Science is exact, regular, arranged knowledge. Of common knowledge savages and barbarians have a vast deal, indeed the struggle of life could not be carried on without it. The rude man knows much of the properties of matter, how fire burns and water soaks, the heavy sinks and the light floats, what stone will serve for the hatchet and what wood for its handle, which plants are food and which are poison, what are the habits of the animals that he hunts or that may fall upon him. He has notions how to cure, and much better notions how to kill. In a rude way he is a physicist in making fire, a chemist in cooking, a surgeon in binding up wounds, a geographer in knowing his rivers and mountains, a mathematician in counting on his fingers. All this is knowledge, and it was on these foundations that science proper began to be built up, when the art of writing had come in and society had entered on the civilized stage. We have to trace here in outline the rise and progress of science. And as it has been especially through counting and measuring that scientific methods have come into use, the first thing to do is to examine how men learnt to count and measure.

Even those who cannot talk can count, as was well shown by the deaf-and-dumb lad Massieu, who wrote down among the recollections of his childhood before the Abbé Sicard educated him, “I knew the numbers before my instruction; my fingers had taught me them.” We ourselves as children began arithmetic on our fingers and now and then take to them still, so that there is no difficulty in understanding how a savage whose language has no word for a number above three will manage to reckon perhaps a list of fifteen killed and wounded, how he will check off one finger for each man, and at last hold up his hand three times to show the result. The next question is, how numeral words came to be invented. This is answered by many languages, which show in the plainest way how counting on fingers and toes led to making numerals. When a Zulu wants to express the number six, he says tatisitupa, which means “taking the thumb;” this signifies that the speaker has counted all the fingers of his left hand, and begun with the thumb of the right. When he comes to seven, for instance when he has to express that his master bought seven oxen, he will say u kombile, that is, “he pointed”; this signifies that in counting he had come to the pointing-finger or forefinger. In this way the words “hand,” “foot,” “man,” have in various parts of the world become numerals. An example how they are worked may be taken from the language of the Tamanacs of the Orinoco; here the term for five means “whole hand,” six is “one of the other hand,” and so on up to ten or “both hands”; then “one to the foot” is eleven, and so on to “whole foot” or fifteen, “one to the other foot” or sixteen, and thence to “one man,” which signifies twenty, “one to the hands of the next man” being twenty-one, and the counting going on in the same way to “two men” which stands for forty, &c. &c. Now this state of things teaches a truth which has sometimes been denied, that the lower races of men have, like ourselves, the faculty of progress or self-improvement. It is evident that there was a time when the ancestors of these people had in their languages no word for fifteen or sixteen, nor even for five or six, for if they had they could not have been so stupid as to change them for their present clumsy phrases about hands and feet and men. We see back to the time when, having no means of reckoning such numbers except on their fingers and toes, they found they had only to describe in words what they were doing, and such a phrase as “both hands” would serve them as a numeral for ten. Then they would keep up these as numerals after their original sense was lost, like the Vei negros who called the number twenty mo bande, but had forgotten that this must have meant “a person finished.” The languages of nations long civilized seldom show such plain meaning in their numerals, perhaps because they are so ancient and have undergone such change. But all through the languages of the world, savage or civilized, with exceptions too slight to notice here, there is ineffaceable proof that the numerals arose out of the primitive counting on fingers and toes. This always led men to reckon by fives, tens, and twenties, and so they reckon still. The quinary kind of counting (by fives) is that of tribes like the negros of Senegal, who count one, two, three, four, five, five-one, five-two, &c.; we never count numbers thus in words, but we write them so in the Roman numerals. The decimal counting (by tens) is the most usual in the world, and our ordinary counting is done by it, thus eighty-three is “eight tens and three.” The vigesimal counting (by twenties) which is the regular mode in many languages, has its traces left in the midst of the decimal counting of civilized Europe, as in English “fourscore and three,” French “quatre-vingt trois,” that is “four twenties and three.” Thus it can hardly be doubted that the modern world has inherited direct from primitive man his earliest arithmetic worked on nature’s counting-board—the hands and feet. This also explains (p. 18) why the civilized world uses a numeral system based on the inconvenient number ten, which will not divide either by three or four. Were we starting our arithmetic afresh, we should more likely base it on the duodecimal notation, and use dozens and grosses instead of tens and hundreds.

To have named the numbers was a great step, but words hardly serve beyond the very simplest arithmetic, as any one may satisfy himself by trying to multiply “seven thousand eight hundred and three” by “two hundred and seventeen” in words, without helping himself by turning them in thought into figures. How did men come to the use of numeral figures? To this question the beginning of an answer may be had from barbaric picture-writing, as where a North American warrior will make four little marks //// to show that he has taken four scalps. This is very well for the small numbers, but becomes clumsy for higher ones. So already when writing was in its infancy, the ancients had fallen upon the device of making special marks for their fives, tens, hundreds, &c., leaving the simple strokes to be used only for the few units over. This is well seen in Fig. 76 which shows how numeration was worked in ancient Egypt and Assyria. Nor has this old method died out in the world, for the Roman numerals I., V., X., L., still in common use among ourselves, are arranged on much the same principle. Another device, which arose out of the alphabet, was to take the letters in their order to stand for numbers. Thus the sections of Psalm CXIX. are numbered by the letters of the Hebrew alphabet, and the books of the Iliad by the letters of the Greek alphabet. By these various plans the arithmetic of the ancient civilized nations made great progress. Still their numeration was very cumbrous in comparison with that of the modern world. Let us put down MMDCLXIX. and multiply by CCCXLVIII., or β͵χʹξʹθʹ by τʹμʹηʹ, and a few minutes’ trial will not fail to convince us of the superiority of our ciphers.

