340 ARTSTOTELES. M.-ithematics, 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 Swofxei (according to poten- tiality) in the abstractive operation of the under- standing, and is therefore no independent exist- ence, nothing substantiaL We arrive at the cognition of its peculiar definitions not from the idea, but only by means of separation (e. g. auxili- ary 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 abstractixm, 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. (Pfit/s. 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 inmuitabi- lity 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, Poli- tics (with Oeconomics as an appendix), and Poetics (Aesthetics, Philosophy of Art). 1. Ethics. 1. General Definilions.f — The highest and last purpose of all action, according to Aristotle, is luippiness {(vScufwvia. Kth. Nic. i. 2 — 7, x. b' — 8, and elsewhere). This he defines to be the energy {fV(pyiia) 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 dur- ing a separate independent period of existence. {Eth. Nic. i. 7.) The two principal component parts of this definition are virtue, and external
- The only mathematical work of Aristotle
(/Lia07jjuaT</coj/, Diog. Laert. v. 24) quoted by an- cient writers is lost. The method which was fol- lowed at a later time for mathematics, re-sts alto- gether on the doctrine of proof given in the Ana- Ij'tics. 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 Pytha- goreans, laid great stress on the separation of arithmetic and geometrv. (Anal. post. i. 27, Met. + In this review of the ethical system of Aris- totle we follow of course the progress of the Nico- niachean Ethics, as being the principal work. The first two books contain the general part of etliics, 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 ijitellectual virtues (5io- vot]TiKai), or moral virtues {-qQiKai)., according to the distinction between the reasoning faculty, and that in the soul which obeys the reason. Accord- ing to this distinction, the origin of the virtues, which Aristotle points out in the second book of the Ethics, is also different. The intellectual vir- tues may be learnt and taught, the ethical virtues are acquired by practice. In the case of these, therefore, we must liave 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. n. 2.) To be able to refrain from sensual desires with pleasure is to be ieinfKraie. The intemperate man experiences pain at such abstinence, when he is compelled to prac- tise it. By the practice of virtue the man becomes good himself; and virtue is therefore a habit, and that too accompanied bj^ fore-choice (e|is irf)oaipr]- rucri), which keeps the medium in our subjective inclinations and impulses {Eth. Nic. ii. 6), and keeps the medium in that way in which the rational man (d (ppovi/jLos) detennines. This me- dium 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 ex- tremes ; they contradict each other, and the proper measiu-e or degree depends on the pai'ticular incli- nations of the individual. 2. Sftccial 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 de- voted 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 {kyKparna) which re- sists pleasure, as in endurance {KapTfpia, 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 tJie essential ?iaiu7-e of pleasure and pai?i. {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 vir- tues, 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 founda- tion 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 exer- cise of each species of activity directed towards the good is accompanied by the feeling of an undis- turbed energy, and this harmony, in which the external and the internal are in accordance, pro-
