CHAPTER II.

THE MATHEMATICIANS.

§ I. ANAXIMANDER OF MILETUS.

"As we now, for the first time in the history of Greek Philosophy, meet with contemporaneous developments, the observation will not perhaps be deemed superfluous that in the earliest times of philosophy, historical evidences of the reciprocal influence of the two lines either entirely fail or are very unworthy of credit; on the other hand, the internal evidence is of very limited value, because it is impossible to prove a complete ignorance in one, of the ideas evolved and carried out in the other; while any argument drawn from an apparent acquaintance therewith is far from being extensive or tenable, since all the olden philosophers drew from one common source-the national habit of thought. When indeed these two directions had been more largely pursued, we shall find in the controversial notices sufficient evidence of an active conflict between these very opposite views of nature and the universe. In truth, when we call to mind the inadequate means at the command of the earlier philosophers for the dissemination of their opinions, it appears extremely probable that their respective systems were for a long time known only within a very narrow circle. On the supposition, however, that the philosophical impulse of these times was the result of a real national want, it becomes at once probable that the various elements began to show themselves in Ionia nearly at the same time, independently and without any external connection."*

* Ritter, i. 265.

The chief of the school we are now about to consider was Anaximander of Miletus, whose birth may be dated in the 42d Olympiad (B. c. 610). He is sometimes called the friend and sometimes the disciple of Thales. We prefer the former rela tion; the latter is at any rate not the one in which this history can regard him. His reputation, both for political and scientific knowledge, was very great; and many important inventions are ascribed to him, amongst others that of the sun-dial and the sketch of a geographical map. His calculations of the size and distance of the heavenly bodies were committed to writing in a small work, which is said to be the earliest of all philosophical writings. He was passionately addicted to mathematics, and framed a series of geometrical problems. He was the leader of a colony to Apollonia; and he is also reported to have resided at the court of the tyrant Polycrates, in Samos, where also lived Pythagoras and Anacreon.

No two historians are agreed in their interpretation of Anaximander's doctrines; few indeed are agreed as to the historical position he is to occupy.

Anaximander is stated to have been the first to use the term apy for the Beginning of things. What he meant by this term principle is variously interpreted by the ancient writers; for, although they are unanimous in stating that he called it the infinite (ò äespov), what he understood by the infinite is yet undecided.*

On a first view, nothing can well be less intelligible than this tenet: "The Infinite' is the origin of all things." It either looks like the monotheism of a far later date,t or like the word-jugglery of mysticism. To our minds it is neither more nor less

* Ritter, i. 267.

Which it certainly could not have been. To prevent any misconception of the kind, we may merely observe that the Infinite here meant, was not even the Limitless Power, much less the Limitless Mind, implied in the modern conception. In Anaxagoras, who lived a century later, we find rò arsipov to be no more than vastness.-See Simplicius, Phys. 33, b, quoted in Ritter.

difficult of comprehension than the tenet of Thales, that "Water is the origin of all things." Let us cast ourselves back in imagination into those early days, and see if we cannot account for the rise of such an opinion.

On viewing Anaximander side by side with his great predecessor and friend, Thales, we cannot but be struck with the exclusively abstract tendency of his speculations. Instead of the meditative Metaphysician, we see a Geometrician. Thales, whose famous maxim, "Know thyself," was essentially concrete, may serve as a contrast to Anaximander, whose axiom, "The Infinite is the origin of all things," is the ultimate effort of abstraction. Let us concede to him this tendency; let us see in him the geometrician rather than the moralist or physicist; let us endeavor to understand how all things presented themselves to his mind in the abstract form, and how mathematics was the science of sciences, and we shall then perhaps be able to understand his

tenets.