To understand how the art of ciphering came to be invented, it is necessary to go back to a ruder state of things. In Africa, negro traders may be seen at market reckoning with pebbles, and when they come to five, putting them aside in a little heap. In the South Sea Islands it has been noticed that people reckoning, when they came to ten, would not put aside a heap of ten things, but only a single bit of coco-nut stalk to stand for ten, and then a bigger piece when they wanted to represent ten tens or a hundred. Now to us it is plain that this use of different kinds of markers is unnecessary, but all that the reckoner with little stones or beans has to do, is to keep separate his unit-heap, his ten-heap, his hundred-heap, &c. This use of such things as pebbles for “counters,” which still survives in England among the ignorant, was so common in the ancient world, that the Greek word for reckoning was psēphizein, from psēphos, a pebble, and the corresponding Latin word was calculare from calculus, a pebble, so that our word calculate is a relic of very early arithmetic. Now to work such pebble-counting in an orderly manner, what is wanted is some kind of abacus or counting-board with divisions. These have been made in various forms, as the Roman abacus with lines of holes for knobs or pegs, or the Chinese swan-pan with balls strung on wires, on which the native calculators in the merchants’ counting-houses reckon with a speed and exactness that fairly beats the European clerk with his pencil and paper. It may have been from China that the Russian traders borrowed the ball-frame on which they also do their accounts, and it is said that a Frenchman noticing it in Russia at the time of Napoleon’s invasion was struck with the idea that it would serve perfectly to teach little children arithmetic; so he introduced it in France, and thence it found its way into English infants’ schools. Now whatever sort of abacus is used, its principle is always the same, to divide the board or tray into columns, so that in one column the stones, beans, pegs, or balls, stand for units, in the next column they are tens, in the next hundreds, and so on, Fig. 77. Here the three stones in the right-hand column stand for 3, the nine in the next column for 90, the one in the fourth column for 1,000 and so on. The next improvement was to get rid of the troublesome stones or beans, and write down numbers in the columns, as is here shown with Greek and Roman numerals. But now the calculator could do without the clumsy board, and had only to rule lines on his paper, to make columns for units, tens, hundreds, &c. The reader should notice that it is not necessary to the principle of the abacus that each column should stand for ten times the one next it. It may be twelve or twenty or any other number of times, and in fact the columns in our account-books for £ s. d. or cwts. qrs. lbs., are surviving representatives of the old method of the abacus. Such reckoning had still the defect that the numbers could not be taken out of the columns, for even when each number from one to nine has a single figure to stand for it, there would still be here and there an empty column (as is purposely left in Fig. 77) which would throw the whole into confusion. To us now it seems a very simple thing to put a sign to show an empty column, as we have learned to do with the zero or 0, so that the number expressed in the picture of the abacus can be written down without any columns, 241093. This invention of a sign for nothing, was practically one of the greatest moves ever made in science. It is the use of the zero which makes the difference between the old arithmetic and our easy ciphering. We give the credit of the invention to the Arabs by using the term Arabic numerals, while the Arabs call them Indian, and there is truth in both acknowledgments of the nations having been scholars in arithmetic one to the other. But this does not go to the root of the matter, and it is still unsettled whether ciphering was first devised in Asia, or may be traced further back in Europe to the arithmeticians of the school of Pythagoras. As to the main point, however, there is no doubt, that modern arithmetic comes out of ancient counting on the columns of the abacus, improved by writing a dot or a round 0 to show the empty column, and by this means young children now work calculations which would have been serious labour to the arithmeticians of the ancient world.

Next as to the art of measuring. Here it may be fairly guessed that man first measured, as he first counted, on his own body. When barbarians tried by finger-breadths how much one spear was longer than another, or when in building huts they saw how to put one foot before the other to get the distance right between two stakes, they had brought mensuration to its first stage. We sometimes use this method still for rough work, as in taking a horse’s height by hands, or stepping out the size of a carpet. If care is taken to choose men of average size as measurers, some approach may be made to fair measurement in this way. That it was the primitive way can hardly be doubted, for civilized nations who have more exact means still use the names of the body-measures. Besides the cubit, hand, foot, span, nail, already mentioned in p. 17, we have in English the ell, (of which the early meaning of arm or fore-arm is seen in el-bow, the arm-bend), also the fathom or cord stretched by the outspread arms in sailors’ fashion, and the pace or double step (Latin passus) of which a thousand (mille) made the mile. But though these names keep up the recollection of early measurement by men’s limbs, they are now only used as convenient names for standard measures which they happen to come tolerably near to, as for instance one may go a long way to find a man’s foot a foot long by the rule. Our modern measurements are made by standard lengths, which we have inherited with more or less change from the ancients. It was a great step in civilization when nations such as the Egyptians and Babylonians made pieces of wood or metal of exact lengths to serve as standards. The Egyptian cubit-rules with their divisions may still be seen, and the King’s Chamber in the Great Pyramid measures very exactly 20 cubits by 10, the cubit being 20·63 of our inches. Our foot has scarcely altered for some centuries, and is not very different from the ancient Greek and Roman feet. The French at the first Revolution made a bold attempt to cast off the old traditional standards and go straight to nature, so they established the metre, which was to be a ten-millionth of the distance from the pole to the equator. The calculation however proved inexact, so that the metre is now really a standard measure of the old sort, but so great is the convenience of using the same measures, that the metre and its fractions are coming more and more into use for scientific work all over the world. The use of scales and weights, and of wet and dry measures, had already begun among the civilized nations in the earliest known times. Our modern standards can even to some extent be traced back to those of the old world, as for instance the pound and ounce, gallon and pint, come from the ancient Roman weights and measures.

