessive weight ? . ' 248 A SHORT HISTORY OF SCIENCE be stated as follows : A motion is said to be uniformly accelerated, whei starting from rest, it acquires, during equal time-intervals, equal in crements of speed. . . . The time in which any space is traversed by a body starting fron rest and uniformly accelerated is equal to the time in which that sam( space would be traversed by the same body moving at a uniform speec whose value is the mean of the highest speed and the speed jusi before acceleration began. . . The spaces described by a body falling from rest with a uniformlj accelerated motion are to each other as the squares of the time- intervals employed in traversing these distances. . . . Galileo passes from falling bodies to pendulums, in which the fric- tion of the inclined plane is absent and air resistance negligible. He appreciates the possibility of utilizing the pendulum for time measurement, and devises a simple apparatus for the purpose, foreshadowing the invention of the clock. He discovers that the time of vibration of the pendulum varies as the square root oi the length. He analyzes correctly the component motions of a projectile, recognizing the law of the parallelogram of motion, as distinguished from the parallelogram of forces discovered by Newton. He shows that whether the initial direction of aim is horizontal or not, the path described is a parabola with axis vertical, explicitly neglecting air resistance and change of direction of vertical force. I now propose to set forth those properties which belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated ; these properties, well worth knowing, I propose to demonstrate in a rigid manner. This is the kind of motion seen in a moving projectile ; its origin I conceive to be as follows : . . . A projectile which is carried by a uniform horizontal motion com- pounded with a naturally accelerated vertical motion describes e path which is a semi-parabola. — Galileo, Two New Sciences. All this Dynamics was practically pioneer work of enormous im- portance for the future of mechanics. PROGRESS OF MATHEMATICS AND MECHANICS 249 In Statics Galileo had somewhat more from the ancients to build upon. To him we owe the formulation of the law of virtual velocities — applying dynamical ideas to problems of statics. If two forces are in equilibrium they are proportional to the cor- responding paths, or : What one by any machine gains in power is lost in distance. The parallelogram or triangle of forces in equilibrium however escapes him, and his ideas about impulse though remarkably in advance of his time were not fully worked out. He investigates strength of materials under tension and fracture, with reference to practical applications in construction. He draws just inferences in regard to the relation between strength and size of plants and animals as well as machines, comparing for example hollow bones and straws with solid bodies of similar mass. He derives an important formula for the stiffness of a horizontal beam supported at one end and regarded as a lever. He discusses the curve formed by a cord suspended between two points, recog- nizing that it is not a parabola. In Hydrostatics he reviews the known work of Archimedes and corrects the error of the Aristotelians in regard to the dependence of floating on specific gravity. He develops the modern theory that the fundamental factor in the mechanics of fluids is that they consist of freely moving particles yielding to the slightest force. He makes effective application of the principle of virtual velocities to fluids. At the close of his third conversation he expresses his modest confidence in the great future of his new ideas. The theorems set forth in this brief discussion will, when they come into the hands of other investigators, continually lead to wonder- ful new knowledge. It is conceivable that in such a manner a worthy treatment may be gradually extended to all the realms of nature — a prediction magnificently fulfilled in succeeding genera- tions. Among other branches of physics in which Galileo accomplished work of value may be mentioned the expansion by heat — the beginnings of thermometry, experiments on the acoustics of 250 A SHORT fflSTORY OF SCIENCE vibrating cords and plates, discovering the dependence of harmon; on the ratio of the rates of vibration, and the relations of length thickness, and tension of cords. He explains resonance and dis sonance. He assumes light to have a finite velocity, but doe; not succeed in measuring it. Let each of two persons take a light contained in a lantern, o: other receptacle, such that by the interposition of the hand, the on( can shut ofF or admit the light to the vision of the other. Next le them stand opposite each other at a distance of a few cubits and prac tice until they acquire such skill in uncovering and occulting theii lights that the instant one sees the light of his companion he will un cover his own. After a few trials the response will be so prompt thai without sensible error the uncovering of one light is immediately followed by the uncovering of the other, so that as soon as one exposes his light he will instantly see that of the other. Having acquired skill at this short distance, let the two experimenters, equipped as before take up positions separated by a distance of two or three miles anc let them perform the same experiment at night, noting carefuUj whether the exposures and occultations occur in the same manner as at short distances ; if they do, we may safely conclude that the prop' agation of light is instantaneous ; but if time is required at a distance of three miles which, considering the going of one light and the coming of the other, really amounts to six, then the delay ought to be easily observable. If the experiment is to be made at still greater distances say eight or ten miles, telescopes may be employed, each observe! adjusting one for himself at the place where he is to make the experi ment at night ; then although the lights are not large and art therefore invisible to the naked eye at so great a distance, they car readily be covered and uncovered since by aid of the telescopes, once adjusted and fixed, they will become easily visible. . . . He seeks to apply to astronomical phenomena the new discoveriei in magnetism. Everywhere the mathematical and inductive method became manifest in this man. Almost all domains of science received there from the most powerful impulse. And above all the whole field o science was freed from the outgrowths of metaphysical modes o thought with which it had been previously so overrun. Galileo'; PROGRESS OF MATHEMATICS AND MECHANICS 251 individual method consisted namely in always conforming to the limits of scientific investigation, and confining his attention to seizing the phenomena sharply in their progress and in their relation with allied processes, without wandering into a fruitless search after the ultimate bases of the phenomena. — Dannemann. Such a limitation has been of the highest value for the renewal of natural science as it followed at the beginning of the seventeenth century. Galileo was not chiefly interested in mathematics, but he em- phasizes the dependence of other sciences upon it. True philosophy expounds nature to us ; but she can be under- stood only by him who has learned the speech and symbols in which she speaks to us. This speech is mathematics, and its sym- bols are mathematical figures. Philosophy is written in this greatest book, which continually stands open here to the eyes of all, but can- not be understood unless one first learns the language and characters in which it is written. This language is mathematics and the characters are triangles, circles and other mathematical figures. He gives an acute discussion of infinite, infinitesimal and con- tinuous quantities leading up to the conclusion "that the attri- butes 'larger,' 'smaller,' and 'equal' have no place either in comparing infinite quantities with each other or In comparing in- finite with finite quantities." Again "the finite parts of a con- tinuum are neither finite nor infinite but correspond to every as- signed number." In commenting on Galileo's achievements, Lagrange the great mathematician of the eighteenth century says : — These discoveries did not bring to him while living as much celebrity as those which he had made in the heavens ; but to-day his work in mechanics forms the most solid and the most real part of the glory of this great man. The discovery of Jupiter's satellites, of the phases of Venus, and the Sun-spots, etc., required only a telescope and assiduity ; but it required an extraordinary genius to unravel the laws of nature in phenomena which one has always under the eye, but the explanation of which, nevertheless, had always baffled the researches of philosophers. 252 A SHORT fflSTORY OF SCIENCE Leonardo da Vinci likens a scientific conquest to a military victory in which theory is the field marshal, experimental facts the soldiers The philosophers who preceded Galileo had, in the main, been tryin; to fight battles without soldiers. — Crew. A Pioneer in Mechanics. Stevinus. — Even before Galileo Stevinus of Bruges (1548-1620), a man who thought independentlj on mechanical problems, made the first really important advance: since Archimedes, eighteen centuries earlier. Besides engaging in mercantile pursuits he was quartermaster-general of th( Dutch army, and an authority on military engineering. He was influential in improving methods of public statistics and account- ing, and advocated decimal weights and measures. Appreciating the possibilities of the decimal fraction he asserted (1585) thai fractions are quite superfluous, and every computation can be made with whole numbers, but he did not realize the simplesi notation. The honor of this great invention he shares with Biirg: of Cassel. Another of his inventions was a sailing carriage carry- ing 28 people and outstripping horses. In a treatise on Statics and Hydrostatics (1586) he introduced comparatively new and powerful geometrical methods for dealing with mechanical problems, Among the most interesting is his discussion of the inclined plane by means of an endless chain hanging freely over a tri- angle with unequal sides. Ex- cluding the inadmissible hy- pothesis of perpetual motion the uniform chain must be ir equilibrium in any position The hanging portion is by itsell in equilibrium, therefore the two inclined sections must bal- ance each other, and eithei would be balanced by a verti- cal force corresponding to th< Stevinua' Triangle. PROGRESS OF MATHEMATICS AND MECHANICS 253 altitude of the triangle. Arriving thus at the parallelogram of forces in equilibrium, he expresses his astonishment by exclaim- ing "Here is a wonder and yet no wonder." In studying pulleys and their combinations he arrives at the far-reaching result that in a system of pulleys in equilibrium "the products of the weights into the displacements they sustain are respectively equal" — a remark containing the principle of virtual displacement. He reaches correct results in regard to basal and lateral pressure by reasoning analogous to that about the chain, and by assuming on occasion that a definite portion of the liquid is temporarily solidified. By ingenious experiments he proves the dependence of fluid pressure on area and depth, and takes proper account of upward and lateral pressure. He studies the conditions of equilibrium for floating bodies, show- ing that the centre of gravity of the body in question must lie in a perpendicular with that of the water displaced by it, and that the deeper the centre of gravity of the floating body the more stable is the equilibrium. In analyzing the lateral pressure of a fluid Stevinus anticipates the calculus point of view by dividing the surface into elements on each of which the pressure lies between ascertainable values. In- creasing the number of divisions, he says it is manifest that one could carry this process so far that the difference between the con- taining values should be made less than any given quantity how- ever small — all quite in harmony with our present definitions of a limit. Stevinus' work and that of Galileo seem to have been quite independent of each other, the former confining his theory to statics, the latter laying a solid foundation for the new science of dynamics. Torricelli, a disciple of Galileo best known for his invention of the mercurial barometer, extended dynamics to liquids, studying the character of a jet issuing from the side of a vessel. Throughout this period the universities lagged. In Italy Galileo lectured to medical students who were supposed to need astronomy for medical purposes — i.e. astrology. At Wittenberg 254 A SHORT HISTORY OF SCIENCE there was a professor for arithmetic and the sphere, and one fo EucUd, Peurbach's planetary theory and the Almagest, but thei students were few. ... So, says a German writer, we face thi extraordinary fact that the most educated of the nation were a helpless in the problems of daily life as a marketwoman of to-day The university lectures in mathematics were mainly confinec to the most elementary computation, — matters taught mon thoroughly in the commercial schools, particularly after the invention of printing. Giordano Bruno (1548-1600). — In the Appendix will b( found the judgment and sentence of the Inquisition upor Galileo, together with his recantation, — one of the darkest page; in the history of Science. Another victim of the Inquisition wa; Bruno, an Italian philosopher, who, having joined the Dominicar order at the age of fifteen, was later accused of impiety and sub- jected to persecution. Bruno fled from Rome to France, anc later to England, where at Oxford he disputed on the rival merit! of the Copernican and the so-called Aristotelian systems of the universe. In 1584 he published an exposition of the Copernicar theory. Bruno, moreover, attacked the established religion, jeerec at the monks, scoffed at the Jewish records, miracles, etc., anc after revisiting Paris, and residing for a time in Wittenberg, rashlj returned to Italy, where he was apprehended by the Inquisitioi and thrown into prison. After seven years of confinement h( was excommunicated and, on Feb. 17, 1600, burnt at the stake In 1889 a statue in his honor was unveiled in Rome at the plac( of his execution, the Square of the Flower Market. Thus wai the end of the sixteenth century illuminated by the flames o martyrdom. Refekences for Reading Ball. Short History of Mathematics, Chapters XII, XIII. Fahie. Life of Galileo. Galileo Galilei. Two New Sciences. (Translated by Crew and De Salvio. HoBSON. John Napier and the Invention of Logarithms. Lodge. Pioneers of Science (Galileo) . Mach. Science of Mechanics (for Galileo and Stevinus). Morley. Life of Cardan. CHAPTER XII NATURAL AND PHYSICAL SCIENCE IN THE SEVEN- TEENTH CENTURY The Circulation of the Blood : Haevey (1578-1657).