Sidebilder
PDF
ePub

as his chief title to fame the inscription of the cylinder within a sphere, and it was this figure which in after years made known his tomb to Cicero.

The development which the science has taken, especially since the introduction of the decimal system, the discovery of logarithms, and the like has been extensive. Beside it the mathematics of the ancients seem in many ways crude enough. The management of their systems of notation, in multiplication, in division, to say nothing of their struggles with fractions, would be to us simply distracting. To-day the schoolboy solves countless problems with an ease which would have filled Archimedes with breathless admiration. The mere introduction of the zero must have simplified operations, as measured by their rapidity, at least a dozen times. It seems incredible now that its advantages should not have been earlier perceived. There is some evidence that the Indian or decimal system was known in Europe before the rise of the Arabs; but it found no general introduction until it was borrowed from the Arabs at about the beginning of the thirteenth century. It does not appear to have been in use among the Hindus earlier than perhaps the fifth or sixth century. It was certainly not known either to the Greek or Latin mathematicians.

We gain a glimpse of the time from the fact that the clumsy abacus was then in universal use, as it is among the Chinese still. It consisted simply of beads or pebbles strung on parallel rods. Our word "calculate" from calx, "a stone," is a survival from this rude counting-machine. They had no algebra even; although Aristotle and even old Ahmes employed letters to represent indeterminate quantities, the swifter use of algebraic equations did not come in until some centuries later. Here, as elsewhere, we perceive the exceeding slowness with which new methods and new devices make their way.

Despite the crudity of their devices, our respect for the ancient mind is heightened rather than dulled by the applications which they made of the means at hand. In this later time we have come to perceive that the knowledge of antiquity, the range of its ideas, was far more extensive than had long been supposed. The high development of Greek philosophy, itself no doubt more or less a rescript of a yet more ancient time, is sufficient evidence of the ancient's capacity for abstract reasoning. In the pages that follow we shall find many acute

and astonishing examples of their powers in the far more difficult analysis of phenomena. Their employment of mathematical methods was constant, its applications were wide.

One of the earliest was the geometry of light. It must have been recognised far back that light travels in straight lines, and that its reflection has a sharp angle, equal in value to the angle of incidence. These facts were very early utilised in fixing the positions of the heavenly bodies and in effecting the beginnings of our cosmic knowledge. They are still the source of by far the larger part of our information as to the world in which we live.

By means of the eclipses men were able to understand that the sun lies back of the moon, and to unravel the mystery that lay in the puzzling and apparently inexplicable motions of the planets. It was by means of the eclipses, as we shall see, that a little later the Greeks were able to gather some idea of the relative distances of the moon and sun, some idea of their respective grandeurs as well.

It must have been very early that attentive minds observed that the light of the sun comes to us in practically parallel lines. On the day of the solstice when the sun seemed to stand still in the heavens, then began its winter retreat, there was a belt of considerable extent, several hundred stadia in width, in which perpendicular objects cast no shadow. This zone lay across the middle reaches of the Nile, just where Egyptian civilisation attained its highest efflorescence. The fact must have been deeply pondered by the observing priests. Its implications could hardly have escaped their wondering minds. A few hundred years later it seems to have been employed by Poseidonius to compute the distance of the sun, with an approximate success that is still amazing.

When we trace out the history of ideas we find as a rule that their lineage is long. In some of the earliest of Greek manuscripts that have come down, and again in the pages of Cicero, of Cleomedes, and a dozen others of that later day, we find perfectly correct notions of the earth and its immediate surroundings. These conceptions, analogy leads us to believe, must have been very old.

Inferences of such moment, deductions of such power, imply that far in the ancient time the idea of fixity in phenomena had been deeply impressed upon at least a slender body of

minds. This was largely fruit from the application of exact, that is to say mathematical, methods of reasoning. To it, one set of phenomena in especial must have powerfully contributed. This was the regularity of recurrence in eclipses. When our knowledge of a given set of phenomena is so certain that we may rise to the prediction of future events, there comes a consciousness of certitude which can be inspired in no other way. The art of eclipse predictions was known among the earliest of the Greek philosophers of whom we have authentic report. The successful issue of a venture of this sort made Thales seem to the simple mortals about him a god-like intelligence. In a larger sense than that in which it was conceived by the marvelling Ionians this expressed a literal truth; for among the attributes which we may conceive of divinity surely a knowledge of the past and of the future must be one. The same story is told of Democritus of Abdera. Both of them had dwelt long in Egypt. Doubtless it was from the Egyptian priesthood that both of them had borrowed their art. It was certainly known among the Chaldeans hundreds of years before.

We know, too, that mathematical methods were very early applied in physical investigations, notably to that of sound. Whewell gives high credit to Plato as having been among the first to teach that phenomena were capable of numerical treatment, as opposed to the empty definitions of philosophers like Aristotle. In this he was but a disciple of Pythagoras, and Pythagoras in turn had drunk of the founts of ancient knowledge, that is to say, of Egypt.

Of far greater moment was the application of geometry to the determination of the figure, and eventually the measure of the earth. It was the first step in a true knowledge of the cosmos-that is to say, towards a rational conception of the world. It carried the mind into regions the feet of man could never traverse, that his eyes might never see.

How, thus hobbled, could he attain to a certainty that all the after flood of years would not disturb ?

CHAPTER V

BION AND THE DOCTRINE OF A ROUND

EARTH

Long centuries, millenniums even, doubtless elapsed between what we may term the tame gorilla stage and that in which the wonder-working slave had become augur, astrologer, alchemist, and austromant. It was long ages later still when from these rude beginnings, more than half mysterious, we may imagine, even to their practisers, that anything like a rational world conception, to say nothing of a mechanical world conception, could arise. Let us look about for the first steps.

Sometimes in the history of ideas the history of a word may shed a deal of light. It is of some curious interest to know that once learning, knowledge, and mathematics were one. This indeed was the meaning of the Greek verb mathemata, "to know." For the rest, one of the most primitive of human needs was a method of counting. If we look about a little we shall perceive that the earliest idea of fixity, of certainty, of inevitable sequence, must have arisen from the sense of fixity of numbers.

When man had learned to put together 2 and 2, to subtract, to multiply, and divide, he must have seemed face to face with a great light. Mere numbers now seem to us such lifeless abstractions, that the idea they could have once been almost the basis of a religion seems absurd. We know, however, that this was true. No doubt a curious sense of coincidence—that is, of mystery-must have come to the first man who noted that the first four figures of the counting scale, put together, sum up ten. Remember that all systems of notation were decimal systems, arising from the chance number of fingers upon the two hands. The Pythagoreans named the first four figures the "grand tetrad." There was a similar mystery about the number "seven," the number of planets or "wanderers observed in the sky. The perfect number was twelve, doubtless because to them there were twelve lunations in the year, and because it divided into so many even parts. Even to Plato, numbers seemed the basis of all things.

Long before this a system of notation had been devised; its beginnings were crude enough. A single stroke counted for one, two strokes set side by side for two, and so on. This grew very clumsy before many digits had been told off, so various combinations were tried. The Babylonian character for ten, Grotefend believes to have been a rude picture of the two hands

« ForrigeFortsett »