From measuring feet in length, men would soon come to reckoning the contents, say of an oblong floor, in square feet. But to calculate the contents of less simple figures required more difficult geometrical rules. The Greeks acknowledged the Egyptians as having invented geometry, that is, “land-measuring,” and there may be truth in the old story that the art was invented in order to parcel out the plots of fertile mud on the banks of the Nile. There is in the British Museum an ancient Egyptian manual of mensuration (the Rhind papyrus), one of the oldest books in the world, originally written more than 1,000 years before Euklid’s time, and which shows what the Egyptians then knew and did not know about geometry. From its figures and examples it appears that they used square measure, but reckoned it roughly; for instance, to get the area of the triangular field ABC Fig. 78 (1) they multiplied half AC by AB, which would only be correct when BAC is a right angle. When the Egyptians wanted the area of a circular field, they subtracted one-ninth from the diameter and squared; thus if the diameter were 9 perches, they estimated that the circle contained 64 square perches, which the reader will find on trial is a good approximation. All this was admirable for the beginnings of geometry, and the record may well be believed that Greek philosophers such as Thales and Pythagoras, when they came to Egypt, gained wisdom from the geometer-priests of the land. But these Egyptian mathematicians, being a priestly order, had come to regard their rules as sacred, and therefore not to be improved on, while their Greek disciples, bound by no such scientific orthodoxy, were free to go on further to more perfect methods. Greek geometry thus reached results which have come down to us in the great work of Euklid, who used the theorems known to his predecessors, adding new ones and proving the whole in a logical series. It must be clearly understood that elementary geometry was not actually invented by means of definitions, axioms, and demonstrations like Euklid’s. Its beginnings really arose out of the daily practical work of land-measurers, masons, carpenters, tailors. This may be seen in the geometrical rules of the altar-builders of ancient India, which do not tell the bricklayer to draw a plan of such and such lines, but to set up poles at certain distances, and stretch cords between them. It is instructive to see that our term straight line still shows traces of such an early practical meaning; line is linen thread, and straight is the participle of the old verb to stretch. If we stretch a thread tight between two pegs, we see that the stretched thread must be the shortest possible; which suggests how the straight line came to be defined as the shortest distance between two points. Also, every carpenter knows the nature of a right angle, and he is accustomed to parallel lines, or such as keep the same distance from one another. To the tailor, the right angle presents itself in another way. Suppose him cutting a doubled piece of cloth to open out into the gore or wedge-shaped piece BAC in Fig. 78 (2). He must cut ADB a right angle, or his piece when he opens it will have a projection or a recess, as seen in the figure. When he has cut it right, so that BDC opens in a straight line, then he cannot but see that the sides AB, AC, and the angles ABC, ACB must exactly match, having in fact been cut out on one another. Thus he arrives, by what may be called tailor’s geometry, at the result of Euklid I. 5, which now often goes by the name of the “asses’ bridge.” Such easy properties of figures must have been practically known very early. But it is also true that the ancients were long ignorant of some of the problems which now belong to elementary teaching. Thus it has just been mentioned how the Egyptian land-surveyors failed to make out an exact rule to measure a triangular field. Yet had it occurred to them to cut out the diagram of a triangle from a sheet of papyrus, as we may do with the triangle ABC in Fig. 78 (3), and double it up as shown in the figure, then they would have found that it folds into the rectangle EFHG, and, therefore, its area is the product of the height by half the base. It would be seen that this is no accident, but a property of all triangles, while at the same time it would appear that the three angles at A, B, C, all folding together at D, make up two right angles. Though the more ancient Egyptian geometers do not seem to have got at either of these properties of the triangle, the Greek geometers had in some way become well aware of them before Euklid’s time. The old historians who tell the origin of mathematical discoveries do not always seem to have understood what they were talking of. Thus it is said of Thales that he was the first to inscribe the right-angled triangle in the circle, and thereupon sacrificed a bull. But a mathematician of such eminence could hardly have been ignorant of what any intelligent carpenter has reason to know, how an oblong board fits into a circle symmetrically; the problem of the right-angled triangle in the semicircle is involved in this, as is seen by (4) in the present figure. Perhaps the story really meant that Thales was the first to work out a strict geometrical demonstration of the problem. The tale is also told of Pythagoras, and another version is that he sacrificed a hekatomb on discovering that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides (Euklid I. 47). The story is not a likely one of a philosopher who forbad the sacrifice of any animal. As for the proposition, it is one which may present itself practically to masons working with square paving-stones or tiles; thus, when the base is 3 tiles long, and the perpendicular 4, the hypothenuse will be 5, and the tiles which form a square on it will just be as many as together form squares on the other two sides. Whether Pythagoras got a hint from such practical rules, or whether he was led by studying arithmetical squares, at any rate he may have been the first to establish as a general law this property of the right-angled triangle, on which the whole systems of trigonometry and analytical geometry depend.