— The blood has always been regarded as one of the principal parts of the body. Hippocrates considered it one of his four great "humors," and in the Hebrew Scriptures it is stated that "the blood ... is the life." Yet up to the seventeenth century nothing definite was known of its movements throughout the body. That it was under pressure must have been known, for it flowed or "escaped" freely from wounds, and flow results only from pressure of some sort, while escape is relief from detention. The arteries had been misinterpreted for centuries and were early considered to be air tubes, because they were studied only after death when as we now know they are empty. Even the dissections of the anato- mists of the sixteenth century had failed to reveal the complete and true office of the arteries, and it remained for Harvey, an English pupil of the Italian anatomist Fabricius, to make — largely through the vivisection of animals and observation of the heart and arteries in actual operation — discoveries of basic importance in anatomy, physiology, embryology, and medicine (see Appendix). While working in Italy, Harvey learned of and doubtless saw the valves in the veins which were discovered by Fabricius. These valves are thin flaps of tissue so placed as to check the flow of blood in one direction while offering no resistance to that flowing the other way. On his return to England, Harvey apparently pondered on the function of these valves and saw that they could be of use only by permitting the flow of the blood in one direction while prevent- ing its movement in the opposite direction. At this time it was supposed that the blood simply oscillated, or moved back and forth 255 256 A SHORT fflSTORY OF SCIENCE like a pendulum, a view which, if the valves had any meaning was now plainly untenable. Harvey therefore set to work t study the beating of the heart and the flowing of the blood, am soon came to the conclusion that there must be a steady flow o streaming in one direction, and not an oscillation back and fort) as was generally supposed. But to prove was here, as always harder than to believe, and much time and labor were required t( settle the question. At length, however, by dissections and vivi sections of the lower animals, and after publishing (in 1628) a bro chure presenting his facts and meeting objections, Harvey sue ceeded, with the result that his name justly stands to-day besid( those of the Greek and Alexandrian Fathers of Medicine, Hippoo rates and Galen. It is one of the ironies of fate that while Harve] rightly reasoned from circumstantial evidence that the blood mus' steadily flow from the arteries to the veins, he himself never actually saw that flowing, — a sight which any schoolboy may now see but impossible before the introduction of the microscope, and firsi enjoyed by Malpighi in 1661, only four years after Harvey's death In embryology, also, Harvey proved himself an original anc penetrating observer. In his day and earlier it was supposed thai the embryo, in the hen's egg, for example, exists even at the verj outset as a perfect though extremely minute chick, with all its parts complete. This "preformation" theory was opposed by Harvey whose doctrine of " epigenesis" was substantially that of modern embryology : viz. that the embryo chick is gradually formed bj processes of growth and differentiation from comparatively simple and undifferentiated matter, somehow set apart and prepared in the body of the parents. Atmospheric Pressure : Torricelli's Barometer. — The problem of the existence and nature of voids and vacua had always been an interesting puzzle for philosophers. The Greeks assumed the existence of empty spaces or "voids," and as late as the age oi Elizabeth it was the orthodox belief that "nature abhors a vacuum." Galileo, even, held to it in 16-38. (Cf. p. 246.) Evangelista Torricelli (1608-1647), inspired by the Dialogues oJ Galileo (1638), published on Motion and other subjects in 1644 NATURAL SCIENCE IN SEVENTEENTH CENTURY 257 He resided with Galileo and acted as his amanuensis from 1641 until Galileo's death. In experimenting with mercury he found that this did not rise to 33 feet, but instead to hardly as many inches. He next proved, by comparing the specific gravity of water and mercury, that the same "pressure" was at work in both cases, and boldly affirmed that this pressure was that of the atmosphere. The tube of mercury used in his experiments was what we now call a barometer (baros, weight), but it was for a long time called "the Torricellian Tube," as the empty space above the mercury is still called the "Torricellian vacuum." This invention or discovery of Torricelli's was one of the most fertile ever made, for at one blow it demolished the ancient super- stition that "nature abhors a vacuum," explained very simply two ancient puzzles (why water rises in a pump, and why it rises only 33 feet), determined accurately the weight of the atmosphere, proved it possible to make a vacuum, and gave to mankind an entirely new and invaluable instrument, the barometer. Torri- celli's results and explanations were received at first with incredu- lity, but were soon confirmed, notably by Pascal (1623-1662) in a treatise. New Experiments on the Vacuum. In one of these Pascal used wine instead of water or mercury in the Torricellian tube, with satisfactory results, and in another, reasoning that if Torricelli were right, liquids in the tube should stand lower on a mountain than in a valley, persuaded his brother-in-law, Perier, to ascend the Puy de Dome (near Clermont, France) in September, 1648, on which mountain the column was found to be much shorter. This and other brilliant work by Pascal have given him a high rank among natural philosophers. Since it was now easy to obtain a vacuum by the Torricellian experiment, fresh attempts were made to produce vacua otherwise. Von Guericke, burgomaster of Magdeburg in Hannover, after many failures, finally succeeded in pumping the air out of a hollow metallic globe. It was in this experiment that the air-pump was introduced. Guericke found that his globe had to be very strong to resist crushing by the atmospheric pressure, and in the popular demonstration now known as that of the Magdeburg hemi- 258 A SHORT HISTORY OF SCIENCE spheres he showed that eight horses on either side were unable t overcome this pressure on a particular globe which he had con structed and exhausted of air. These various experiments and dis coveries relating to atmospheric pressure led to the investigation and laws of Boyle, Mariotte, and others and, less than a centur; later, to the steam-engine of Watt, in which steam was at firs used only to produce a vacuum, — atmospheric pressure beinj employed as the moving force. Further Studies of the Atmosphere : Gases. — Meantim( the chemical composition of the atmosphere was being no les! eagerly studied. Robert Boyle (1627-1691) published at Oxforc in 1660, New Experiments Physico-Mechanical touching th( Spring of the Air and its Effects, and in his Sceptical Chymisi gives an interesting and instructive picture of the chemical idea; of his time. He was the first to insist on the difference betweer compounds and mixtures, and probably the first to use the pneu- matic trough for the collection and study of gases. The word "gas" was introduced by Van Helmont (1577-1644) who by virtue of the following remarkable statement deserves tc be remembered as the principal chemist of the earlier half of the seventeenth century : — Charcoal and in general those bodies which are not immediately re- solved into water, disengage by combustion spiritum sylvestrum. From 62 lbs. of oak charcoal 1 lb. of ash is obtained, therefore the remaining 61 lbs. are this spiritum syhestre. This spirit, hitherto unknown, I call by the new name of gas. It cannot be enclosed in vessels or re- duced to a visible condition. There are bodies which contain this spirit and resolve themselves entirely into it : in these it exists in a fixed or solidified form, from which it is expelled by fermentation, as we observe in wine, bread, etc. It has been well said that this passage is remarkable not only for the explicit mention of car- bonic acid gas (as we now call it) as a product of fermentation, and foi the introduction of the word gas for the first time, but also for its ap- peal to the balance, NATURAL SCIENCE IN SEVENTEENTH CENTURY 259 the formal introduction of which into chemistry was only made a century later by Lavoisier. Van Helmont also points out that his gas syhestre is produced by the action of acids on shells, is en- gendered in putrefaction and combustion, and is present in caves, mines, and mineral waters. In these ideas and passages we find an agreeable departure from the mysticism of the alchemists and the wild surmises of Paracelsus. At the same time Van Helmont's ideas in other directions were crude enough, since he is credited with a recipe for the artificial production of mice from " corn and sweet basil." From Philosophy to Experimentation. — The seventeenth century differs from all before it in the increasing attention paid to experimental science. From the philosophizing of Paracelsus and Gilbert it is agreeable to pass to the experimental work of Harvey, Torricelli and in chemical inquiries to Van Helmont, whose logical successor is Robert Boyle (1627-1691), already mentioned for his work on the resistance, or "spring," of the atmosphere, etc. Among many other ingenious experiments Boyle worked on evapo- ration, in air and in vacuo; on boiling and on freezing; and on the effects of exposing animals to the diminished atmospheric pres- sure produced by the air-pump. In this direction he was the first to prove that fishes require air dissolved in the water in which they live. He also studied the rusting of metals — a problem then widely discussed — and from all his studies con- cludes that there is in the atmosphere some vital substance which plays a principal part in such phenomena as combustion, respira- tion, and fermentation. When this substance has once been con- sumed, flame is instantly extinguished, and yet the air from which it has gone seems nearly intact. He wrote a treatise entitled, Fu-e and Flame weighed in a Balance, in which he described the increase of weight of metals on calcination. But as he got about the same results whether the crucible was open or shut, he was misled into the belief that the air had little to do with his results, which he attributed rather to the fixation of the "fire" by the porous crucibles. In these and Boyle's other experiments it is plain that we are rapidly moving from alchemical and iatro-chemical stages 260 A SHORT fflSTORY OF SCIENCE toward the modern experimental period of chemistry, of which he and Van Helmont are the pioneers. Neither, however, while work- ing on ak greatly advanced our ideas of atmospheric chemistry. The atmosphere in its relation to combustion and respiration was further studied by an English physician, Dr. John Mayow (1645- 1679), who made many experiments upon the shrinkage of air- volume during the burning of camphor and other substances anc during the confinement of mice under a bell-glass. The dying oi the mice and the cessation of the combustion, which after a time ensued, he attributed to the exhaustion of some ingredient in the air indispensable to life and combustion. This ingredient, whict we now call oxygen, Mayow named "fire-air." Very soon, however, a new theory of combustion (and as il turned out a false theory) began to absorb the attention oJ natural philosophers. From Alchemy to Chemistry. — The saying is attributed te Liebig that "Alchemy was never at any time different fron chemistry." In one sense this is undoubtedly true. The searcl for "the philosopher's stone," "the elixir of life," "potable gold,' and the "transmutation of metals," consisted of necessity in the use of processes such as boiling, baking, wetting, drying, evaporat ing, condensing, burning, calcifying, decalcifying, acidifying freezing, melting, and the like, mostly tending towards chemica changes and the formation of new mixtures and compounds. Bu even if Liebig's saying were true, chemistry has passed througl three principal stages ; viz. first, purely empirical experimenting mostly for practical purposes, whether metallurgical or other second, an iatro-chemical or medico-chemical phase ; and finalb the really scientific period of to-day, the way for which may bi said to have been cleared by the Sceptical Chymist of Rober Boyle,^ first published in English in Oxford in 1661. In this re 1 The Hon. Robert Boyle was one of the most active, perhaps the most so, c that remarkable group of scientifio investigators who, in the reign of Charles II raised England to the foremost place among European nations in the pursuit c science, and gave their period a renown which has caused it to be often spoken o; and very justly, as the classical age of English science. . . . Boyle had been sine 1646 engaged in chemical researches in London, being then connected with the earlie NATURAL SCIENCE IN SEVENTEENTH CENTURY 261 markable little book Boyle by means of a dialogue discusses and sharply criticises the chemistry of the "hermetick" {i.e. Aris- totelian) natural philosophers, and also "the vulgar Spagyrists" {i.e. the medico-chemists of the Paracelsus type) and questions the value of terms then hazy in their meaning, such as "element" and "principle," as used in alchemy. He does not himself pro- pound any new theories of consequence, but he does insist on more knowledge, more experimentation, and less groundless specu- lation. We quote from Professor Pattison Muir's valuable intro- ductory essay to the "Everyman" edition : — The Sceptical Chyrwist embodies the reasoned conceptions which Boyle had gained from the experimental investigations of many physi- cal phenomena. . . . The book is more than an elegant and suggestive discourse on chemico-physical matters ; it is an elucidation of the true method of scientific inquiry. ... At that time the alchemical scheme of things dominated most of those who were inquiring into the trans- mutations of material substances. That scheme was based on a magi- cal conception of the world. . . . When a magical theory of nature prevails, the impressions which external events produce on the senses of observers are corrected, not by careful reasoning and accurate experi- mentation, but by inquiring whether they fit into the scheme of things which has already been elaborated and accepted as the truth. Natural events become as clay in the hands of the intellectual potter for whom 'there is nothing good or bad but thinking makes it so.' . . . An al- chemical writer of the seventh century said : ' Copper is like a man ; it has a soul and a body.' . . It is not possible to attach any definite, clear, meanings to alchemical writings about the four elements. Their indefiniteness was their strength. ... As the plain man to-day is soothed and made comfortable by the assurance that certain phrases to which he attaches no definite meanings are really scientific, so, when Boyle lived, the plain man rested happily in the belief that the four elements were the last word of science regarding the structure of the materials of the world. . . . group of scientific inquirers in London known as the ' Invisible College "... Boyle, too, we must observe, was above aU things unprejudiced. He had leanings towards alchemy and never quite repudiated a belief in the possibility of transmuting metals. In medical matters, which greatly interested him, he showed perfect tolerance towards those whom the profession called quacks. — /. F. Payne. 262 A SHORT HISTORY OF SCIENCE Boyle found the same fault with the 'Principles' of the 'Vulga Spagyrists ' as he found with the ' Elements ' of the ' hermetick philos ophers.' 'Tell me what you mean by your Principles and your Ele ments,' he cried ; ' then I can discuss them with you as working in struments for advancing knowledge.' 'Methinks the Chymists in their search after truth are not unlike the navigators of Solomon's Tarshish Fleet, who brought home fron their long and perilous voyages not only gold and silver and ivori but apes and peacocks too : for so the writings of several (I say not all of your hermetick philosophers present us, together with diverse sub stantial and noble experiments, theories which, either like peacocks feathers, make a great show, but are neither solid nor useful, or elsi like apes, if they have some appearance of being rational, are blemishec with some absurdity or other that, when they are attentively con sidered, makes them appear ridiculous.' The fact that at the middle of the seventeenth century criticisn of this sort seemed to Boyle to be needed shows how little rea progress toward modern scientific chemistry had even then beei made; and, as often happens, truth had to be reached througl further error. A False Theory op Combustion : Phlogiston. — Tw< German contemporaries of Boyle, Becher (1625-1682), and Stah (1660-1734), as a result of studies on combustion and the calcininj of metals, departed from the four elements of antiquity anc assumed the participation in these processes of a something dis pelled by heating. To this something Stahl gave the nami ■phlogiston, " the combustible substance, a principle of fire, but no fire itself." And because from a metallic calx (oxide) the meta could be recovered by burning with charcoal, the metal was hek to have absorbed " phlogiston " in the process from the charcoal which, having mostly disappeared, was regarded as almost pur( phlogiston. Conversely, when the metal was calcined (or oxi dized) by burning without charcoal, it was held to have lost it; phlogiston. This theory, which to-day seems bizarre, satisfiec the chief requirement imposed on any new theory : viz. that of ac counting for the facts (as then known), and was therefore naturall;; NATURAL SCIENCE IN SEVENTEENTH CENTURY 263 accepted and advocated by natural philosophers for the next hundred years. It was not until new facts had been accumulated which were not explained by the theory of Becher and Stahl, and especially the fact revealed by the use of the balance, that sub- stances calcined often gained weight (making it necessary to assume that phlogiston possessed negative gravity, or " levity " since its loss increased weight), that the theory became plainly untenable and was abandoned. This, however, only happened late in the eighteenth century, and before this time much progress had been made in chemistry in other du-ections. Meanwhile, in spite of its falsity, the theory of phlogiston had done good service. It had, for example, effectually turned the attention of chemists away from magic, from potable gold, and from the making of medicines, to speculations on composition, decomposition, and chemical change, — topics not only more worthy but more fruitful. Beginnings of Organic Chemistry. — Meantime, a kind of organic chemistry was initiated by Hermann Boerhaave (1668- 1738), a physician of Leyden. In the seventeenth and eighteenth centuries the term "organic" stood more than it does to-day for the living world and its products which were then regarded as things altogether apart from the lifeless or inorganic world. To- day organic chemistry hardly means more than the chemistry of the carbon compounds, but at that time it meant the chemistry of bodies found in or produced by living things. Medical men had long been interested in alchemy, and in more modern times in iatro- chemistry, so that it was natural enough that Boerhaave, a physi- cian, should undertake to subject organic substances to chemical processes. And this he did, though more in the fashion of the pharmaceutical, than the analytical, chemist of to-day. Boer- haave was a famous teacher of medicine and of botany, and crowds of students attended his lectures, thereby testifying to the now rapidly growing popularity of scientific learning. His Elements of Chemistry, published in 1732, was widely used and marks an epoch in the history of chemistry. At about the same time. Dr. Stephen Hales (1677-1761), an English clergyman of a strongly scientific bent, did