Thales, in searching for the origin of things, was led, as we have seen, to maintain water to be that origin. But Anaximander, accustomed to view things in the abstract, could not accept so concrete a thing as Water: something more ultimate in the analysis was required. Water itself, which in common with Thales, he held to be the material of the universe, was it not subject to conditions? What were those conditions? This Moisture, of which all things are made, does it not cease to be moisture in many instances? And can that which is the origin of all, ever change, ever be confounded with individual things? Water itself is a thing; but a Thing cannot be All Things.

These objections to the doctrine of Thales caused him to reject, or rather to modify, that doctrine. The dpx, he said, was not Water; it must be the Unlimited All, rò nepov.

Vague and profitless enough this theory will doubtless appear. The abstraction "All" will seem a mere distinction in words. But in Greek Philosophy, as we shall repeatedly notice, distinctions in words were generally equivalent to distinctions in things.

And if the reader reflects how the mathematician, by the very nature of his science, is led to regard abstractions as entities,-to separate form, and treat of it as if it alone constituted body,there will be no difficulty in conceiving Anaximander's distinction between all Finite Things and the Infinite All.

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It is thus only we can explain his tenet; and this explanation seems borne out by the testimony of Aristotle and Theophrastus, who agree, that by the Infinite he understood the multitude of elementary parts out of which individual things issued by separation. By separation:" the phrase is significant. It means the passage from the abstract to the concrete,-the All realizing itself in the Individual Thing. Call the Infinite by the name of Existence, and say, "There is existence per se, and Existence per aliud; the former is Existence, the ever-living fountain whence flow the various existing Things." In this way we may, perhaps, make Anaximander's meaning intelligible.

Let us now hear Ritter. Anaximander "is represented as arguing that the primary substance must have been infinite to be all-sufficient for the limitless variety of produced things with which we are encompassed. Now, although Aristotle especially characterizes this infinite as a mixture, we must not think of it as a mere multiplicity of primary material elements; for to the mind of Anaximander it was a Unity immortal and imperishable -an ever-producing energy. This production of individual things he derived from an eternal motion of the Infinite."

The primary Being, according to Anaximander, is unquestionably a Unity. It is One yet All. It comprises within itself the multiplicity of elements from which all mundane things are composed; and these elements only need to be separated from it to appear as separate phenomena of nature. Creation is the decomposition of the Infinite. How does this decomposition originate? By the eternal motion which is the condition of the Infinite. "He regarded," says Ritter, "the Infinite as being in a constant state of incipiency, which, however, is nothing but a constant secretion and concretion of certain immutable ele

ments; so that we might well say, the parts of the whole are constantly changing, while the whole is unchangeable.”

The idea of elevating an abstraction into a Being-the origin of all things-is baseless enough; it is as if we were to say, "There are numbers 1, 2, 3, 20, 80, 100; but there is also Number in the abstract, of which these individual numbers are but the concrete realization: without Number there will be no numbers." Yet so difficult is it for the human mind to divest itself of its own abstractions, and to consider them as no more than as abstractions, that this error lies at the root of the majority of philosophical systems. It may help the reader to some tolerance of Anaximander's error to learn that celebrated philosophers of modern times, Hegel and others, have maintained precisely the same tenet, though somewhat differently worded: they say, that Creation is God passing into activity, but not exhausted by the act; in other words, Creation is the mundane existence of God; finite Things are but the eternal motion, the manifestation of the All.

Anaximander separated himself from Thales by regarding the abstract as of higher significance than the concrete: and in this tendency we see the origin of the Pythagorean school, so often called the mathematical school. The speculations of Thales tended towards discovering the material constitution of the universe; they were founded, in some degree, upon an induction from observed facts, however imperfect that induction might be. The speculations of Anaximander were wholly deductive; and, as such, tended towards mathematics, the science of pure deduction.

As an example of this mathematical tendency we may allude to his physical speculations. The central point in his cosmopœia was the earth; for, being of a cylindrical form, with a base in the ratio 1: 3 to its altitude, it was retained in its centre by the aid and by the equality of its distances from all the limits of the world.

From the foregoing exposition the Reader may judge of the propriety of that ordinary historical arrangement which places