The early history of mathematics seems so far clear, that its founders were the Egyptians with their practical surveying, and the Babylonians whose skill in arithmetic is plain from the tables of square and cube numbers drawn up by them, which are still to be seen. Then the Greek philosophers, beginning as disciples of these older schools, soon left their teachers behind, and raised mathematics to be, as its name implies, the “learning” or “discipline” of the human mind in strict and exact thought. In its first stages, mathematics chiefly consisted of arithmetic and geometry, and so had to do with known numbers and quantities. But in ancient times the Egyptians and Greeks had already begun methods of dealing with a number without as yet knowing what it was, and the Hindu mathematicians, going further in the same direction, introduced the method now called algebra. It is to be noticed that the use of letters as symbols in algebra was not reached all at once by a happy thought, but grew out of an earlier and clumsier device. It appears from a Sanskrit book that the venerable teachers began by expressing unknown quantities by the term “so-much-as,” or by the names of colours, as “black,” “blue,” “yellow,” and then the first syllables of these words came to be used for shortness. Thus if we had to express twice the square of an unknown quantity, and called it “so much squared twice,” and then abbreviated this to so sq 2, this would be very much as the Hindus did in working out the following problem, given in Colebrooke’s Hindu Algebra: “The square root of half the number of a swarm of bees is gone to a shrub of jasmin: and so are eight-ninths of the whole swarm: a female is buzzing to one remaining male, that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.” This Hindu equation is worked out clumsily from the want of the convenient set of signs = + -, which were invented later in Europe, but the minus numbers are marked, and the solution is in principle an ordinary quadratic. The Arab mathematicians learnt from India this admirable method, and through them it became known in Europe in the middle ages. The Arabic name given to it is al-jabr wa-l-mukabalah, that is, “consolidation and opposition,” this meaning what is now done by transposing quantities on the two sides of an equation; thence comes the present word algebra. It was not till about the 17th century in Europe that the higher mathematics were thoroughly established, when Descartes worked into a system the application of algebra to geometry, and Galileo’s researches on the path of a ball or flung stone brought in the ideas which led up to Newton’s fluxions and Leibnitz’s differential calculus, with the aid of which mathematics have risen to their modern range and power. Mathematical symbols have not lost the traces of their first beginnings as abbreviated words, as where n still stands for number and r for radius, while √, which is a running-hand r, does duty for root (radix), and ∫, which is an old-fashioned s, stands for the sum (summa) in integration.

Mechanics and Physics, worked mathematically, now form the very foundation of our knowledge of the universe. But in the old barbaric life, men had only rudimentary notions of them. The savage understands the path of a projectile well enough to aim it, and how to profit by momentum when he mounts his axe on a long rather than a short handle. But he hardly comes to bringing these practical ideas to a principle or law. Even the old civilized nations of the East, though they could lift stones with the lever, set their masonry upright with the plumb-line, and weigh gold in the balance, are not known to have come to scientific study of mechanical laws. What makes this more sure is that if they had, the Greeks would have learnt it of them, whereas it is among the Greek philosophers that the science is found just coming into existence. In Aristotle’s time they were thinking about mechanical problems, though by no means always rightly; it was considered that a body is drawn toward the centre of the world, but the greater its weight the faster it will fall. The chief founder of mechanical science was Archimedes, who worked out from the steel-yard the law of the lever, and deduced thence cases of all the particles of a body balancing on a common centre, now called its centre of gravity; he even gave the general theory of floating bodies, which mathematicians far on in the middle ages could hardly be brought to understand. Indeed, mechanical science, after the classical period, shared the general fate of knowledge during the long dead time when so much was forgotten, and what was left was in bondage to the theology of the schoolmen. It sometimes surprises a modern reader that the “wisdom of the ancients” should still now and then be set up as an authority in science. But the scholars of the middle ages, who on many scientific points knew less than the ancient Greeks, might well look up to them. It is curious to look at the book of Gerbert (Pope Sylvester II.) who was a leading mathematician in the tenth century, and who bungles like an early Egyptian over the measurement of the area of a triangle, though the exact method as stated by Euklid had been well known in classical times. Physical science might almost have disappeared if it had not been that while the ancient treasure of knowledge was lost to Christendom, the Mohammedan philosophers were its guardians, and even added to its store. For this they have not always had due praise. A pretty story is told of Galileo inventing the pendulum, being led to it by watching the great hanging lamp in the cathedral of Pisa swinging steadily to and fro; but as a matter of fact, it appears that six centuries earlier Ebn Yunis and other Moorish astronomers were already using the pendulum as a time-measurer in their observations. Of all the services which Galileo did for science, perhaps the greatest was his teaching clearer ideas of force and motion. People had of old times been deceived by the evidence of their senses into the belief that the force of a moving body would gradually become exhausted and it would stop of itself, but this idea of force was changed by the new principle that force is as much required to stop a moving body as to set it in motion, and that did no opposing force retard the arrow or the wheel, the one would fly and the other roll on for ever. In that age of mathematics applied to science new discoveries followed fast. If Archimedes could have come to life again, he would have seen progress going on at last, when the pressure of the air was weighed with Torricelli’s barometer, and Stevin of Bruges made out the principle of the parallelogram of forces. The notion of an attractive force had come into the minds of philosophers by observing how the magnet attracts iron at a distance, and glass and other substances when rubbed become attractive. Thus the way was open for Newton to calculate the effect of gravitation as such an attractive force, and by it to explain the movements of the heavenly bodies, thus bringing the visible world within the sway of one universal law. In the present day, among the great laws which have been established in physical science, is that of the conservation of energy, that power is not created and destroyed in the processes of nature or the machines of man, but is transformed into new manifestations equivalent to those which were before. Philosophers’ minds used often to be set on the invention of a perpetual moving power, that should go on creating its own force. But nowadays this idea is so discarded that, when some projector plans an absurd machine, he is sufficiently answered by being shown that if his machine could work, the perpetual motion would be possible. The modern mechanician has only to apply in the most desirable way the stores of force placed at his disposal by nature, and within this well-understood boundary his business flourishes more and more.