similar work 264 A SHORT HISTORY OF SCIENCE in England. In addition, Hales made important studies on th( atmosphere, and invented the manometer, which he appHed to th( measurement of the arterial blood pressure in the horse, and th( upward root pressure in plants, besides accomplishing much othei good work. Hales will perhaps be longest remembered in chem istry for his skilful use of the pneumatic trough, a simple but in dispensable laboratory appliance for the easy collection of gase; in a closed vessel over water, and especially for his studies on air Medical Science and Medical Theory in the Seventeente Century. Thomas Sydenham. — In the middle of the seven- teenth century medical theory took a long step forward unde: the influence of Thomas Sydenham (1624-1689), often called "th( English Hippocrates" because of the naturalism and rationalisn which he urged in medicine and because of the sanity of his opinion: and theories. Setting aside magic, mysticism, and the medica' chemistry of Paracelsus, and insisting on a material basis {materiet morhi) for the causes of disease, Sydenham laid the foundations of modern scientific medical philosophy and practice. He was £ close friend of Locke, the philosopher, — by whose materialistic and rationalistic ideas he was doubtless influenced, — and was also a correspondent of Boyle. His famous definition of disease as, " An effort of nature, striving with all her might to restore tht patient by the elimination of morbific matter," is still interesting for its implication of the modern idea of disease as a struggle foi existence between pathogenic matters (such as microbes) and thf inner forces of the body. It is, however, to Vesalius and Harvey, to Leeuwenhoek and Kircher and Malpighi and the other microscopists of the seven- teenth century, and their successors, i.e. to the experimenters and laboratory workers, rather than to Sydenham or his successors, that medical science is chiefly indebted, since no great progress could be made in sound medical theory or rational medical prac- tice until anatomy, physiology and microscopy had paved the way for a more scientific pathology. The Beginning op Modern Ideas op Light and Optics. — The nature of light, darkness and vision are very old problems NATURAL SCIENCE IN SEVENTEENTH CENTURY 265 It is easy, even for savages, to account for daylight as sunlight, and the corresponding nightlight as moonlight and starlight, but for more higlily developed man to explain just what the light is which comes from sun, moon and stars, is not so easy. Obviously, since " luminaries " — sun, moon, stars, firebrands and torches — produce light which is weaker as distance from the source in- creases, a kind of "emission" theory of light is natural and reason- able. It was even held by the ancients that we see by means of light emitted from our own eyes, and that light is a more or less palpable substance. A similar error was held concerning heat, which until the end of the eighteenth century was generally re- garded as a peculiar material body or substance, "caloric," which when absorbed from other bodies produced a state of heat, and when emitted caused, by its absence, cold. How men could have believed for ages that objects are rendered visible by something projected from the eye itself — so that the organ of sight was supposed to be analogous to the tentacula of insects, and sight itself a mere species of touch — is most puzzling. They seem not till about 350 B.C. to have even raised the question : If this is how we see. Why cannot we see in the dark ? or, more simply ; What is darkness? The former of these questions seems to have been first put by Aristotle. The ancients probably understood that light travels in straight lines, and they must have known something about reflection and refraction of light, for they knew about images in still water, and had mirrors of polished metal, and burning glasses of spherical glass shells, or balls of rock crystal. To Hero of Alexandria we owe the important deduction from the Greek geometers that the course of a reflected ray is the shortest possible (p. 123). The perfection of gem cuttmg among the ancients has also been held to prove their acquaintance with lenses. But it was not until the seventeenth century that modern ideas of light and optics began to be formulated, with the work of Snellius, Descartes and Newton on reflection and refraction, and of Romer on the velocity, of light. 266 A SHORT HISTORY OF SCIENCE Every student should read the earher parts of Newton's Optics in which are described the fundamental experiments on the decomposi- tion of white light. — Lobd Rayleigh. The work of Christian Huygens, towards the end of the seven- teenth century, second only to that of Newton, both in extent and importance, touched upon a great variety of subjects, including some in the natural sciences. As a young man he wrote upon geometry ; in early middle life he invented the cycloidal pendu- lum. He was the first to apply pendulums to clocks and spiral springs to watches, and to devise the achromatic eye-piece which still bears his name. He also made a telescope and, finally, at the age of fifty, observed the phenomena of polarization and, most important of all, proposed the modern wave theory of light. The First Scientific Instruments: Telescope, Barom- eter, Thermometer, Air-Pump, Microscope, Manometer. — The complete history of the origin of the telescope, the ther- mometer and the microscope is not known. The account usually given of the invention of the telescope makes it accidental and due to the children of a Dutch spectacle maker, named Jansen, who while at play happened to bring together two lenses in such a way that a distant church spire seen through them looked mag- nified and near. The father, whose attention was drawn to the phenomenon, seeing in the arrangement a source of profit, there- upon made and sold the combination as a toy or "wonder," under which form it was on sale in 1609, becoming known to Galileo, who instantly realized its importance and made improvements in it. It appears that soon after 1609 Galileo had a fairly good instrument, magnifying 8 diameters, with which he was quickly and easily able to make some of his most splendid astronomical discoveries. The early history of the telescope shows that the effect of com- bining two lenses was understood by scientists long before any partic- ular use was made of this knowledge; and that those who are accredited with introducing perspective glasses to the public hit by accident upon the invention. Priority was claimed by two firms of spectacle-makers in Middelburg, Holland, namely Zacharias, miscalled PHYSICAL SCIENCE IN SEVENTEENTH CENTURY 267 Jansen, and Lippershey. Galileo heard of the contrivance in July 1609 and soon furnished so powerful an instrument of discovery that . he was able to make out the mountains in the moon, the satel- lites of Jupiter in rotation, the spots on the revolving sun . . . About 1639, Gascoigne, a young Englishman, invented the mi- crometer which enables an observer to adjust a telescope with very great precision. The history of the microscope is closely connected with that of the telescope. In the first half of the seventeenth century the simple microscope came into use. It was developed from the convex lens . Leeuwenhoek before 1673 had studied the structure of minute animal organisms and ten years later had even obtained sight of bac- teria. Very early in the same century Zacharias had presented Prince Maurice, the commander of the Dutch forces, and the Arch- duke Albert, Governor of Holland, with compound microscopes. Kircher (1601-1680) made use of an instrument that represented microscopic forms at one thousand times larger than their actual size. — LiBBT. Introduction to the History of Science. The name of Galileo goes also with the invention of the ther- mometer, an air, or more strictly a water, thermometer having been introduced by him about 1597. Mercury was not substituted for water until 1670, but alcohol thermometers, also introduced by Galileo, were used much earlier. The freezing and boiling of water were supposed to take place at variable temperatures and it was not until the end of the seventeenth century that it was realized that the freezing and the boiling points are invariable. (For Galileo's other work in physics, see pp. 246-250.) Pendulum clocks, " aerial " telescopes and the achromatic eye-pieces which bear his name were introduced by Huygens, the first in 1657 and the others about 1680. The invention of the barometer by Torricelli has already been described above (p. 257) . The air-pump, though merely the appli- cation of an ordmary pump to air uistead of water, was so rich in its results that it deserves a high place among the other and more important inventions of this remarkable scientific era. About the origin of the (compoimd) microscope there is much 268 A SHORT HISTORY OF SCIENCE the same obscurity as about that of the telescope. Simple micro- scopes such as "magnifiers," burning glasses, spectacles, and other lenses, had long been known, — some of them from antiquity, — but the compound microscope, which consists of two lenses or combinations of lenses so placed as to cooperate in the produc- tion of one highly magnified image of a near and minute object (the telescope doing the same for large and distant objects), first ap- pears about 1650. Some of the earliest microscopists are Kircher, Leeuwenhoek, Malpighi, and Grew. The two former apparently saw with the microscope and made drawings of bacteria, besides many other micro-organisms and cellular structures. The two latter are the founders of microscopic anatomy, INIalpighi of that of animals. Grew of that of plants. Malpighi's work is especially notable, since he for the first time actually observed the passage of blood cells from arteries to veins, and that in 1661 only four years after Harvey's death. Malpighi's name is also familiar to students of human anatomy and physiology in connection with those parts of the kidneys and the spleen which bear his name. The versatile and accomplished Englishman Dr. Robert Hooke (1635-1703), who flourished in this century and did ingenious, extensive, and often remarkable work at the basis of almost every branch of modern science, was the first to discover by the microscope the cellular structure of living things. Hooke was one of the original members of the Royal Society, with which Leeuwenhoek also corresponded. The most remarkable fact connected with the invention of the compound microscope is that, because of its physical imperfec- tions, and in spite of some use as just described, it was virtually abandoned for almost a century and a half, and only re-introduced after the invention and perfection of the achromatic objective in the first quarter of the nineteenth century. The truth appears to be that owing to excessive spherical and chromatic aberration the compound microscope of the seventeenth and eighteenth centuries was of limited value, and that microscopists often preferred the less powerful, but more perfect, simple microscope. The manometer was apparently first used by Stephen Hales, who measured with it the blood pressure of a horse, the root pres- PHYSICAL SCIENCE IN SEVENTEENTH CENTURY 269 sure of plants, etc. It is described in his Statical Essays (1727) and Haemostaticks (1733). Organization of the First Scientific Academies and So- cieties. — The Academy of Plato (fifth century B.C.), and the Lyceum of Aristotle, the Museum at Alexandria (third century B.C.), and the so-called Academy of Alcuin (in the eighth century a.d.) may be regarded as precursors of the academies and societies of the Renaissance, but — with the possible exception of an academy formed by Leonardo da Vinci in the fifteenth century — the first devoted chiefly to science was probably that founded by della Porta at Naples in 1560 and named Academia Secretoruvi Naturae. The requirement for membership was to have made some dis- covery in natural science. Della Porta fell under ecclesiastical suspicion as a practitioner of the black arts, and though acquitted was ordered to close his "Academy." The Accademia dei Lincei (of the Lynx), founded at Rome in 1603, included both della Porta and Galileo among its early members, and still flourishes. Its de- vice is a lynx with upturned eyes. The Royal Society of London, like many other societies, was the outgrowth of meetings of friends for discussion and was chartered in 1662. (For Boyle's Invisible College see above, p. 261.) Among the earlier members of the Royal Society were Boyle and Hooke, Mayow, Huygens, Ray, Grew, Malpighi, Leeuwen- hoek, and Isaac Newton. A well-known passage quoted by Huxley from Dr. Wallis, one of the first members, is of special interest since it shows what subjects were most dwelt upon by men of science at the time of Cromwell and the Restoration :^ Some twenty years before the outbreak of the plague (1665), says Huxley, a few calm and thoughtful students banded themselves to- gether for the purpose, as they phrased it, of 'improving natural knowledge. ' The ends they proposed to attain cannot be stated more clearly than in the words of one of the founders of the organisation : — 'Our business was (precluding matters of theology and state affairs) to discourse and consider of philosophical enquiries, and such as re- lated thereunto : — as Physick, Anatomy, Geometry, Astronomy, Navigation, Staticks, Magneticks, Chymicks, Mechanicks, and 270 A SHORT HISTORY OF SCIENCE Natural Experiments ; with the state of these studies and their cultiva- tion at home and abroad. We then discoursed of the circulation of the blood, the valves in the veins, the vence Icwtew, the lymphatic vessels, the Copernican hypothesis, the nature of comets and new stars, the satellites of Jupiter, the oval shape (as it then appeared) of Saturn, the spots on the sun and its turning on its own axis, the inequalities and selenography of the moon, the several phases of Venus and Mer- cury, the improvement of telescopes and grinding of glasses for that purpose, the weight of air, the possibility or impossibility of vacuities and nature's abhorrence thereof, the Torricellian experiment in quick- silver, the descent of heavy bodies and the degree of acceleration therein, with divers other things of like nature, some of which were then but new discoveries, and others not so generally known and em- braced as now they are ; with other things appertaining to what hath been called the New Philosophy, which from the times of Galileo at Florence, and Sir Francis Bacon (Lord Verulam) in England, hath been much cultivated in Italy, France, Germany, and other parts abroad, as well as with us in England.' The learned Dr. Wallis, writing in 1696, narrates in these words what happened half a century before, or about 1645. Among the first publications of the Royal Society of London were the works of Malpighi, the Italian microscopical anatomist, in 1669, and others by Leeuwenhoek, the Dutch microscopist. The French Academy {Academie des sciences) began its meetings in 1666, and the corresponding Berlin Academy in 1700. The oldest American association for the promotion of science is the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, proposed by Benjamin Franklin in 1743 and finally organized in 1769. Franklin himself presided over it from 1769 until his death in 1790. The New Philosophy: Bacon and Descartes. — It has been shown above how the all-inclusive philosophy of their predecessors began with Plato and Aristotle to be divisible into general and " natural " philosophy, — a differentiation which continued to exist and to increase slowly through the Middle Ages and the Renaissance. We have also shown how the mariner's compass, the invention BACON AND DESCARTES 271 of printing, the discovery of the New World, the heliocentric hypothesis, the idea of the earth as a magnet, the exploration of the human body, the Reformation, and the progress of mathe- matical science, were already widely opening men's minds, so that by the end of the sixteenth century it is not surprising to find the new knowledge reacting upon the old philosophy. With this movement two great names will always be associated : viz. those of Francis Bacon and Rene Descartes. Bacon, because of his ojBBcial position and immense philosophical and literary ability, was able to draw universal attention to the methods of science and especially to the method of investigation by induction, so that his indirect service to science was great. Bacon's true place in science was, however, well understood by his contemporaries, for one of the greatest, Harvey, discoverer of the circulation of the blood, remarks that, "the Lord Chancellor writes of science like — a Lord Chancellor." Descartes, far more important than Bacon in respect to his con- tributions to various branches of science, likewise stirred the in- tellect of Europe and helped to bring about those changes in the old philosophy which in the minds of many made it new. Des- cartes was not only a mathematician of the first rank but an in- genious and original worker in many branches of scientific inquiry such as music, anatomy, physiology, optics, etc. It is to him that we owe the first ideas of mechanism in living bodies, his notion of a "man machine" being highly original and suggestive. Science, says Descartes, may be compared to a tree ; metaphysics is the root, physics the trunk, and the three chief branches are mechanics, medicine, and morals. Here are my books, he is reported to have