A Dictionary of Greek and Roman biography and mythology. By various writers. Ed. by William Smith. Illustrated by numerous engravings on wood.

340 ARISTOTELES.
Mathematics, therefore, stand half-way between
physics and metaphysics. (Met. i. 6, p. 20, 23,
i. 9, p. 33, 23, xi. 1. p. 212, 22.) Mathematical
existence exists only 3vvaiLs (according to potentiality) in the abstractive operation of the understanding, and is therefore no independent existence, nothing substantial. We arrive at the
cognition of its peculiar definitions not from the
idea, but only by means of separation (e. g. auxiliary lines in figures for proof). On that account,
neither motion nor the idea of purpose occurs in
mathematics. (Met. iv. 2, Phys. ii. 9.) In this
science, that which is simple, as an abstractum,
forms the starting-point, and its necessity depends
on our advancing from the simple to the composite,
or from the basis to that which is based upon it.
(Phys. ii. 9.) Respecting the axioms from which
the mathematical sciences proceed, mathematics
can therefore say nothing (Met. iv. 3), because
these belong to every existing thing as such.-*
Respecting the view taken by Aristotle of the
mathematical sciences, see Biese, ii. pp. 225-234.
B.
THE PRACTICAL SCIENCES.
Mathematics, restricted as the science is to the
quantitative, can exhibit the good and the beautiful
only as they manifest themselves in that immutability which consists in the fixed order and harmony
of the quantitative. But the way in which these
two, the good and the beautiful, acquire existence
in the department of the mind, is considered and
pointed out by the practical sciences, Ethics, Politics (with Oeconomics as an appendix), and Poetics
(Aesthetics, Philosophy of Art).
1. Ethics.
1. General Definitions.t-The highest and last
purpose of all action, according to Aristotle, is
happiness (satiastovia. Eth. Nic. i. 2-7, x. 6-8,
and elsewhere). This he defines to be the energy
(ievp-yTa) of life existing for its own sake (perfect
life), according to virtue existing by and for itself
(perfect virtue). As the highest good, it must be
pursued for its own sake; as the highest human
good, its essence must be derived from the peculiar
destination of man. Accordingly, happiness is the
activity of the soul in accordance with virtue during a separate independent period of existence.
(Eth. Nie. i. 7.) The two principal component
parts of this definition are virtue, and external
S The only mathematical work of Aristotle
(pjaOrivarnTiKLv, Diog. Laert. v. 24) quoted by ancient writers is lost. The method which was followed at a later time for mathematics, rests altogether on the doctrine of proof given in the Analytics. Aristotle probably composed no separate
treatises on arithmetic and geometry. In his
Organon he frequently borrows examples from
geometry. Aristotle, as an opponent of the Pythagoreans, laid great stress on the separation of
arithmetic and geometry. (Anal. post. i. 27, Met.
v. 6.)
+ In this review of the ethical system of Aristotle we follow of course the progress of the Nicomachean Ethics, as being the principal work. The
first two books contain the general part of ethics,
the remaining eight books carry out the definitions
of this portion more closely.

ARISTOTELES.
good circumstances as means of virtue. Virtues
are of two kinds, either intellectual virtues (iLavo',ruiai), or moral virtues (7'itcal), according to
the distinction between the reasoning faculty, and
that in the soul which obeys the reason. According to this distinction, the origin of the virtues,
which Aristotle points out in the second book of
the Ethics, is also different. The intellectual virtues may be learnt and taught, the ethical virtues
are acquired by practice. In the case of these,
therefore, we must have regard to the practice of
them in particular cases; therefore, only quite
general directions admit of being given respecting
them. Youth must be accustomed and trained
"to rejoice and be sorry in the proper way," for
grief and joy are the criteria of virtue, inasmuch
as it is the proper medium between excess and
deficiency. (Eth. Nic. ii. 2.) To be able to
refrain from sensual desires with pleasure is to be
temperate. The intemperate man experiences pain
at such abstinence, when he is compelled to practise it. By the practice of virtue the man becomes
good himself; and virtue is therefore a habit, and
that too accompanied by fore-choice (Zis lrpoaimp)rn/), which keeps the medium in our subjective
inclinations and impulses (Elh. Nic. ii. 6), and
keeps the medium in that way in which the
rational man ( (ppviuos) determines. This medium assumes different forms according to the
several impulses, under the influence of which the
actor has reference either solely to himself, or to
others also. The medium is opposed to the extremes; they contradict each other, and the proper
measure or degree depends on the particular inclinations of the individual.
2. Special part. - Virtue is based upon free,
self-conscious action. Aristotle, therefore, before
developing the several virtues specially, defines
the idea of responsibility (iii. 1-7), and then and
not before gives the development of the ethical
(iii. 8, v. extr.) and logical (vi.) virtues. As now,
in the definition of happiness, virtues and the
means of virtue formed the chief parts, so the
second section of the special part of ethics is devoted to the internal and external circumstances of
life, which become the means of virtue through
the good manifesting itself in them as the purpose.
Continuance in a course of virtue is connected
chiefly with firmness of character, which exhibits
itself as well in abstinence (4yKpdTreia) which resists pleasure, as in endurance (icaupepia, a Platonic
idea: see Plat. Laches), which remains unshaken,
even by the attacks of pain. (Eth. Nic. vii. 1-12.)
This firmness therefore manifests itself especially
in the manner in which a man demeans himself
towards pleasure and pain. This leads to the
investigation of the essential nature of pleasure and
pain. (Eth. Nic. vii. 12, &c.) Farther, in the
social life of men, friendship, which is itself a
virtue (viii. 1), and indeed the crown of all virtues, is a principal means for a steady continuance
in virtue. Aristotle, therefore, in the 8th and 9th
books, treats of friendship with the most careful
explicitness. He shews that it forms the foundation for all kinds of unions, and contributes to the
realization of the good in the smaller and larger
circles of social life. Lastly, the unrestricted exercise of each species of activity directed towards the
good is accompanied by the feeling of an undisturbed energy. and this harmony, in which the
external and the internal are in accordance, pro