Among the forms or manifestations of energy are sound, light, heat, electricity. The classic philosophers knew in a vague way that sound spreads like waves; and the relation between the length of a harpstring and its note was laid down in arithmetical rule by Pythagoras, who measured it with the instrument we still use, the monochord. But it was the moderns who measured the velocity of sound, explained musical pitch by the rate of vibration, and made the science of tone. About light the ancients knew more. Their polished metal mirrors, flat and curved, had taught them the first principles of reflexion. Nor were they ignorant of refraction; they already knew the familiar experiment of putting a ring in a basin and pouring in water till it becomes visible. A rock-crystal lens has been dug up at Nineveh, and the Greeks and Romans were well acquainted with glass lenses. One is surprised that neither the Arab astronomers, who knew a good deal of optics, nor Roger Bacon, who in the thirteenth century gave an intelligent account of their science, ever seem to have combined two lenses into a telescope. It was not till the seventeenth century that a telescope is plainly mentioned in Holland, and Galileo, hearing of it, made the famous instrument with which he saw Jupiter’s moons, and revolutionized men’s ideas of the universe. The microscope and telescope may be called inverted forms of one another, and their inventions came nearly together. By these two instruments the range of man’s vision has been so vastly extended beyond his unaided eyesight, that animalcules under a ten-thousandth of an inch long can now be watched through all the stages of their life, while stars whose distance from the earth is hundreds of thousands of billions of miles, are within the maps of the universe. The rainbow led to the problem of the decomposition of light and the theory of colour. The doctrine that light was as it were bright particles emitted in straight lines from the luminous body, failed to explain effects such as light extinguishing light by interference, and it has yielded to the undulatory theory, of ethereal light-waves of extreme smallness and speed. In our own day the lines of the spectrum have become the means of recognising a glowing substance, so that the astronomer whose telescope reveals the faint shine of a nebula in the depths of the heavens, may test its composition with the spectroscope, as if it were a gas-jet on the laboratory table. Closely connected with the science of light, is the science of heat. Not only do heat and light proceed together from the sun or fire, but the two were seen to be subject to the same laws, when it was noticed that the mirror or lens which concentrated a bright spot of light, also brought to the same focus heat that would set wood on fire. The great step in the study of heat was the invention of the heat-measurer or thermometer. Who first made it is not known, but it was about three centuries ago, and its earliest form may have been the air-flask with its tube in which coloured water rises and falls, which is still the most striking way of showing a class the principle of thermometers. The doctrine of heat as due to vibration explains how heat is transformed force, so that the steam-hammer worked by the heat used in the furnace can be set to beat cold iron till it is white-hot; thus part of the force which came from heat has gone back into heat, and with the heat re-appears the other form of radiant energy, light. Lastly, the history of electricity comes from the time when the ancients wondered to see amber when rubbed pick up morsels of straw, and the loadstone draw bits of iron. The pointing of the loadstone south and north seems to have been earliest noticed by the Chinese, whence in the middle ages came its world-wide use in navigation. The electrical machine is only an enlarged form of the old experiment of rubbing the bit of amber. But the discoveries associated with the name of Volta and Galvani brought in a new method of generating electricity by chemical action in the battery. Franklin’s kite proved the lightning-flash to be but a great electric spark. Oersted’s current-wire deflecting a magnetic needle showed the relation between electricity and magnetism, and set on foot the line of invention to which the world owes the electric telegraph and much besides.

Next, as to chemistry. Its beginnings lie in practical processes such as smelting metal from the ore, fusing sand and soda into glass, and tanning leather with astringent pods or bark. The oldest civilized nations knew these and many other chemical arts, which not only were learnt by the artificers of Greece and Rome, but from time to time new processes were added to the store of knowledge, as when we hear of their distilling mercury from cinnabar, or treating copper with vinegar to make verdigris. In early civilized ages also there arose beside these practical recipes the first dim outlines of scientific chemistry. The Greek philosophers expressed their ideas of the states of matter by the four elements, fire, air, water, earth; and they also had learnt or invented the doctrine of matter being made up of atoms—a principle now more influential than ever in modern lecture-rooms. The successors of the Greeks were the Arabic alchemists, and their disciples in mediæval Christendom. Their belief that matter might be transmuted or transformed led many of them to spend their lives among their furnaces and alembics in the attempt to turn baser metals into gold. To modern chemists, who would not be surprised to find all the many so-called elements proved to be forms of one matter, the alchemists’ idea does not seem quite unreasonable in itself, and practically it led them to the pursuit of truth by experiment, so that though they found no philosophers’ stone, they were repaid by discoveries such as alcohol, ammonia, sulphuric acid. Their method, being founded on trials of real fact, cleared itself more and more from the magical folly it had grown up with, and the alchemist prepared the way for the later chemist. What of all things brought on the new chemical knowledge, was the explanation of what takes place in burning, rusting, and breathing. How is it that the air in a receiver is spoilt by a burning candle or a mouse within, so that it no longer allows flame or life? How is it that while some substances, like charcoal, seem to be dissipated by fire, others, like lead or iron, turn into matter heavier than before? The answers to such questions led the way to clearer notions of chemical combination, but it was long before it was understood by what fixed laws of affinity and proportion this combination takes place. The advanced student of chemistry may spend an instructive hour in looking over old chemistry books, where the catalogue of substances is a confused chaos, not as yet brought into form and order on the lines of Dalton’s atomic theory.