told a visitor, as he pointed to the animals which he had dissected. The conservation of health, he writes in 1646, has always been the principal end of my studies. Bacon and Descartes were methodologists, both urging the fundamental importance to progress, of method and its right use in investigation and inquiry, and Descartes, younger by almost a 272 A SHORT HISTORY OF SCIENCE generation, admired and to some extent imitated his predecessor in this direction. Progress of Natural and Physical Science in the Seventeenth Century. — A mere glance at the Tabular View of Chronology in the Appendix will sufBce to show the immense superiority of the seventeenth to any preceding century in the number as well as the productivity of the workers devoted to the mathematical, and likewise to the natural and physical, sciences. The achievements of this century in natural philosophy are especially notable both for their fundamental character and their wide range. A century which began with a Galileo and ended with a Huygens and a Newton; which witnessed the introduction of the telescope, the barometer, the thermometer, the air-pump, the manometer, and the microscope, as well as the organization of the greatest and most useful scientific societies the world has hitherto known, must be forever famous. And when to the names and works of Galileo and Huygens and Newton we add those of Kepler, Harvey, Torricelli, Halley, Descartes, Boyle, Hales, Boerhaave, Leeuwenlioek, and Malpighi, we have a brilliant com- pany indeed. References for Reading Francis Bacon. Essay in Great Englishmen of the Sixteenth Century, by Sidney Lee. Robert Boyle. Sceptical Chymist. (Everyman's Library.) Brewster's Life of Newton, and Ldves of Eminent Persons. R. Descartes. Life, by Haldane. R. Descartes, Discourse touching the method of using one's reason rightly and of seeking scientific truth. Cf. Huxley. Methods and Results, 1896. G. E. Hale. National Academies and the Progress of Research. William Harvey On The Movement of the Heart and the Blood. (Everyman's Library. ) William Harvey. By D'Arcy Power. (Masters of Medicine Series.) Herschel's Familiar Lectures. Thomas Sydenham. By J. F. Payne. (Masters of Medicine Series.) CHAPTER XIII BEGINNINGS OF MODERN MATHEMATICAL SCIENCE .... All the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics. And a proof that it far surpasses in facility and importance the sciences which de- pend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides. ... — Descartes. As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection. — Lagrange. The application of algebra has far more than any of his meta- physical speculations, immortalized the name of Descartes, and con- stitutes the greatest single step ever made in the progress of the exact sciences. — Mill. The idea of coordinates which forms the indispensable scheme for making all processes visible, with its many-sided and stimulating applications in all branches of daily life, — whether medicine, physi- cal geography, political economy, statistics, insurance, the technical sciences — the first beginnings of the calculus in their historical evolution, the development of the ideas of function and limit in connection with the elementary theory of curves, these are things without which in the present day not the slightest comprehension of the phenomena of nature can be attained, of which, however, the knowledge enables us as by magic to gain an insight with which in depth and range, but above all in certainty, scarcely any other can be compared. — Voss. How many celebrate the names of Newton and Leibnitz ! How few have a real appreciation of that which these men have created of permanent value ! Here lie the roots of our present-day knowledge, here the true continuation of the strivings of antique wisdom. — Lindemann. T 273 274 A SHORT HISTORY OF SCIENCE The invention of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a high level. — Whitehead. Mathematical Philosophy. Analytic Geometey. Des- cartes. — The invention of analytic geometry by Descartes in 1637 and the almost contemporary introduction of integral calculus as the method of " indivisibles" may be regarded as the real beginning of modern mathematical science. Thanks to these fruitful ideas the science has during the three centuries that have since elapsed made extraordinary progress both in its own internal development and in its application throughout the range of the physical sciences. Descartes was born in Touraine in 1596, and after the education appropriate for a youth of family and some years of fashionable life in Paris, entered the army, then in Holland. His military career continued till 1621 with incidental opportunity for his favorite speculations in mathematics and philosophy. Some of his most fruitful ideas dated from dreams and his best thinking was habitually done before rising. It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery — Columbus when he first saw the Western shore, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of coordinate geometry. — Whitehead. In order to devote himself more completely to his favorite studies he settled in Holland in 1629, devoting the next four years to writing a treatise, entitled Le Monde, upon the universe. In 1637 he published his great Discourse on the Method of Good Reasoning and of Seeking Truth in Science.^ This begins : — ' Discours de la Mfethode pour bien conduire sa raison et cheroher la vferitfe dans les sciences. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 275 If this discoiirse seems too long to be read all at once, it can be divided into six parts. In the first will be found various considera- tions concerning the sciences ; in the second, the chief rules of the method which the author has sought ; in the third, some of those of ethics which he has deduced by this method ; in the fourth, the reasons by which he proves the existence of God and of the human soul, which are the foundations of his metaphysics ; in the fifth, the order of questions of physics which he has sought, and particularly the explanation of the movement of the heart and of some other difficulties which belong to medicine ; also the difference which exists between our soul and that of the beasts ; and in the last, what things he believes necessary in order to go farther in the investiga- tion of nature than has been done, and what reasons have made him write. Good sense is the most widely distributed commodity in the world, for every one thinks himself so well supplied with it that even those who are hardest to satisfy in every other respect are not accus- tomed to desire more of it than they have. In this it is not prob- able that all men are mistaken, but rather this testifies that the power of good judgment and of discriminating between the true and the false, which is properly what one calls good-sense or reason, is naturally equal in all men ; and thus that the diversity of our opinions is not due to the fact that some are more reasonable than others, but only that we conduct our thought along different channels, and do not consider the same things. For it is not enough to have a good mind, but the principal thing is to apply it well. The greatest souls are capable of the greatest vices as well as of the greatest virtues : and those who only progress very slowly can advance much more, if they follow always the straight road than do those who run, depart- ing from it. His four cardinal precepts were : — Never to receive anything for true which he did not recognize to be evidently so; that is, to avoid carefully precipitancy and prejudg- ment. Second, to divide each of the difficulties which he should examine into as many pieces as possible. Third, to conduct his thoughts in order, beginning with the simplest objects. The last, to make everywhere enumerations so complete and reviews so general that he should be assured of omitting nothing. 276 A SHORT fflSTORY OF SCIENCE Three appendices dealt with optics, meteors, and geometry, the last containing the beginnings of analytic geometry. The relation of his philosophy to mathematics may be indicated in the following passages. Considering that, among all those who up to this time made dis- coveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations — that is to say, at proofs cer- tain and evident — I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons. When ... I asked myself why was it then that the earliest phi- losophers would admit to the study of wisdom only those who had studied mathematics, as if this science was the easiest of all and the one most necessary for preparing and disciplining the mind to com- prehend the more advanced, I suspected that they had knowledge of a mathematical science different from that of our time. . . . I believe I find some traces of these true mathematics in Pappus and Diophantus, who, although they were not of extreme antiquity, lived nevertheless in times long preceding ours. But I willingly be- lieve that these writers themselves, by a culpable ruse, suppressed the knowledge of them ; like some artisans who conceal their secret, they feared, perhaps, that the ease and simplicity of their method, if become popular, would diminish its importance, and they preferred to make themselves admired by leaving to us, as the product of their art, certain barren truths deduced with subtlety, rather than to teach us that art itself, the knowledge of which would end our admiration. Those long chains of reasoning, quite simple and easy, which geom- eters are wont to employ in the accomplishment of their most difficult demonstrations, led me to think that everything which might fall under the cognizance of the human mind might be connected together in a similar manner, and that, provided only that one should take care not to receive anything as true which was not so, and if one were always careful to preserve the order necessary for deducing one truth from another, there would be none so remote at which he might not at last arrive, nor so concealed which he might not discover. BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 277 Descartes had attempted the solution of a historic geometrical problem propounded by Pappus. From a point P perpendiculars are dropped on m given straight lines and also on n other given lines. The product of the m perpendiculars is in a constant ratio to the product of the n; it is required to determine the locus of P. Pappus had stated without proof that for m = ?i = 2 the locus is a conic section, Descartes showed this algebraically, — Newton afterwards conquering the difBculty by unaided geometry. Descartes distinguished geometrical curves for which x and y may be regarded as changing at commensurable rates, or as we should say, curves for which the slope is an algebraic function of the coordinates, from ciu-ves which do not satisfy this condition. These he called "mechanical," and did not discuss further. For the accepted definition of a tangent as a line between which and the curve no other line can be drawn, he introduced the modern notion of limiting position of a secant. In connection with this he considered a circle meeting the given curve in two consecutive points, a perpendicular to the radius of the circle being a common tangent to the circle and the given curve. The circle was not however that of curvature, but had its centre on an axis of sym- metry of the given curve. He recognized the possibility of ex- tending his methods to space of three dimensions, but did not work out the details. His geometry contained also a discussion of the algebra then known, and gave currency to certain important inno- vations, in particular the systematic use of a, b, and c, for known, X, y, and z, for unknown quantities ; the introduction of exponents ; the collection of all terms of an equation in one member ; the free use of negative quantities ; the use of undetermined coefficients in solving equations ; and his rule of signs for studying the number of positive or negative roots of equations. He even fancied that he had found a method for solving an equation of any degree. It is important to distinguish just what Descartes contributed to mathematics in his analytic geometry. Neither the com- bination of algebra with geometry nor the use of coordinates was new. From the time of Euclid quadratic equations had been solved geometrically, while latitude and longitude involving a 278 A SHORT HISTORY OF SCIENCE system of coordinates are of similar antiquity. The great stej made by Descartes was his recognition of the equivalence of ar equation and the geometrical locus of a point whose coordinate; satisfy that equation. On this foundation facts known or ascer tainable about geometry may be translated into algebra and con- versely. The advantage is comparable with that conferred by th( possession of two arms or eyes, or even two senses, under a commoi will. The intricate but powerful machinery of algebra become: available for solving geometrical problems, while, on the othei hand, the geometrical illustration makes the algebra visible anc concrete. Later works dealt with philosophy and physical science, in par ticular with a theory of vortices. Descartes enunciates tei natural laws, the first two corresponding with the first two o Newton's. He argues that all matter is in motion and that thii must result in the formation of vortices. The sun is the centre o one great vortex, each planet of its own, thus approximatin; vaguely the future nebular hypothesis. Newton thought it wortl while to refute this theory, which was chiefly notable as a bolt attempt to interpret the phenomena of the universe by means o a single mechanical principle. Lord Kelvin has expressed, with all his force, that the sole satis factory explanation of the phenomena of nature is that which lead; them back in the last analysis to motion in a continuous incompres sible fluid. This however was the guiding thought with Descartes. — Timerding. Descartes's achievements in mathematics leave no doubt of hi; exceptional intellectual power. He had neither the data nor th( scientific method for accomplishing similar results in other branche; of science, and in mathematics he would doubtless have accom plished much more had he not expended his energies so widel; in over-confident reliance on his logical method. He died a Stockholm in 1650. Indivisibles. Cavalieri. — While Descartes was thus as i were incidentally laying the foundations of modern geometrica BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 279 analysis, his Italian contemporary, Cavalieri (1598-1647) was rendering a similar service to the integral calculus in developing his theory of indivisibles. The problem of measuring the length of a curve or the area of a figure having a curved boundary, or the volume of a solid bounded by a curved surface goes back indeed to comparatively ancient Greek times. Most notable in this direction was the work of Archimedes. Kepler, attempting to resolve astronomical difiiculties by the hypothesis of elliptical orbits, is confronted at once with the problem of determining the circumference of an ellipse. He gives the approximation t {a + b) where a and b are the semi- axes. This is close if a and b are nearly equal, as in most of the planetary orbits. Interesting himself in current methods of measuring the capacity of casks, he published in 1615 his Nova Stereometria Doliorum Vinariorum, in which he determines the volumes of many solids bounded by surfaces of revolution. The Greek method had in case of the circle, etc., depended on an "exhaustion" process of inscribing and circumscribing polygons differing less and less from the curve both in boundary and in area. Kepler however divided his solid into sections, determined the area of a section and then sought the sum. He lacked an adequate system of coordinates, a clearly defined conception of a limit, and an effective method of summation. In view of the intrinsic difficulty of this important problem, however, the extent of his success is remarkable. He also sought to determine the most economical proportions for casks, etc., expressing his view of the underlying mathematical theory by the theorem " In points where the transition from a less to the greatest and again to a less takes place, the difference is always to a certain degree imperceptible." Cavalieri, in 1635, adopted the form of statement that a line consists of an infinite number of points, a surface of an infinity of lines, a solid of an infinity of surfaces, but later revised this on the basis of the assumption "that any magnitude may be divided into an infinite number of small quantities which can be made to bear any required ratios one to the other." On this basis, open 280 A SHORT fflSTORY OF SCIENCE as it was to criticism, were solved simple area problems involving the parabola and the hyperbola. The principle of comparing areas by comparing lengths of s system of parallel lines crossing them is easily illustrated in the case of the ellipse by comparing it with the circle having as its diameter the (horizontal) major axis of the ellipse. If a and I are the semi-axes of the ellipse the twc curves are known to be so related that everj vertical chord of the circle is in a fixed ratio a : & to the part of it lying within the ellipse, The area of the circle must bear the same relation to the area of the ellipse. The transition from length to area while nol DE _ CB _ _a_ rigorously worked out by Cavalieri does no1 ^^ ^ necessarily involve the false assumption thai area consists of the sum of parallel lines. A similar method ij evidently applicable to volumes. Thus was anticipated one of the most interesting and important processes of modern mathematics — integration as a summation. Similarly Cavalieri determined volumes by a consideration ol the thin sections or elements into which they may be resolved bj parallel planes. The principle that "two bodies have the same volume if sections at the same level have the same area" is stil known by his name. Descartes's work with tangents seems not to have led him tc develop the fundamental ideas of the differential calculus, and ii appeared that the integral calculus would be evolved first from the work of Cavalieri. Projective Geometry : Desargues. — Hardly less interesting than the new ideas of Descartes and Cavalieri are those of thei contemporary Desargues (1593-1662), an engineer and architec of Lyons, who made important researches in geometry. But fo: the still more brilliant geometrical achievements of Descartes these might have led to the immediate development of projectiv geometry, the elements of which are contained in Desargues' work. In general this geometry instead of dealing with definit BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 281 triangles, polygons, circles, etc., in the Euclidean manner, is based on a consideration of all points of a straight line, of all lines through a common point and of the possible effects of setting up an orderly one-to-one correspondence between them. In particular, Des- argues makes a comparative study of the different plane sections of a given cone, deducing from known properties of the circle anal- ogous results for the other conic sections. In his chief work Desargues enunciates the propositions : — 1. A straight line can be considered as produced to infinity and then the two opposite extremities are united. 