From the chemical nature of matter we pass to the nature of living things. The more evident parts of biology or the science of life, have come under man’s attentive observation from the first. So far as zoology and botany consist in noticing the forms and habits of animals and plants, savages and barbarians are skilled in them. Such people, for instance, as the natives of the South American forests, have names for each bird and beast, whose voices, resorts, and migrations they know with an accuracy that astonishes the European naturalist whom they guide through the jungle. The catalogue of the Brazilian native names of animals and plants, often curiously descriptive of their natures, would make a small book. Thus the jaguara pimina or painted jaguar is distinguished from the jaguarete or great jaguar; the capybara signifies the creature “living in the grass,” the ipe-caa-goene, or “little wayside-plant-emetic,” is our ipecacuanha. Mankind everywhere possesses this sort of popular Natural History. So it is with anatomy. When the savage kills a deer, cuts it up, cooks the joints, heart, and liver, makes clothes and straps of the hide, cuts harpoon-heads and awls out of the long bones, and uses the sinews for thread, it stands to reason that he must have a good rough knowledge of the anatomy of an animal. The barbaric warrior and doctor have beyond such butchers’ anatomy an acquaintance with the structure of man’s body, as may be seen in the description of the wounds of the heroes in the Iliad, where the spear takes one in the diaphragm below the heart, and another has the shoulder-tendon broken which makes his arm drop helpless. Among the Greeks such rough knowledge passed into the scientific stage when Aristotle wrote his book on animals, and Hippokrates took medicine away from the priests and sorcerers to make it a method of treatment by diet and drugs. The action of the body came to be better understood during this classical period, as, for instance, is seen in the nerves leading to and from the brain being no longer confounded with the sinews which pull the limbs, although the same Greek word neuron (nerve) still continued to be used for both. It is curious how long it took the ancients to get at the notion of what muscle is, and how it acts. They never understood the circulation of the blood, though they had ideas about it, as in Plato’s celebrated passage in the Timaios which compares the heart to a fountain sending the blood round to nourish the body, which is like a garden laid out with irrigating channels. Imperfect as ancient knowledge was, it may be plainly seen how modern science is based upon it. Thus the medical terms of Galen’s system, such as the diagnosis of disease, are still used; and indeed many old physician’s words have passed into common talk, as when one is said to be in a sanguine humour, which carries us back to the time when the humours or fluids of the body were thought to cause the state of mind, the humour which is sanguine, or “of the blood,” being lively and impetuous. But in knowledge of the body the moderns have left the ancients quite behind, now that the microscope shows its minute vessels and tissues, and there have been made out the circulation of the blood, the process of respiration, the chemistry of digestion, and the travelling of currents along the nerves. Natural History still goes on the principles of Aristotle, when he traces life on from lifeless matter through the series of plants and animals. Modern naturalists like Linnæus so improved the old classification, that it became possible to take a plant or animal one had never seen before and did not know the name of, and make out by examination that it must belong to such and such a genus and species. Moreover, naturalists have long been seeking to understand why the thousands of species should arrange themselves in groups or genera, the species in each genus being connected by a common likeness, and the genera themselves falling into higher groups, or orders. The thought that the likeness among the species forming a genus is a family likeness, due to these species being in fact the varied descendants of one race or stock, is the foundation of that theory of development or evolution which for many ages has been in the minds of naturalists, and now so largely prevails. This is not the place to discuss the doctrine of descent or development (see page 38), but it is worth while to remember that the very word genus meant originally birth or race, so that the naturalist who sets down the horse, ass, zebra, quagga, as all belonging to one genus Equus, is really suggesting that they are all descended from one kind of animal, and are in fact distant cousins, which is the first principle of the development-theory.

The world we live in is the subject of astronomy, geography, geology. It seems plain how the rudiments of these sciences began from the evidence of men’s senses. Children living unschooled in some wild woodland would take it as a matter of course that the earth is a circular floor, more or less uneven, arched over with a dome or firmament springing from the horizon. Thus the natural and primitive notion of the world is that it is like a round dish with a cover. Rude tribes in many countries are found thinking so, and working out the idea so as to account for such phenomena as rain, which is water from above dripping in through holes in the sky-roof. This firmament is studded with stars, and is a few miles off. There is nothing to suggest to the savage that the sun should be enormously more distant than the cloud it seems to plunge into. The sun seems to go down in the west into the sea, or through an opening in the horizon, and to rise in like manner in the east, so that sunset and sunrise force on the minds of the first rude astronomers the belief in an under-world or infernal region, through which the sun travels in the night, and which to many a nation has seemed also the abode of departed souls, when after their bright day of life they sink like the sun into the night of death. The sun and moon move as living gods in the heaven, or at least are drawn or driven by such celestial powers, while the presence of living beings in the sky seems peculiarly manifest in eclipses, when invisible monsters seize or swallow the sun and moon. All this is very natural, so natural indeed that more correct astronomy has not yet rooted it out of Europe. Not many years ago a schoolmaster who ventured to lecture on astronomy in the west of England roused the displeasure of the country folk, that this young man should tell them the world was round and went about, when they had lived on it all their lives and knew it was flat and stood still. One part of the earliest astronomy, which was so sound as to have held its own ever since, was the measurement of time by the sun, moon, and stars. The day and the month fix themselves at once. In a less exact way the seasons of the year, such as the rainy season, or the icy season, or the growing season, furnish a means of reckoning, as where a savage tells of his father’s death having been three rains or three winters ago. Rude tribes, who observe the stars to find their way by, notice also that the rising and setting of particular stars or constellations mark the seasons. Thus the natives of South Australia call the constellation Lyra the Loan-bird, for they notice that when it sets with the sun, the season for getting loan-birds’ eggs has begun. It stands to reason that the great facts of the year’s course, the change of the sun’s height at noon, and the lengthening and shortening of the days, would be noticed, so that even among people who have not as yet measured them with any accuracy, there exists in a loose way the notion of the year. Within the year, too, the successive moons or months come to be arranged with some regularity, as where the Ojibwas reckoned in order the wild-rice moon, the leaves-falling moon, the ice-moon, the snow-shoes moon, and so forth. But such lunar months have to be got into the year as they best may. Indeed what distinguishes the uncivilized calendar, is that though days, months, and years are known, the days are not yet fitted regularly into the months, nor is it settled how many months, much less how many days, the year is to consist of.

When we look from this to the astronomy of the ancient cultured nations, we find great progress made in observing and calculating. Yet the astronomer-priests who for ages watched and recorded the aspect of the heavens, had not yet cut themselves free from the ideas of their barbarian forefathers as to what the world as a whole was like. In the Egyptian Book of the Dead, the departed souls descend with the sun-god through the western gate, and travel with him among the fields and rivers of the under-world, and the Assyrian records also tell of the regions below, where Ishtar descends into the dark abode of fluttering ghosts, the house men enter but cannot depart from. Yet the Egyptians who held to this primitive astronomy had set the Great Pyramid by the cardinal points with remarkable exactness. In reckoning the year, they not only added to the 12 solar months of 30 days 5 intercalary days to make 365, but becoming aware that even this was not accurate, they recorded its variation till it should come round in a cycle of 1,461 years, as determined by the rising of Sirius. Even more advanced was the astronomy of the Chaldæans, with its records of eclipses extending over 2,000 years. In the astronomy of barbarians the five planets Mercury, Venus, Mars, Jupiter, and Saturn, are not thought much of in comparison with the Sun and Moon. But among the Chaldæans all the seven planets were classed together as objects of worship and observation, starting the ideas of the sacred number seven, which thence pervaded the mystical philosophy of the ancients. It may have been among the Babylonian astronomers that the study of the motions of the planets led to the theory that they were carried round on seven crystal spheres; to this day people talk of being “in the seventh heaven.” The next and great step in astronomy was when the long-treasured knowledge of Babylon and Egypt was taken up by the Greeks, to be carried on by the exact methods of the geometer. The Greek astronomers were familiar with the idea of the earth being a sphere; they calculated its circumference, and usually taking it as the centre of the universe, they measured the apparent movements of the heavenly bodies. This system, in its most perfect form known as the Ptolemaic, held its place into the middle ages, when it came into rivalry with the Copernican system of a central sun round which revolve the earth and other planets. How this became in the hands of Kepler and Newton a mechanical theory of the universe, and how man was at last stripped of the fond conceit that his little planet was the centre of all things, need not be re-told here.

Geography is a practical kind of knowledge in which the rudest tribes are well skilled, so far as it consists in the lie of their own land, the course of the streams, the passes over the mountains, how many days’ marches through forest and desert to reach some distant hunting-ground, or the hill-side where hard stone for hatchets is to be found. However uncivilized a people may be, they name their mountains and rivers in such terms as “red hill” or “beaver brook.” Indeed the atlas contains hundreds of names of places that once had meanings in tongues which no man any longer speaks. Scientific geography begins when men come to drawing maps, an art which perhaps no savage takes to untaught, but which was known to the early civilized nations; the oldest known map is an Egyptian plan of the gold-mines of Æthiopia. The earliest known mention of a geographer attempting a map of the world is by Herodotus, who tells of Aristagoras’s bronze tablet inscribed with the circuit of the whole earth, the sea and all rivers. But to the ancients the known world was a very limited district round their own countries. It brings the growth of geography well before our minds to look at the map in Gladstone’s Juventus Mundi, representing the world according to the Homeric poems, with its group of nations round the Mediterranean, and the great Ocean River encircling the whole. Later, in the world as known to geographers such as Strabo, the lands of men form a vast oval, reaching from the pillars of Herakles across to far India, and from tropical Africa up to polar Europe. How land and sea came to lie as they do, it is the business of geology to explain. This is among the most modern of sciences, yet its problems had long set rude men thinking. Even the Greenlanders and the South Sea Islanders have noticed the fossils inland and high on the mountains, and account for them by declaring that the earth was once tilted over, or that the sea rose in a great flood and covered the mountains, leaving at their very tops the remains of fishes. In the infancy of Greek science, Herodotus speculated more rightly as to how the valley of Egypt had been formed by deposits of mud from the Nile, while the shells on the mountains proved to him that the sea had once been where dry land now is. But two thousand years had to pass before these lines of thought were followed up by the modern geologists, to whom the earth is now revealing the long history of the deposit and removal, rising and sinking of its beds, and the succession of plants and animals which from remote ages have lived upon it.

From this survey of the various branches of science, it is clear that their progress has been made in age after age by facts being more fully observed and more carefully reasoned on. Reasoning or logic is itself a science, but like other sciences, it began as an art which man practised without stopping to ask himself why or how. He worked out his conclusions by thinking and talking, untold ages before it occurred to him to lay down rules how to argue. Indeed, speech and reason work together. A language which distinguishes substantive, adjective, and verb, is already a powerful reasoning-apparatus. Men had made no mean advance toward scientific method when their language enabled them to class wood as heavy or light, and to form such propositions as, light wood floats, heavy wood sinks. The rise of reasoning into the scientific stage was chiefly due to the Greek philosophers, and Aristotle brought argument into a regular system by the method of syllogisms. Of course the simpler forms of these had always belonged to practical reasoning, and a savage, aware that red-hot coals burn flesh, would not thank a logician for explaining to him that in consequence of this principle a particular red-hot coal will burn his fingers. It must not be supposed that the introduction of logic as a science had the effect of at once stopping bad argument, and it was rather by setting practically to work on exact reasoning, especially in mathematics, that the Greeks brought on a general advance in knowledge. The importance of science was recognised when the famous Museum of Alexandria flourished, the type of later universities, with its great libraries, its laboratories, its zoological and botanical gardens. Hither students came by thousands to follow mathematics, chemistry, anatomy, under professors who resorted there at once to teach others and to learn themselves. Looking at the history of science for eighteen hundred years after this flourishing time, though some progress was made, it was not what might have been expected, and on the whole things went wrong. The so-called scholastic period which prevailed in Europe was unfavourable, partly because excessive reverence for the authority of the past fettered men’s minds, and partly because the learned successors of Aristotle had come to believe so utterly in argumentation as to fancy that the problems of the world could be dealt with by arguing about them, without increasing the stock of real knowledge. The great movement of modern philosophy with which the name of Bacon is associated as a chief expounder, brought men back to the sound old method of working experience and thought together, only now the experience was more carefully sought and observed, and thought arranged it more systematically. We who live in an age when every week shows new riches of nature’s facts, and new shapeliness in the laws that connect them, have the best of practical proof that science is now moving on a right track.