2. Parallel lines are lines meeting at infinity and conversely. 3. A straight line and a circle are two varieties of the same species. On these he bases a general theory of the plane sections of a cone. Desargues contented himself with enunciating general princi- ples, remarking : — "He who shall wish to disentangle this prop- osition will easily be able to compose a volume." He met Descartes while employed by Cardinal Richelieu at the siege of Rochelle, and they with others met regularly in Paris for the discussion of the new Copernican theory and other scientific problems. He says ' I freely confess that I never had taste for study or re- search either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proxi- mate causes .... for the good and convenience of life, in maintaining health, in the practice of some art, . . . having observed that a good part of the arts is based on geometry, among others the cutting of stones in architecture, that of sun-dials, that of perspective m particular.' Perceiving that the practitioners of these arts had to burden them- selves with the laborious acquisition of many special facts in geometry, he sought to relieve them by developing more general methods and printing notes for distribution among his friends. An interesting theorem bearing his name and typical of pro- jective geometry is as follows : — If two triangles ABC and A'B'C 282 A SHORT fflSTORY OF SCIENCE are so related that lines joining corresponding vertices meet in a point O, then the intersections of corresponding sides will lie in a straight line A"B"C". It remained for Monge, the inventor oi descriptive geometry (p. 335) and others more than a century later to carry this development forward. Desargues's work was indeed prac- tically lost until Poncelet in 1822 proclaimed him the Monge of his century. Theory of Numbers and Probability : Feemat, Pascal. — But little younger than Descartes and Cavalieri was Pierre de Fermat (1601-1665) a man of quite exceptional position in mathe- matical history. Devoting to mathematics such leisure as his public duties afforded, he nevertheless published almost noth- ing, many of his results being known to us only in the form ol brief marginal notes without proof. In editing Diophantus he enunciated numerous theorems on integers, for example. An odd prime can be expressed as the difference of two square integers in one and only one way. No integral values of x, y, z can be found to satisfy the equatior a;" + 2/" = s", if re be an integer greater than 2. This seemingly simple theorem has been verified for so wide a range of values of n, that its truth can hardly be doubted, but nc general proof has yet been given in spite of a prize of 100,000 mark; awaiting him who either proves or disproves it. Some writers ever credit Fermat with a substantial share in the invention of the nevi analytic geometry, in which he had certainly done independem work for some years before Descartes's publication. Laplace in deed calls Fermat "the true inventor of the differential calculus.' He discusses problems of maxima and minima, and passing t( concrete phenomena, enunciates the interesting theorem : tha Nature, the great workman which has no need of our instru ments and machines, lets everything happen with a minimum o BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 283 outlay, — an idea not indeed strange to some of the Greeks. The law of refraction of a ray of light he deals with correctly as a particular case of the principle of economy, a principle which exerted a potent influence in the scientific philosophy of the following century. Thus for example Euler says in 1744 : — Since the organization of the world is the most excellent, nothing is found in it, out of which some sort of a maximum or minimum property does not shine forth. Therefore no doubt can exist, that all action in the world can be derived by the method of maxima and minima as well as from the actual operating causes. Fermat's work in the theory of probability is fundamental. He discusses the case of two players, A and B, where A wants two points to win and B three points. Then the game will certainly be decided in the course of four trials. Take the letters a and b, and write down all the combinations that can be formed of four letters. These combinations are 16 in number, namely aaaa, aaab, aaba, aabb, abaa, abab, abba, abbb, baaa, baab, baba, babb, bbaa, bbab, bbba, bbbb. Now every combination in which a oc- curs twice or oftener represents a case favorable to A, and every combination in which b occurs three times or oftener represents a case favorable to B. Thus, on counting them, it will be found that there are 11 cases favorable to A, and 5 cases favorable to B ; and, since these cases are all equally likely, A's chance of win- ning the game is to B's chance as 11 is to 5. Like Descartes, Pascal (1623-1662) devoted but a fraction of his great talent to mathematical science. I have spent much time in the study of the abstract sciences, — but the paucity of persons with whom you can communicate on such subjects gave me a distaste for them. When I began to study man, I saw that these abstract studies were not suited to him, and that in diving into them, I wandered farther from my real track than those who were ignorant of them, and I forgave men for not having at- tended to these things. But I thought at least I should find many companions in the study of mankind, which is the true and proper study of man. Again I was mistaken. There are yet fewer students of Man than of Geometry. 284 A SHORT HISTORY OF SCIENCE Learning geometry surreptitiously at 12 years, he had at 1! written an essay on conic sections and constructed the first com puting machine. While most of his later life was devoted to re ligion, theology, and literature, he undertook a wide range o physical experimentation, and made important contributions t( the then new theories of numbers and probability, besides i discussion of the cycloid. The juvenile essay on conic section contains the beautiful theorem since named for him that th opposite sides of a hexagon inscribed in a conic section meet in i straight line. Of geometry and logic Pascal says : — Logic has borrowed the rules of geometry without understanding its power. ... I am far from placing logicians by the side of geom- eters who teach the true way to guide the reason. . . The methoc of avoiding error is sought by every one. The logicians profess t( lead the way, the geometers alone reach it, and aside from their science there is no true demonstration. His work on probability connected itself with the problem oi two players of equal skill wishing to close their play, ofwhicl Fermat's solution has been given above. The following is my method for determining the share of eacl player when, for example, two players play a game of three point! and each player has staked 32 pistoles. Suppose that the first player has gained two points and the seconc player one point ; they have now to play for a point on this condition that if the first player gain, he takes all the money which is at stake namely 64 pistoles ; while if the second player gain, each player ha: two points, so that they are on terms of equality, and if they leav( off playing, each ought to take 32 pistoles. Thus if the first playe gain, then 64 pistoles belong to him, and if he lose, then 32 pistole; belong to him. If therefore the players do not wish to play this game but separate without playing it, the first player would say to th second, ' I am certain of 32 pistoles, even if I lose this point, and a for the other 32 pistoles, perhaps I shall have them and perhaps yoi will have them ; the chances are equal. Let us then divide these 3! pistoles equally, and give me also the 32 pistoles of which I am certain, Thus the first player will have 48 pistoles and the second 16 pistoles BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 285 By similar reasoning he shows that if the first player has gained two points and the second none, the division should be 56 to 8 ; while if the first has gained one point, the second none, it should be 44 and 20. The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experi- mentalist, and the statesman. From the time when Pascal and Fermat established its first principles, it has rendered, and continues daily to render, services of the most eminent kind. It is the calculus of probabilities, which, after having suggested the best arrangements of the tables of population and mortality, teaches us to deduce from those numbers, in general so erroneously interpreted, conclusions of a precise and useful character ; it is the calculus of probabilities which alone can regulate justly the premiums to be paid for assurances; the reserve funds for the disbursements of pensions, annuities, dis- counts, etc. It is under its influence that lotteries and other shameful snares cunningly laid for avarice and ignorance have definitely disap- peared. — Arago. With this work connected itself his arithmetical triangle in which successive diagonals contain the coefEcients which occur in ex- pansions by the binomial theorem, which Newton was soon to generalize. 1 I 1 1 I 1 12 3 4 5 1 3 6 10 1 4 10 1 5 1 He applies the method of indivisibles successfully to the cycloid (the curve generated by a point on the rim of a rolling wheel). Pascal invented in 1645 an arithmetical machine, writing the Chancellor in regard to it : Sir : If the pubhc receives any advantage from the invention which I have made to perform all sorts of rules of arithmetic in a manner as novel as it is convenient, it will be under greater obligation to your 286 A SHORT HISTORY OF SCIENCE Highness than to my small efforts, since I should only have been abli to boast of having conceived it, while it owes its birth absolutely t( the honor of your commands. The length and difficulty of the ordi nary means in use have made me think on some help more prompt anc easy to relieve me in the great calculations with which I have beei occupied for several years in certain affairs which depend on th( occupations with which it has pleased you to honor my father for th( service of his Majesty in Normandy. I employed for this investi gation all the knowledge which my inclination and the labor of my firsi studies in mathematics have gained for me, and after profound re flection, I recognized that this aid was not impossible to find. Mechanics and Optics : Htjtgens. — Most notable amon^ the successors of Galileo in mechanics before we reach Newtor was Huygens of Holland (1629-1695) who combined mathematica power with exceptional practical ingenuity. He first (in 1655^ explained as a ring the excrescences of Saturn which had beer misunderstood by Galileo and others, publishing his discovery ir the occult form a^c^d^e^g%H''l^^n^o*p^qVsH^u^ {Annulo cingitm tenui, -piano, nusquam cohcerente ad eclipticam inclinato.) He alsc discovered Saturn's largest moon. About the same time he made his great invention of the pendulum clock. Accepting a call tc Paris by Colbert at the founding of the French Academy, h( remained there from 1666 to 1681. In optics he developed and maintained even In opposition tc the authority of Newton the undulatory or wave theory whicl only found general acceptance a century later. The velocity o: light Galileo had failed to measure by means of signal lanterns and Descartes had likewise been unable to ascertain it by compar- ing the observed and computed instants of a lunar eclipse Huygens points out that even this latter test does not prove In stantaneous transmission. Romer's conclusive report on observa tions of a satellite of Jupiter dates from 1675. On this basi; Huygens estimated the velocity of light at 600,000 times that o sound, — a result about one-third too small. The medium In which light waves travel Huygens named tb ether, attributing to Its particles three properties in compariso: ■ ^^H ^^^m ^^-'"^-^ ^ ^m^H^^i^ ^l^^^^^l ^^^^^^^^^H ^^^^^r ^^t'M '^i '^ ^wBB "- :: ^1^^^ ^^^^^^^^^^^^B ^^H ^V "^-P^Bkiil^i:"' 1^^^ H^l ^^V <*' s^r'- '-^Hi^£K->, "^fSSk ^ V 1 j^l!^mH ^^^^^■^^^^^^1 V -^^^^^^^Elm W^^ Hl^^I ^V' '-''^~'^'^/^^^HIi ' "^^ ^^SBKm ^^^^^^H H .n>;"^.. '"i^al^Hli^^^.''-' ^^HH ^^HK^^H ^B- ?j'^ '"RIhB^HIB^/^I^ "^^hHH ^^^H^^l ^B ^^PJ^M^BWI ^^^^fc-^^tTJ^^BW ^^^B^H ■ "' IfiMi^TOiiii ' ^'^TM ^hI I : -^: jn ^gists became anxious to entertain no opinion whatever on the causes )f phenomena, and were inclined to scepticism even where the con- tusions deducible from observed facts scarcely admitted of reasonable loubt. Geological Society of London. — But although the reluctance to heorize was carried somewhat to excess, no measure could be more alutary at such a moment than a suspension of all attempts to form ^'hat were termed "theories of the earth." A great body of new data vere required, and the Geological Society of London, founded in .807, conduced greatly to the attainment of this desirable end. To nultiply and record observations, and patiently to await the result it some future period, was the object proposed by them, and it was ;heir favourite maxim that the time was not yet come for a general lystem of geology, but that all must be content for many years to be ;xclusively engaged in furnishing materials for future generalizations. 3y acting up to these principles with consistency, they in a few years lisarmed all prejudice, and rescued the science from the imputation )f being a dangerous, or at best but a visionary pursuit. Modern Progeess of Geology Study of Organic Remains. — Inquiries were at the same time )rosecuted with great success by the French naturalists, who devoted ;heir attention especially to the study of organic remains. They ihewed that the specific characters of fossil shells and vertebrated mimals might be determined with the utmost precision, and by their exertions a degree of accuracy was introduced into this department )f science, of which it had never before been deemed susceptible. It vas found that, by the careful discrimination of the fossil contents of itrata, the contemporary origin of different groups could often be APPENDIX H: LYELL 431 established, even where all identity of mineralogical character was wanting, and where no light could be derived from the order of super- position. The minute investigation, moreover, of the relics of the animate creation of former ages, had a powerful effect in dispelling the illusion which had long prevailed concerning the absence of analogy between the ancient and modern state of our planet. A close comparison of the recent and fossil species, and the inferences drawn in regard to their habits, accustomed the geologist to contemplate the earth as having been at successive periods the dwelling-place of animals and plants of different races, some of which were discovered to have been terrestrial, and others aquatic — some fitted to live in seas, others in the waters of lakes and rivers. By the consideration of these topics, the mind was slowly and insensibly withdrawn from imaginary pic- tures of catastrophes and chaotic confusion, such as haunted the imagination of the early cosmogonists. Numerous proofs were dis- covered of the tranquil deposition of sedimentary matter and the slow development of organic life. If many still continued to main- tain, that " the thread of induction was broken, " yet in reasoning by the strict rules of induction from recent to fossil species, they vir- tually disclaimed the dogma which in theory they professed. The adoption of the same generic, and, in some cases, even the same specific, names for the exuviae of fossil animals, and their living ana- logues, was an important step towards familiarising the mind with the idea of the identity and unity of the system in distant eras. It was an acknowledgment, as it were, that a considerable part of the ancient memorials of nature were written in a living language. The growing importance then of the natural history of organic remains, and its general apphcation to geology, may be pointed out as the character- istic feature of the progress of the science during the present century. This branch of knowledge has already become an instrument of great power in the discovery of truths in geology, and is continuing daily to unfold new data for grand and enlarged views respecting the former changes of the earth. When we compare the result of observations in the last thirty years with those of the three preceding centuries, we cannot but look forward with the most sanguine expectations to the degree of excel- lence to which geology may be carried, even by the labours of the present generation. Never, perhaps, did any science, with the excep- 432 A SHORT HISTORY OP SCIENCE tion of astronomy, unfold, in an equally brief period, so many novel and unexpected truths, and overturn so many preconceived opinions. The senses had for ages declared the earth to be at rest, until the astronomer taught that it was carried through space with incon- ceivable rapidity. In like manner was the surface of this planet regarded as having remained unaltered since its creation, until the geologist proved that it had been the theatre of reiterated change, and was still the object of slow but never-ending fluctuations. The discovery of other systems in the boundless regions of space was the triumph of astronomy — to trace the same system through various transformations — to behold it at successive eras adorned with dif- ferent hills and valleys, lakes and seas, and peopled with new inhabi- tants, was the delightful meed of geological research. By the geom- eter were measured the regions of space, and the relative distances of the heavenly bodies — by the geologist myriads of ages were reck- oned, not by arithmetical computation, but by a train of physical events — a succession of phenomena in the animate and inanimate worlds — signs which convey to our minds more definite ideas than figures can do, of the immensity of time. Whether our investigation of the earth's history and structure will eventually be productive of as great practical benefits to man- kind, as a knowledge of the distant heavens, must remain for the decision of posterity. It was not till astronomy had been enriched by the observations of many centuries, and had made its way against popular prejudices to the establishment of a sound theory, that its application to the useful arts was most conspicuous. The cultiva- tion of geology began at a later period ; and in every step which it has hitherto made towards sound ethical principles, it has had to contend against more violent prepossessions. The practical advan- tages already derived from it have not been inconsiderable : but our generalizations are yet imperfect, and they who follow may be expected to reap the most valuable fruits of our labour. Meanwhile the charm of first discovery is our own, and as we explore this magnificent field of inquiry, the sentiment of a great historian of our times may con- tinually be present to our minds, that "he who calls what has van- ished back again into being, enjoys a bliss like that of creating." . . . APPENDIX H: LYELL 433 Assumption of the Discordance of the Ancient and Existing Causes of Change Unphilosophical . . . For more than two centuries the shelly strata of the Sub- Apennine hills afforded matter of speculation to the early geologists of Italy, and few of them had any suspicion that similar deposits were then forming in the neighboring sea. They were as unconscious of the continued action of causes still producing similar effects, as the astronomers, in the case supposed by us, of the existence of certain heavenly bodies still giving and reflecting light, and performing their movements as in the olden time. Some imagined that the strata, so rich in organic remains, instead of being due to secondary agents, had been so created in the beginning of things by the fiat of the Al- mighty ; and others ascribed the imbedded fossil bodies to some plastic power which resided in the earth in the early ages of the world. At length Donati explored the bed of the Adriatic, and found the closest resemblance between the new deposits there forming, and those which constituted hills above a thousand feet high in various parts of the peninsula. He ascertained that certain genera of living testacea were grouped together at the bottom of the sea in precisely the same manner as were their fossil analogues in the strata of the hills, and that some species were common to the recent and fossil world. Beds of shells, moreover, in the Adriatic, were becoming incrusted with cal- careous rock; and others were recently enclosed in deposits of sand and clay, precisely as fossil shells were found in the hills. This splen- did discovery of the identity of modern and ancient submarine opera- tions was not made without the aid of artificial instruments, which, like the telescope, brought phenomena into view not otherwise within the sphere of human observation. In like manner, in the Vicentin, a great series of volcanic and marine sedimentary rocks were examined in the early part of the last century ; but no geologist suspected, before the time of Arduino, that these were partly composed of ancient submarine lavas. If, when these enquiries were first made, geologists had been told that the mode of formation of such rocks might be fully elucidated by the study of processes then going on in certain parts of the Mediterranean, they would have been as incredulous as geometers would have been before the time of Newton, if any one had informed them that, by making experiments on the motion of bodies on the earth, they 2f 434 A SHORT HISTORY OF SCIENCE might discover the laws which regulated the movements of distant planets. The establishment, from time to time, of numerous points of identi- fication, drew at length from geologists a reluctant admission, that there was more correspondence between the physical constitution of the globe, and more uniformity in the laws regulating the changes of its surface, from the most remote eras to the present, than they at first imagined. If, in this state of the science, they still despaired or reconciling every class of geological phenomena to the operations of ordinary causes, even by straining analogy to the utmost limits of credibility, we might have expected, that the balance of probabihty at least would now have been presumed to incline towards the identity of the causes. But, after repeated experience of the failure of attempts to speculate on different classes of geological phenomena, as belong- ing to a distinct order of things, each new sect persevered systematic- ally in the principles adopted by their predecessors. They invariably began, as each new problem presented itself, whether relating to the animate or inanimate world, to assume in their theories, that the economy of nature was formerly governed by rules quite independent of those now established. Whether they endeavoured to account for the origin of certain igneous rocks, or to explain the forces which elevated hills or excavated valleys, or the causes which led to the extinction of certain races of animals, they first presupposed an orig- inal and dissimilar order of nature ; and when at length they approxi- mated, or entirely came round to an opposite opinion, it was always with the feeling, that they conceded what they were justified a priori in deeming improbable. In a word, the same men who, as natural philosophers, would have been greatly surprised to find any deviation from the usual course of Nature in their oiun time, were equally sur- prised, as geologists, not to find such deviations at every period of the past. The Huttonians were conscious that no check could be given to the utmost license of conjecture in speculating on the causes of geo- logical phenomena, unless we can assume invariable constancy in the order of Nature. But when they asserted this uniformity with- out any limitation as to time, they were considered, by the majority of their contemporaries, to have been carried too far, especially as they applied the same principle to the laws of the organic, as well as of the inanimate world. APPENDIX H: LYELL 435 We shall first advert briefly to many difficulties which formerly appeared insurmoimtable, but which, in the last forty years, have been partially or entirely removed by the progress of science; and shall afterwards consider the objections that still remain to the doc- trine of absolute uniformity. In the first place, it was necessary for the supporters of this doc- trine to take for granted incalculable periods of time, in order to ex- plain the formation of sedimentary strata by causes now in diurnal action. The time which they required theoretically, is now granted, as it were, or has become absolutely requisite, to account for another class of phenomena brought to light by more recent investigations. It must always have been evident to unbiassed minds, that succes- sive strata, containing, in regular order of superposition, distinct beds of shells and corals, arranged in families as they grow at the bottom of the sea, could only have been formed by slow and insen- sible degrees in a great lapse of ages, yet, until organic remains were minutely examined and specifically determined, it was rarely possible to prove that the series of deposits met with in one country was not formed simultaneously with that found in another. But we are now able to determine, in numerous instances, the relative dates of sedi- mentary rocks in distant regions, and to show, by their organic re- mains, that they were not of contemporary origin, but formed in succession. We often find, that where an interruption in the consecu- tive formations in one district is indicated by a sudden transition from one assemblage of fossil species to another, the chasm is filled up, in some other district, by other important groups of strata. The more attentively we study the European continent, the greater we find the extension of the whole series of geological formations. No sooner does the calendar appear to be completed, and the signs of a succession of physical events arranged in chronological order, than we are called upon to intercalate, as it were, some new periods of vast duration. A geologist, whose observations have been confined to England, is accustomed to consider the superior and newer groups of marine strata in our island as modem, and such they are, compara- tively speaking; but when he has travelled through the Italian peninsula and in Sicily, and has seen strata of more recent origin forming mountains several thousand feet high, and has marked a long series both of volcanic and submarine operations, all newer than any of the regular strata which enter largely into the physical struc- 436 A SHORT HISTORY OF SCIENCE ture of Great Britain, he returns with more exalted conceptions of the antiquity of some of our modern deposits, than he before enter- tained of the oldest of the British series. We cannot reflect on the concessions thus extorted from us, in regard to the duration of past time, without foreseeing that the period may arrive when part of the Huttonian theory will be combated on the ground of its departing too far from the assumption of uniformity in the order of nature. On a closer investigation of extinct volcanos, we find proofs that they broke out at successive eras, and that the eruptions of one group were often concluded long before others had commenced their activity. Some were burning when one class of organic beings were in existence, others came into action when dif- ferent races of animals and plants existed, — it follows, therefore, that the convulsions caused by subterranean movements, which are merely another portion of the volcanic phenomena, occurred also in succession, and their efforts must be divided into separate sums, and assigned to separate periods of time ; and this is not all : when we examine the volcanic products, whether they be lavas which flowed out under water or upon dry land, we find that intervals of time, often of great length, intervened between their formation, and that the effects of one eruption were not greater in amount than that which now results during ordinary volcanic convulsions. The ac- companying or preceding earthquakes, therefore, may be considered to have been also successive, and to have been in like manner inter- rupted by intervals of time, and not to have exceeded in violence those now experienced in the ordinary course of nature. Already, therefore, may we regard the doctrine of the sudden eleva- tion of whole continents by paroxysmal eruptions as invalidated ; and there was the greatest inconsistency in the adoption of such a tenet by the Huttonians, who were anxious to reconcile former changes to the present economy of the world. It was contrary to analogy to suppose that Nature had been at any former epoch parsimonious of time and prodigal of violence — to imagine that one district was not at rest while another was convulsed — that the disturbing forces were not kept under subjection, so as never to carry simultaneous havoc and desolation over the whole earth, or even over one great region. If it could have been shown, that a certain combination of circum- stances would at some future period produce a crisis in the subter- ranean action, we should certainly have had no right to oppose our APPENDIX H: LYELL 437 experience for the last three thousand years as an argument against the probabiUty of such occurrences in past ages ; but it is not pre- tended that such a combination can be foreseen. In speculating on catastrophes by water, we may certainly antici- pate great floods in future, and we may therefore presume that they have happened again and again in past times. The existence of enormous seas of fresh water such as the North American lakes, the largest of which is elevated more than six hundred feet above the level of the ocean, and is in parts twelve hundred feet deep, is alone sufficient to assure us, that the time will come, however distant, when a deluge will lay waste a considerable part of the American continent. No hypothetical agency is required to cause the sudden escape of the confined waters. Such changes of level, and opening of fissures, as have accompanied earthquakes since the commencement of the present centm^y, or such excavation of ravines as the receding cataract of Niagara is now effecting, might breach the barriers. Notwith- standing, therefore, that we have not witnessed within the last three thousand years the devastation by deluge of a large continent, yet, as we may predict the futiu^e occurrence of such catastrophes, we are authorized to regard them as part of the present order of Nature, and they may be introduced into geological speculations respecting the past, provided we do not imagine them to have been more frequent or general than we expect them to be in time to come. The great contrast in the aspect of the older and newer rocks, in their texture, structure, and in the derangement of the strata, ap- peared formerly one of the strongest grounds for presuming that the causes to which they owed their origin were perfectly dissimilar from those now in operation. But this incongruity may now be regarded as the natural result of subsequent modifications, since the difference of the relative age is demonstrated to have been so immense, that, however slow and insensible the change, it must have become im- portant in the course of so many ages. In addition to the volcanic heat, to which the Vulcanists formerly attributed too much influence, we must allow for the effect of mechanical pressure, of chemical affinity, of percolation by mineral waters, of permeation by elastic fluids, and the action, perhaps, of many other forces less understood, such as electricity and magnetism. In regard to the signs of up- raising and sinking, of fracture and contortion in rocks, it is evident that newer strata cannot be shaken by earthquakes, unless the sub- 438 A SHORT HISTORY OF SCIENCE jacent rocks are also affected; so that the contrast in the relative degree of disturbance in the more ancient and the newer strata, is one of many proofs that the convulsions have happened in different eras, and the fact confirms the uniformity of the action of subter- ranean forces, instead of their greater violence in the primeval ages. The science of Geology is enormously indebted to Lyell — more so, as I believe, than to any other man who ever lived. — Darwin. Autobiography. Pour juger de ce qui est arrive, el meme de ce qui arrivera, nous n'avons qu'a examiner ce qui arrive. — Buffon. Theorie de la Terre. I. SOME INVENTIONS OF THE EIGHTEENTH AND NINE- TEENTH CENTURIES. APPLIED SCIENCE AND ENGINEERING He who seeks for immediate practical use in the pursuit of science, may be reasonably sure that he will seek in vain. Complete knowledge and complete understanding of the action of the forces of nature and of the min d, is the only thing that science can aim. at. The individual investigator must find his reward in the joy of new discoveries ... in the consciousness of having contributed to the growing capital of knowledge. Who could have imagined, when Galvani observed the twitching of the frog muscles as he brought various metals in contact with them, that eighty years later Europe would be