The student who wishes to compare the mental habits of rude and ancient peoples with our own, may look into a subject which has now fallen into contempt from its practical uselessness, but which is most instructive in showing how the unscientific mind works. This is Magic. In the earlier days of knowledge men relied far more than we moderns do on reasoning by analogy or mere association of ideas. In getting on from what is known already to something new, analogy or reasoning by resemblance always was, as it still is, the mind’s natural guide in the quest of truth. Only its results must be put under the control of experience. When the Australians picked up the bits of broken bottles left by the European sailors, the likeness of the new material to their own stone flakes at once led them to try it for teeth to their spears; experience proved that in this case the argument from analogy held good, for the broken glass answered perfectly. So the North American Indian, in default of tobacco, finds some more or less similar plant to serve instead, such as willow-bark. The practical knowledge of nature possessed by savages is so great, that it cannot have been gained by mere chance observations; they must have been for ages constantly noticing and trying new things, to see how far their behaviour corresponded with that of things partly like them. And where the matter can be brought to practical trial by experiment, this is a thoroughly scientific method. But the rude man wants to learn and do far more difficult things—how to find where there is plenty of game, or whether his enemies are coming, how to save himself from the lightning, or how to hurt some one he hates, but cannot safely throw a spear at. In such matters beyond his limited knowledge, he contents himself with working on resemblances or analogies of thought, which thus become the foundation of magic. On looking into the “occult sciences,” it is easy to make out in them principles which are intelligible if one can only bring one’s mind down to the childish state they belong to. Nothing shows this better than the rules of astrology, although this is far from the rudest kind of magic. According to the astrologers, a man born under the sign Taurus is likely to have a broad brow and thick lips, and to be brutal and unfeeling, but when enraged, violent and furious. If he had been born under the sign Libra, he would have had a just and well-balanced mind. All this is because two particular groups of stars happen to have been called the bull and the balance; the child whose hour of birth has some sort of astronomical relation to these constellations is imagined to have a character resembling that of a real bull or a real pair of scales. So with the planets. He over whom Mars presides in his better aspect will be bold and fearless, but where the planet is “ill-dignified,” then he will be a boastful shameless bully, ready to rob and murder. Had he but been born when Venus was in the ascendant, how different would he have been, with dimpled cheek and soft voice apt to speak of love. Practically foolish as all this is, it is not unintelligible. There is in it a train of thought which can be followed quite easily, though it is a train of thought hardly strong enough for a joke, much less for a serious argument. Yet such is the magic which still pervades the barbaric world. The North American Indian, eager to kill a bear to-morrow, will hang up a rude grass image of one and shoot it, reckoning that this symbolic act will make the real one happen. The Australians at a burial, to know in what direction they may find the wicked sorcerer who has killed their friend, will take as their omen the direction of the flames of the grave-fire. The Zulu who has to buy cattle may be seen chewing a bit of wood, in order to soften the hard heart of the seller he is dealing with. The accounts of such practices would fill a volume, and they do not seem broken-down remains of old ideas, for there is no reason to suppose they ever had more sense in them than is to be plainly seen now. They may be derived from some such loose savage logic as this:—Things which are like one another behave in the same way—shooting this image of a bear is like shooting a real bear—therefore, if I shoot the image I shall shoot a real bear. It is true that such magical proceedings, if tested by facts, prove to be worthless. But if we wonder that nevertheless they should so prevail among mankind, it may be answered that they last on even in our own country among those who are too ignorant to test them by facts—the rustics who believe a neighbour’s ill-wishing has killed their cow, and who, on true savage principles, try to punish the evil-doer by putting a heart spitefully stuck full of pins up the chimney to shrivel in the smoke, that in like manner sharp pangs may pierce him and he may waste away.

In another and very different way the student of science is interested in magic. Loose and illogical as man’s early reasonings may be, and slow as he may be to improve them under the check of experience, it is a law of human progress that thought tends to work itself clear. Thus even the fancies of magic have been sources of real knowledge. Few magical superstitions are more troublesome than the Chinese geomancy or rules of “wind and water,” by which a lucky site has to be chosen for building a house. Absurd as this ancient art is, its professors appear to have been the earliest to use the magnetic compass to determine the aspects of the heavens, so that it seems the magician gave the navigator his guide in exploring the world. What exact science owes to astrology is well known, how in Chaldæa the places of the stars were systematically observed and recorded for portents of battle and pestilence, and registers of lucky and unlucky days. The old magical character hung to astronomy even into modern ages, when astrologers like Tycho Brahe and Kepler, who believed that the destinies of men were foretold by the planets, helped by their observation and calculation to foretell the motions of the planets themselves. Thus man has but to go on observing and thinking, secure that in time his errors will fall away, while the truth he attains to will abide and grow.