overspun with wires which transmit messages from Madrid to St. Petersburg with the rapidity of lightning, by means of the same principle whose first manifestations this anatomist then observed. — Helmholtz. The place of inventions in the history of science is hard to define. Conditioned as they doubtless are by a favorable environment — at least for survival — they do not always obviously arise as a direct or logical consequence of preceding discoveries, or even of known principles, but seem sometimes to spring almost de novo from the brain of the inventor. And yet such an origin is probably more apparent than real. The steam-engine could hardly have come from Watt without Newcomen and Black as his predecessors, the telegraph from Morse or the telephone from Bell except after Franklin, Oersted and Faraday. Probably the truth is that if we only knew all the facts, instead of only some of them, we should find every invention the natural descendant, near or remote, of science already existing. And as inheritance often seems to skip a generation or two and children APPENDIX I: INVENTIONS 439 sometimes show, no discoverable resemblance to their immediate forbears, so inventions may come without disclosing any resemblance to parent inventions or ideas, while yet really intimately related to knowledge that has gone before. Nor is it easy to estimate the reciprocal debt of science to inventions and the arts. That this debt is large there can be no doubt. To illustrate this fact it is hardly necessary to do more than mention examples, such as the service of the compass to the sciences of geog- raphy, navigation and surveying ; of the telescope and the chronom- eter to astronomy ; of the microscope to biology ; of the air pump to natural philosophy ; or of the abacus or the Arabic numerals to arithmetic. Among the more notable of the inventions of the nineteenth cen- tury were the locomotive, the steamboat, the friction match, the sewing-machine, the steel pen, the telegraph, the telephone and the phonograph ; labor-saving machinery ; explosives ; and the internal combustion engine, with its numerous offspring (motor vehicles, air- planes, motor boats, etc.). PowEE : Its Soueces and Significance. — The recent progress of science and of civilization has been accompanied by a remarkable extension of man's control over his environment, which has come largely with his ability to develop, transmit, and utilize chemical, gravitational and electrical energy or power. The ancients and the men of the Middle Ages used chiefly the power of man and other ani- mals and of winds (windmills) and to some extent water {i.e. gravi- tation), as in water-wheels, but knew little of heat power or chemical power and nothing of electrical power, or of power transmission of any kind, — except in moving herds, treadmills, or marching armies. In past times the chief store of national power was manual labor : to-day it is the machine that does the work. — K. Pearson. The first step in the modem direction was apparently toward chemi- cal power, in the invention of gunpowder. GuNPOWDEE, NiTEOGLYCEEiNE, Dttstamite. — Gunpowder is be- lieved to have been known to the Chinese long belong it appeared in Europe. An explosive mixture of charcoal, sulphur, and nitre was apparently also known to the Arabians, but the first important appearance of gunpowder in Europe was about the fourteenth cen- tury, and since the sixteenth it has played an all-important part in 40 A SHORT HISTORY OF SCIENCE ?ar and in peace. Its effects upon society and civilization have been irofound, and with society and civihzation the progress of science 3 always closely bound up. The manufacture of gunpowder marks the beginning of the manu- acture of power, if we may describe the controlled accumulation, torage and liberation of energy by that convenient term. In 1845 ^n-cotton was invented by Schonbein, and in 1847 nitroglycerine by lobrero, and both explosives were found to be far more copious and lowerful sources of energy than gunpowder. It was Alfred Nobel, lowever, a Swedish engineer, who after mixing nitroglycerine with unpowder first made practical use of this for blasting. It was also ^obel who in 1867 made nitroglycerine less dangerous by diluting it fith inert substances such as silicious earth, — mixtures to which he ave the name dynamite.' The manufacture of power from gravitational sources, such as .^ater-power and wind power, goes back to the earliest times — ails, wind-mills and water-wheels being of very ancient origin, 'ower from fuel begins with Newcomen, Watt and the steam- ngine. Electrical power is at present chiefly derived indirectly rom gravitational (hydraulic) or from chemical (fuel) sources. The Steam-engine. — The last half of the eighteenth century was LOt merely an era of great revolutions : it was also an age of great [iventions and among these, first in importance as well as first to rise, was the steam-engine. Various and more or less successful attempts to utilize heat or team as a source of power had been made before Watt's time, such, or example, as those of Hero in Alexandria (120 B.C.) the Marquis if Worcester (1663) and Newcomen (1705). Of these only New- omen's need be dwelt upon here. In Newcomen's engine a vertical ylinder with piston was used, the piston-rod, also vertical, being fixed bove to one end of a walking-beam of which the other end carried a larallel rod. Thus the rise and fall of the piston caused a corre- ponding fall and rise of a parallel rod, which could be attached to ny thing, e.g. to a pump. The cylinder was connected with a steam 1 Nobel died in 1896, bequeathing his fortune, estimated at $9,000,000, to the funding of a fund which supports the international "prizes" — -usually $40,000 ach — which bear his name and are annually awarded to those who have most ontributed to "the good of humanity." Five prizes have been usually given: iz. one in physics, one in chemistry, one in medicine or physiology, one in terature and one for the promotion of peace. APPENDIX I: INVENTIONS 441 boiler by a pipe fitted with a stopcock, and was filled with steam below the piston by opening the stopcock. The steam pressing upon the boiler raised the piston and depressed the parallel (pump) rod. The stopcock was then closed, a "vent" in the cylinder was opened, cold water was introduced from another pipe to condense the steam, where- upon a vacuum formed, and the atmospheric pressure depressed the piston and lifted the pump rod. By having the various stopcocks carefully worked by hand a certain regularity of operation could be obtained, but before long improvements were made and the stop- cocks were caused to work automatically. But since the cold (con- densing) water chilled the cylinder, much heat was necessarily wasted. Watt began by inventing (in 1765) a separate condenser, for cooling the steam without cooling the cylinder, — thus saving a vast amount of heat. He next abandoned altogether the use of atmospheric pres- sure for depressing the piston, employing steam above as well as below the piston, to lower as well as to lift it : and with these improve- ments, to which he added many others, he soon had in his possession a serviceable and automatic steam-engine, rudimentary in many respects, but not essentially unlike that of to-day. The Spinning Jenny, the Watek-Feame and the Mule. — In 1770 James Hargreaves patented the spinning jenny, a frame with a number of spindles side by side, by which many threads could be spun at once instead of only one, as in the old, one-thread, distafF or the spinning wheel. In 1771 Arkwright operated successfully in a mill a patent spinning machine which, because actuated by water power, was known as the "water-frame." In 1779 Crompton com- bined the principles involved in Hargreaves' and Arkwright's machines into one, which, because of this hybrid origin, became known as the spinning "mule." This proved so successful that by 1811 more than four and a half milhon spindles worked as "mules" were in operation in England. A similar machine for weaving was soon urgently needed, and in 1785 the "power loom" of Cartwright appeared, although it required much improvement and was not widely used before 1813. The Cotton Gin (engine). — With the inventions just described facilities arose for the manufacture of cotton as well as woollen, but the supply of raw cotton was limited, chiefly because of the difficulty of separating the staple (fibres) from the seeds upon which they are borne. Cotton had for centuries been grown and manufactured in 442 A SHORT HISTORY OF SCIENCE India, the fibres being separated from the seeds by a rude hand ma- chine known as a churka, used by the Chinese and Hindus. By this it was impossible to clean cotton rapidly. The invention there- fore in 1793 by Eli Whitney of Connecticut of the saw cotton- gin which enormously facilitated this separation was one of the most important inventions ever made. This consisted in a series of saws revolving between the interstices of an iron bed upon which the cotton was so placed as to be drawn through while the seeds were left behind. The value of the saw gin was instantly recognized and the output of cotton in America was rapidly and immensely increased by its use. Steam Transpoktation. — Boats and ships propelled by man power or by the wind have been used from time immemorial, and parallel rails for wheeled conveyors moved by animal power or by gravity preceded the steam locomotive. The steamboat and the steam vehicle appeared at (or in the case of the latter even before) the opening of the nineteenth centiu-y. The first practically successful steamboat was a tug, the Charlotte Dundas, built and operated in Soctland for the towing of canal boats by Symmington in 1802. The first commercially successful steam- boat was Fulton's Clermont, on the Hudson, in 1807. The first steam-engine to run on roads appears to have been Cugnot's in France in 1769. The first to run on rails was Trevithick's, in 1804, built to fit the rails of a horse railway. This engine also discharged its exhaust steam into the funnel to aid the draught of the furnace, — a device of fundamental importance to the further development of the loco- motive. The first practically successful locomotive was Stephenson's Rocket (1829). The compound (double or triple expansion) engine, which dates from 1781 (Hornblower), 1804 (Woolf), and 1845 (McNaughton), embodies what is perhaps the greatest single improvement in the steam-engine in the nineteenth century. The turbine has begun to replace the reciprocating engine only very recently (1900). The Achromatic Compound Microscope. — The compound microscope, after its introduction about the middle of the seventeenth century, and its use by Malphigi, Kircher, Leeuwenhoek, Grew, and others, was of only limited value because of the spherical, and espe- cially the chromatic, aberration of its lenses. This remained true until long after Huygens had perfected the eye-piece of the telescope. APPENDIX I: INVENTIONS 443 and Hall and DoUand had succeeded in correcting chromatic aberra- tion in telescope objectives by the combination of crown and flint glass, in the eighteenth century. Amici, of Modena, in 1812, Fraunhofer of Munich in 1816, TuUy of London ui 1824, J. J. Lister in 1830 and others gradually per- fected the achromatic microscope objective, so that about 1835 really excellent instruments became accessible to microscopical investiga- tors. The numerous discoveries in cellular biology and in pathology which soon followed testify to the extent and importance of these improvements. Illuminating Gas, — made by the destructive distillation of coal, was invented and introduced in 1792 by Wilham Murdock, who in 1802 had so far perfected the process that even the exterior of his factory in Birmingham was illuminated with gas in celebration of the peace of Amiens. Friction Matches, — were preceded early in the nineteenth century by splinters of wood coated with sulphur and tipped with a mixture of chlorate of potash and sugar. These when touched with sulphuric acid ignited. It was not, however, until 1827 that practical friction matches were made and sold. These were known, after their inventor, as "Congreves" and consisted of wooden splints coated with sulphur and tipped with a mixture of sulphide of antimony, chlorate of potash, and gum. When subjected to severe friction, specially arranged for, these took fire. The phosphorus friction match was introduced commercially in 1833. The Sewing-Machine. — Very few labor-saving inventions sur- pass in efficiency sewing-machines. These also were invented in the nineteenth century and had a gradual development, in which various inventors participated. The first which need be mentioned was that of a French tailor, named Thimonier, patented in 1830. It is said that although made of wood and clumsy, eighty of these machines were in use in Paris in 1841, when an ignorant mob wrecked the establishment in which they were located and nearly murdered the inventor. The most important ideas embodied in modern ma- chines are, however, of strictly American origin, the work of Walter Hunt of New York, and of Elias Howe of Spencer, Massachusetts being of principal importance (1846). Other Americans, especially Singer, Grover, Wilson and Gibbs, afterwards contributed to the present excellence and variety of the sewing-machine. 444 A SHORT HISTORY OF SCIENCE Photography. — Scheele, the Swedish chemist, appears to have been the first to study the efPeet of sunhght on silver chloride. Others, including Rumford and Davy, observed the chemical properties of light, but it was Wedgwood who, in 1802, made the first photograph by throwing shadows upon white paper moistened with nitrate of silver. Wedgwood was unable, however, to fix his prints. Daguerreotypes, taken on silver plated copper, date from 1839, and were made by covering the copper with a thin film of silver iodide, — a compound sensitive to light. The image was developed by mer- cury vapor and fixed by sodium hyposulphite. The discovery of the fixing power of hyposulphite was in itself alone of immense impor- tance. With the name of Daguerre, who began experimenting in 1826, that of a fellow countryman and partner, Niepce, is intimately asso- ciated. The subsequent development of photography is due to a host of workers. The collodion film which underlies all modern work was first introduced in 1850. It is said to be a practically perfect medium because totally unaifected by silver nitrate. Anesthesia. The Ophthalmoscope. — Anaesthesia, or insen- sibility to pain, during dental surgical operations was introduced, if not discovered, by Wells, a dentist of Hartford, Connecticut, who himself took nitrous oxide gas for anaesthesia in 1844. The first public demonstration of surgical anaesthesia under ether was made by a dentist, Morton, and a sm-geon, Jackson, at the Massachusetts General Hospital in Boston in 1846. Antesthesia by chloroform was introduced by Simpson of Edinburgh, in 1847. The ophthalmoscope, an instrument for examination of the inte- rior of the eye, of inestimable value to medicine, was invented by Helmholtz in 1851. It is said that when von Graefe, an eminent ophthalmologist, first saw with it the interior of the eye he cried out, "Helmholtz has unfolded to us a new world." India-rubber, — the coagulated and dried juice of the rubber tree, first reported by Herrera, " who in the second voyage of Columbus observed that the inhabitants of Hayti played a game with balls made ' of the gum of a tree ' and that the balls although large were lighter, and bounced better, than the windballs of Castile," was at the end of the eighteenth century still a curiosity, employed by Priestley, among others, as an eraser or "rubber." Rubber is a hydrocarbon soft when pure but readily hardened by APPENDIX I: INVENTIONS 445 ' vulcanization," i.e. treatment with sulphur or certain sulphur com- pounds (chloride, carbon bisulphide), a process introduced by Good- year in 1839. Electrical Apparatus; Telegraph, Telephone, Electric Lighting, Electric Machinery. — The first important applica- tion of electricity to the service of man was the telegraph. This is too well known to require more than the briefest description. An electric circuit in a wire "made" or "broken" at one point is likewise made or broken at all other points. Hence, it is only necessary to employ a preconcerted system of make-and-break signals to dispatch messages. This plan was first employed in 1836 by S. F. B. Morse, a native of Charlestown, Massachusetts, and the first telegraph line between two cities was installed between Baltimore and Washington in 1844. The first transatlantic cable was laid in 1858. The telephone, invented by Alexander Graham Bell, is even more familiar. This, also, depends on the making and breaking of an elec- tric circuit, not (as is usual in the telegraph) by a key manipulated by the finger, but by sound waves of the human voice impinging upon a delicate membrane (the transmitter) and reproduced at a distance by corresponding vibrations of another delicate membrane (the receiver). Wireless telegraphy and wireless telephony differ from ordinary telegraphy and telephony merely in the use of signal waves set up in the ether instead of signal waves {i.e. making and breaking) set up in the current carried by a wire. Both arts are inventions of very recent date. The electric light, which had long been known as a laboratory experiment, became of practical utility about 1880, with the inven- tion of the incandescent lamp, first the carbon arc and then the car- bon filament, the former by Brush, the latter by Edison. The phonograph was invented by Edison in 1876, and was the culmination of attempts extending over many years to record and reproduce sound waves. In these attempts Young, Konig, Fleeming Jenkin and many others participated. Food Preserving by Canning and Refrigeration. — In 1810 Appert of France succeeded in preserving foods in closed vessels by heating and sealing while hot. In 1816 a small amount of food pre- served in this way found its way into the British Navy, where its value was recognized to some extent as a preventive of scurvy. It was not, however, until after the American Civil War that the industry 446 A SHORT HISTORY OF SCIENCE began to assume anything like the vast extent and importance it has since reached. Refrigeration in various forms has been used for food preserving probably from the earliest times, but the present enormous industry of cold storage has all grown up since the middle of the nineteenth century with the invention and development of refrigerators (domestic and commercial) and especially of machines for producing and dis- tributing compressed air or other vapors or brine ammonia and other liquids at very low temperatures. These have been perfected rather rapidly since 1860, but did not becomje common before 1880. The first cargo of fresh meat successfully exported from America to Europe was shipped in March, 1879, and from New Zealand to Europe in February, 1880, arriving after a passage of 98 days in excellent condition. The Internal-Combustion Engine. — For a century or there- abouts the steam-engine stood without a rival as a thermodynamic machine and prime mover. Innumerable attempts had been made meantime to construct other kinds of engines to convert heat more directly into power for mechanical work ; but it was not until 1876 that the internal-combustion engine as improved by Otto became a practical success. In the steam-engine, the furnace in which the heat is generated is external to the cylinder in which that heat does its work, the steam being merely an intermediary. It is therefore an external-combus- tion engine. Obviously, if the fuel burned is made to liberate its heat in the cylinder instead of the furnace, the steam can be dis- pensed with. This is what actually happens in the internal-com- bustion engine. The present enormous extent of the use of such engines for motors of all kinds, testifies to the importance of this invention. Aniline, — ■ was first obtained from indigo in 1826 by Unverdorben and named by him crystalline. In 1834 Runge prepared a similar substance from coal tar, and in 1841 Fritsche obtained from indigo an oil which he called aniline, — a word derived from the Sanskrit Nila, the indigo plant. The commercial importance of aniline in the dye-stuffs industry dates from the discovery of mauve by Perkin in 1858. This was the first of the notable series of aniline dyes now so well known, and the forerunner of the immense color industry of to-day. The Manufactuee of Steel; Bessemer. — The making of steel APPENDIX I: INVENTIONS 447 by the decarbonization of cast-iron, a process which initiated what has been called the "age of steel," was introduced by Bessemer (1813- 1898) in 1856. Bessemer's attention was drawn to the subject by his recognition of the necessity of improving gun-metal. Bessemer's process was at first only partially successful, but since others have shown how to improve it (by the addition of spiegeleisen, etc.) it has reached enormous proportions. Agricultueal Apparatus and Inventions. — Beginning about 1850 an era of improved agricultural apparatus began, of which one result has been the opening of vast tracts of farm lands which might otherwise have remained unproductive. Steel plows, better harrows, mowing-machines, horse-power rakes, haymaking machinery, and especially harvesters of ingenious design for cereal crops (first intro- duced by McCormick in 1834), threshing-machines and spraying- machines are to-day common, where these were almost unknown before 1875. Machinery has also been applied to dairying, first to the making of butter and cheese, and more recently even to the milking of cows. Progress has also been made in the preservation of milk and of eggs by condensing, drying, freezing, etc. by new and economical processes invented and applied since that time. Applied Science. Engineering. — Very much as discoveries and inventions blend together and as both spring from a common source, manifested as curiosity, inquiry, experimentation and cor- relation (i.e. from science), so applied science, including engineering, comes from a common ancestry, i.e. from correlated knowledge, — which is science. Both terms are loosely used and both cover to-day a multitude of diversified human activities. With the progress of science, arts and invention, engineering and other forms of applied science have developed so that these frequently have their own schools, either with or apart from universities and colleges ; the school for miners at Freiberg, in Saxony, begun in 1765, being now only one of hundreds of technological and scientific schools for the training of engineers and others. Up to 1850 most engineers in America were trained in military schools and were primarily military engineers. But from that time forward the civil, as opposed to the military, engineer began to appear, and from the parent stem of civil engineering we now have mechanical, mining, electrical, sanitary, chemical, marine and other branches of engineering, often highly specialized. The term " engineer" is now very widely employed, with 448 A SHORT HISTORY OF SCIENCE more or less appropriateness, to occupations remote from those of the mihtary or civil engineer, as for example, the "illuminating engineer," the " efficiency engineer," the " public health engineer," etc. We may soon expect to have added to these many others, such as the agricul- tural engineer, the forest engineer and even the fishery engineer. An historical sketch of applied science and engineering would ob- viously include the work of Archimedes, Vitruvius, Frontinus, and Leonardo, and proceed with the applications made of the discov- eries and inventions of the Renaissance and modern times. Some of this ground is covered in the present volume, and more of it in the series of books by Smiles entitled Lives of the Engineers. — There is scarcely a department of science or art which is the same, or at all the same, as it was fifty years ago. A new world of inventions — of railways and of telegraphs — has grown up around us which we cannot help seeing; a new world of ideas is in the air and affects us, though we do not see it. — Bagehot. Physics and Politics (1868). — Only since continental ideas and influences have gained ground in this country (Great Britain) has the word science gradually taken the place of that which used to be termed natural philosophy or simply philosophy. One reason why science forms such a prominent feature in the culture of this age is the fact that only within the last hundred years has scientific research approached the 'more intricate phenomena and the more hidden forces and conditions which make up and govern our everyday life. The great inventions of the sixteenth, seventeenth and eighteenth centuries were Tnade without special scientific knowledge, and frequently by persons who possessed skill rather than learning. They greatly influenced science and promoted knowledge, but they were brought about mare by accident or by the prac- tical requirements of the age than by the power of an unusual insight acquired by study. But in the course of the last hundred years the scientific investigation of chemical and electric phenomena has taught us to disentangle the intricate web of the elementary forces of nature, to lay bare the many interwoven threads, to break up the equilibrium of actual existence, and to bring within our power and under our control forces of undreamed-of magnitude. The great inventions of former ages were made in countries where practical life, industry and commerce were most advanced ; but the great inventions of the last fifty years in chemistry and electricity and the science of heat have been made in the scientific laboratory: the former were stimulated by practical wants; the latter themselves produced new practical requirements, and created new spheres of labor, industry, and commerce. Science and knowledge have in the course of this century overtaken the march of practical life in many directions. — Merz. SKETCH MAP SHOWING PLACES IMPORTANT IN ANCIENT AND MEDIAEVAL SCIENCE «u.l HO.ff nit. I Some Important Names, Dates and Events in the History of Science and Civilization (For certain earlier events, see Chapters I and II.) c. = circa, about. 1 General History, Literatcre, Science I 1 Art, etc. c. 2000-1700 . Ahmes Papyrus. c. 1100. c. 1000. c. 850. 0. 800. Gades {Cadiz} founded by the Phoenicians. Homer. David. Solomon. Carthage founded. Hesiod. c. 753. Mome founded (legend- & ary). 3 c. 700. Nineveh flourishes under 4> Sennacherib. 5 c. 640-546. Thales. c. 660. Byzantixim founded. in c. 611-545. Anaximander. 610. Sappho and other Greek 5 poets. -4-» O Necho II undertakes to 1 connect Miver Nile and S Bed Sea by Canal. His sailors circumnavigate Africa- o c. 606. Nineveh destroyed. n c. 588-524. Anaximenes. c. 600. Marseilles founded. c. 582-500. Pythagoras. c. 560. CrcBsus and Solon. 1 c. 576-480. Xenophanes. c. 550-478. Confucius. o c. 540-475. Heraclitus. c. 538. Babylon taken by Cyrus. f 03 c. 539. Parmenides. 525-456. jEsohylus. c. 500. Alomaeon. c. 500. Carthaginians explore c. 500-428. Anaxagoras. west coast of Africa. c. 470. Hippocrates of Chios 490-429. Pericles. (Mathematician). 490. Marathon, Battle of. d 469-399. Socrates. c. 484-425. Herodotus. m c. 465. Empedooles. 480. Thermopylm, Battle of. t? 5 c. 460. Leucippus. 480. Salamis, Battle of § 0. 460-370. Democritus. o c. 460. Hippocrates of Cos 480-406. Euripides. Phidias. a CPhvsician"). 450-385. Aristophanes. c. 428-347. 427-347. c. 420. 0. 408-? 2g Archytas. Plato. Hippias. Eudoxus. 450-400. Thucydides. , 434-359. Xenophon. . 430. The plague at Athens. 449 450 A SHORT HISTORY OF SCIENCE Geneeal History, Literatdkb, Science Art, etc. c. 400. Meton (Calendar). c. 400. Motion of Earth (Philolaus). 384-322. c. 370. Demosthenes. Diogenes. Scopas. Prax- 384-322. Aristotle. iteles. ^ 375-325. Menfechmus. 356-323. Alexander the Great. 3 4-> c. 375. Heraclides of Pontus. 338. Chxronea, Battle of. a 372-287. Theophrastus. A c. 350-260. Zeno (Stoic). 332. Alexandria founded. ^1 a 342-270. Epicurus. & c. 330-275. Euclid. c. 326. Eudemus. 323-30. 0. 300. 300. 283. The Ptolemies, I-VI. Museum and Library of Alexandria. Epicurus. The Pharos built at c. 300. Herophilus. Erasistra^ Alexandria. o tus. 280. The Colossus of Rhodes. 3 287-212. Archimedes. c. 276-194. Erastosthenes. 4> 270- Aristarchus. c. 286. Theocritus. •2 c. 260-200. Apollonius. IS 238. Decree of Canopus 269. Silver money first coined H {Leap Tear). c. 210. in Borne. The Great Chinese Wall begun. Paper made in China. d c. 170. Polybius. m c. 166. Terence. ^ 161. Philosophers and Bheto- ■s c. 135. Ctesibius. ricians banished from c. 146-126. Hipparchus. Borne. a 146. Carthage destroyed {re- 1 built in 123). CHRONOLOGY 451 General History, Literature, Science Art, etc. 98-55. Lucretius. 106-43. Cicero. c. 70- Geminus. 102-44. C»sar. 0. 63 B.c- 24A.D. 59 B.C.-17 A.D. u Strabo. Livy. Varro. 54 B.C.-3C »A.D. B 4^- Julian Calendar. Seneca. g 14. VitruYius. Be Archi- 47. Cassar takes Alexandria. •4-* tectura. 39. Pollio founds First CO ■s. Public Library. 27. End of Boman Republic. Golden Age of Boman Literature. (Horace, Virgil, Livy, etc.) a Nicomachus. < 1" 23-79. PUny. g c. 40-103 u Frontinus. |c.75. Hero. & ■a c. 130. Galen. > 410-485. Proolus. i O .a to 476. Brahmagupta. Martianus Capella {Liberal Arts) . 476. Fall of Borne. g^c 480-524. Boethius. 630. Arya-bhata. 529. 569-632. Edict of Justinian. Schools of Athens closed. Mohammed. a o CO 781-790. Schools of Alcuin. o c. 830. Algebra of Alkarismi. 940-1003. Gerbert (Pope Syl- vester II). 980-1037. Aviceima. a o 1000. Bhaskara. 1038. Alhazen. 622. 641. 711. 732. c. 742-814. The Hegira. Fall of Alexandria. Moorish Conquest of Spain. Moorish Invasion of Western Europe checked by Charles Martel. Charlemagne. 962. Holy Soman Empire. Abelard. 1066. Battle of Hastings. 1096-1270. Crusades. CHRONOLOGY 453 1 General History, Literature, Science Art, etc. Arabic numerals. 5 1113-1160. Translations of Greek Classics g from Arabic. .a 1126-1198. Averroes. H Jordanus Nemora- 1 rius. H 1176. 1206-1280. Pisano (Fibonacci)- Albertus Magnus. 1210. Aristotle^s Physics 1216. Magna Gharta. proscribed in c. 1254-1324. Marco Polo. Paris. 1266-1321. Dante Alighieri. jCj 1214-1294. Roger Bacon. ■s ( ,. 1219. University of Bo- 4> logna. c. 1300. Spectacles invented. V4 2 1249. University College, H 1284. Oxford. Peterhouse College, Cambridge. Mongolian Observa- tory at Meraga. 1304-1374. Petrarch. 5 1337-1453. The Hundred Tears' 1.364. University of War. € Vienna. c. 1340-1400. Chaucer. 3 o 1340-1450. The Black Death. (ii 1401-1464. 142.3-1461. Nicolas of Cusa. Peurbaoh. 1379-1446. Brunelleschi. ^ 1436-1476. Regiomontanus. 1444-1611. Bramante. 3 Tartar Observatory c. 1450. Invention of Printing. a {Samarcand). 1453. Fall of Constantinople ^ 1452-1519. Leonardo da Vinci. to the Turks. g 1473-1543. Copernicus. 1471-1528. Diirer. 1486-1567. Stifel. 1492. Discovery of America. ■1-1 1490-1855. Agrioola. 1497. Vasco da Gama rounds 1493-1541. Paracelsus. Cape of Good Hope. 1501-1576. Cardan. V 1503. ' Margarita philoso- io phica. 454 A SHORT HISTORY OF SCIENCE Science General Histoet, Liteeatuke, Aet, etc. 1510-1558. c. 1606-1559. 1510-1589. 1512-1694. 1514-1564. 1514-1576. 1616-1565. 1522-1565. 1540-1603. ij 1543. 154.3-1615. 1544-1603. 1546-1601. 1548-1600. 1548-1620. 1650-1617. ^ 1560-1621. I 1561-1626. ^ 1564-1642. 1571-1630. 1575-1660. 1577-1644. 1578-1657. 1682. 1591-1626. 1593-1662. 1596-1650. 1598-1647. 1601-1665. 1602-1686. 1608-1647. >. 1616-1703. B 1623-1662. 3 1624-1689. 3 1627-1691. S 1629-1696. S 1630-1677. I 1685-1703. " 1635-1672. 1642-1727. 1646-1716. Recorde. Tartaglia. Palissy. Mercator. Vesalius. Rheticus. Gesner. Ferrari. Vieta. De Bevolutionibus of Copernicus. Baptista deUa Porta. Gilbert. Tj'cho Brahe. Bruno. Stevinus. Napier. Harriott. Francis Bacon. Galileo. Kepler. Oughtred. Van Helmont. W. Harvey. Gregorian Calen- dar. Snellius. Desargues. Descartes. Cavalieri. Fermat. Von Guerioke. Torricelli. Wallis. Pascal. Sydenham. Boyle. Huygens. Barrow. Hooke. Willughby. Newton. Leibnitz. 1513. 1517. 1519-1522. 1524-1580. 1530. 1547-1616. c. 1552-1599. 1564-1616. 1573-1637. Balboa reaches Pacific Ocean. Protestant Beforma- tion. First Circumnaviga- tion of the Globe by Magellan. Camoens. Spinning wheel. Cervantes. Spenser. Shakespeare. Ben Jonson. 1588. 1598. 1600-1681. Calderon, Defeat of the Spanish Armada. Edict of Nantes. 1605. 1607. 1608-1774. 1618-1648. 1622-1673. 1631-1700. 1636. 16.38-1715. 1639-1699. Don Quixote. First Permanent Eng- lish Colony in America. Milton. Thirty Tears' War. Molifere. Dryden. Harvard College founded. Louis XIV. Racine. CHRONOLOGY 455 So lENOE General History, Litebathre, Art, etc. 1637. Discours sur la Me- 1649-1660. English Common- I thode. {Analytic wealth. Geometry.) fi 1644-1710. Eoemer. s 1656-1742. Halley. 1660-1731. Be Foe. 1660-1734. Stahl. 3 4^ 1668-1738. Boerhaave. 1672-1726. Peter the Great. i 1677-1761. Hales. 5 1687. Principia of New- 1683. Siege of Vienna by I ton. Turks. f 1698-1746. Maclaurin. 1688-1744. Pope. > 1699-1739. Dufay. 1694-1778. Voltaire. m 1699-1777. 1700-1782. 1705. 1706-1790. 1707-1778. 1707-1783. 1707-1788. Jussieu. Bernouilli, D. Newcomen^s En- gine. Franklin. Linnseus. Euler. BuHon. 1707-1777. Haller. 1709-1784. Samuel Johnson. 1717-1783. d'Alembert. 1711-1776. Hume. 1726-1797. Hutton. 1728-1793. Hunter. 1728-1774. Goldsmith. 1731-1810. Cavendish. 1732-1790. Washington. t 1733-1804. Priestley. 3 1736-1813. 1736-1819. Lagrange. "Watt. 1737-1794. Gibbon. 5 1738-1822. Herschel, F. W. 9> 1742-1786. Scheele. I 1743-1794. Lavoisier. s 1744-1829. Lamarck. 1746-1818. 1749-1827. Monge. Laplace. 1749-1832. Goethe. 1750-1817. 1762-1833. 1763-1814. Werner. Legendre. Rumford. 1759-1796. 1759-1805. Bums. Schiller. 1764. Watt's Steam En- gine. 1766-1844. Dalton. 1767. Spinning Jenny. 1769-1832. 1769-1859. Cuvier. Humboldt. 1769-1821. Napoleon I. 456 A SHORT HISTORY OF SCIENCE Science General Histort, Literature, Art, etc. 1769. Spinning Frame. 1770-1850. Wordsworth. 1773-1829. Young. 1771-1832. Scott. 1774. Discovery of Oxy- 1772-1834. Coleridge. gen. 1773-1859. Mettemich. 1775-1836. Ampfere. 1775-1781. American Revolution. 1776-1839. Treviranus. 1777-1855. Gauss. ig- 1778-1829. Davy. 1778-1841. De Candolle. 1779-1848. Berzelius. 1 1781-1848. Stephenson. 1781. 1783. Discovery of Ura- nus. Air Balloon. u 1784-1846. Bessel. 1789-1794. French Sevolution. Parallax of stars. 1791-1867. Faraday. J3 1791-1872. Morse. M 1792. 1792-1876. 1793-1856. Cotton Gin. von Baer. Lobatchewski. 1792-1822. Shelley. 1794. Ecole Polytech- 1795-1821. Keats. nique. 1795-1881. Carlyle. 1796. Vaccination. 1796-1882. Camot. 1799-1853. St. Hilaire. 1801-1858. Miiller. 1803-1873. Liebig. 1802-1885. "Victor Hugo. 1807-1873. Agassiz, L. 1802-1894. Kossuth. 1809-1882. Darwin. 1803-1882. Emerson. 1810-1888. Gray, Asa. 1804-1865. Cobden. 1811-1877. Leverrier. 1805. Battle of Trafalgar. 1811-1890. Bunsen. 1805-1872. Mazzini.