I.Speculation(θεωρία).—Since things are individuals, and there is nothing, and nothing universal, beyond them, there are two kinds of knowledge (γνῶσις), sense (αἴσθησις) of individuals, intellect (νοῦς) of universals. Both powers know by being passively receptive of essence propagated by an efficient cause; but, while in sense the efficient cause is an external object (ἔξωθεν), in intelligence it is active intellect (νοῦς τῷ ποιεῖν) propagating its essence in passive intellect (νοῦς παθητικός). Nevertheless, without sense there is no knowledge. Sense receives from the external world an essence,e.g.of white, which is really universal as well as individual, but apprehends it only as individual,e.g.this white substance: intellect thereupon discovers the universal essence but only in the individuals of sense. This intellectual discovery requires sensation and retention of sensation; so that sense (αἴσθησις) receives impressions, imagination (φαντασία) retains them as images, intellect (νοῦς) generalizes the universal, and, when it is intelligence of essence, is always true.This is the origin of knowledge, psychologically regarded (in theDe Anima). Logically regarded, the origin of all teaching and learning of an intellectual kind is a process of induction (ἐπαγωγή) from particulars to universal, and of syllogism (συλλογισμός) from universal to further particulars; induction, whenever it starts from sense, becomes the origin of scientific knowledge (ἐπιστήμη); while there is also a third process of example (παράδειγμα) from particular to particular, which produces only persuasion. In acquiring scientific knowledge, syllogism cannot start from universals without induction, nor induction acquire universals without sense. At the same time, there are three species of syllogism, scientific, dialectical and eristical or sophistical; and in consequence there are different ways of acquiring premisses. In order to acquire the knowledge of the true and primary principles of scientific knowledge, and especially theintelligence of the universal essence of the subject, which is always true, the process of knowledge consists of (1) sense (αἴσθησις), which receives the essence as individual, (2) memory (μνήμη), which is a retention of sensible impression, (3) experience (ἐμπειρία), which consists of a number of similar memories, (4) induction (ἐπαγωγή), which infers the universal as a fact (τὸ ὅτι), (5) intellect (νοῦς), which apprehends the principle (ἀρχή); because it is a true apprehension that the universal induced is the very essence and formal cause of the subject: thereupon, scientific syllogism (ἐπιστημονικὸς συλλογισμός), making the definition (ὁρισμος) of this essence the middle term (τὸ μέσον), becomes a demonstration (ὁρισμος) of the consequences which follow from the essence in the conclusion. Such then is science. In order to acquire the probabilities (τὰ ἔνδοξα) of opinion (δόξα), which are the premisses of dialectical syllogism, the process is still induction, as in science, but dialectical induction by interrogation from the opinions of the answerers until the universal is conceded: thereupon the dialectical syllogism (διαλεκτικὸς συλλογισμός) deduces consequent opinions in the conclusion. Nor does the process of acquiring the premisses of eristical syllogism, which is fallacious either in its premisses or in its process, differ, except that, when the premisses are fallacious, the dialectical interrogations must be such as to cause this fallacy. Hence, as science and dialectic are different, so scientific induction and syllogism must be distinguished from dialectical induction and syllogism. Dialectic is useful, for exercise, for conversation and for philosophical sciences, where by being critical it has a road to principles. But it is by a different process of sense, memory, experience, induction, intelligence, syllogism, that science becomes knowledge of real causes, of real effects, and especially of real essences from which follow real consequences, not beyond, but belonging to real substances. So can we men, not, as Plato thought, by having in our souls universal principles innate but forgotten, but by acquiring universal principles from sense, which is the origin of knowledge, arrive at judgments which are true, and true because they agree with the things which we know by sense, by inference and by science. Such is Aristotle’s psychological and logical realism, contained in theDe Animaand logical treatises.2.Practice(πρᾶξις).—In this natural world of real substances, human good is not an imitation of a supernatural universal form of the good, but is human happiness; and this good is the same both of the individual as a part and of the state as a whole. Ethics then is a kind of Politics. But in Ethics a man’s individual good is his own happiness; and his happiness is no mere state, but an activity of soul according to virtue in a mature life, requiring as conditions moderate bodily and external goods of fortune; his virtue is (1) moral virtue, which is acquired by habituation, and is a purposive habit of performing actions in the mean determined by right reason or prudence; requiring him, not to exclude, but to moderate his desires; and (2) intellectual virtue, which is either prudence of practical, or wisdom of speculative intellect; and his happiness is a kind of ascending scale of virtuous activities, in which moral virtue is limited by prudence, and prudence by wisdom; so that the speculative life of wisdom is the happiest and most divine, and the practical life of prudence and moral virtue secondary and human. Good fortune in moderation is also required as a condition of his happiness. Must we then, on account of misfortunes, look with Solon at the end, and call no man happy till he is dead? Or is this altogether absurd for us who say that happiness is an activity? Virtuous activities determine happiness, and a virtuous man is happy in this life, in spite of misfortunes unless they be too great; while after death he will not feel the misfortunes of the living so much as to change his happiness. Still, for perfect happiness a man should prefer the speculative life of divine intellect, and immortalize (ἀθανατίζειν) as far as possible. For intellect is what mainly makes a man what he is, and is divine and immortal.To turn from Ethics to Politics, the good of the individual on a small scale becomes on a large scale the good of the citizen and the state, whose end should be no far-off form of good, and no mere guarantee of rights, but the happiness of virtuous action, the life according to virtue, which is the general good of the citizen. Hence, the citizen of the best state is he who has the power and the purpose to be governed and govern for the sake of the life according to virtue.A right government is one which aims at the general good, whereas any government which aims at its own good is a deviation. Hence governments are to be arranged from best to worst in the following order:—I. Right governments (ὀρθαὶ πολιτείαι), aiming at the general good:—i. Monarchy, of one excelling in virtue:ii. Aristocracy, of a class excelling in virtue:iii. Commonwealth, of the majority excelling in virtue.II. Deviations (παρεκβάσεις), aiming at the good of the government:—i. Democracy, aiming at the good of the majority:ii. Oligarchy, aiming at the good of the few:iii. Tyranny, aiming at the good of one.Such is Aristotle’s practical philosophy, contained in his maturedNicomachean Ethics, and his unfinishedPolitics.3.Production(ποίησις).—Production differs from practice in being an activity (ἐνέργεια;e.g.building) which is always a means to a work (ἔργον;e.g.a house) beyond itself. Productive science, or art, is an intellectual habit of true reasoning from appropriate principles, acquired from experiences, and applied to the production of the work which is the end of the art. All the arts are therefore at once rational and productive. They are either for necessity (e.g.medicine) or for occupation (e.g.poetry), the former being inferior to the latter. Rhetoric is a faculty on any subject of investigating what may be persuasive (πιθανόν), which is the work of no other art; its means are artificial and inartificial evidences (πίστεις), and, among artificial evidences, especially the logical arguments of example and enthymeme. Poetry is the art of producing representations; (1) in words, rhythm and harmony (ἁρμονία, “harmony” in the original sense); (2) of men like ourselves, or better as in tragedy, or worse as in comedy; (3) by means of narrative as in epic, or by action as in the drama. The cause of poetry is man’s instinct of representation and his love of representations caused by the pleasure of learning. Comedy is representation of men inferior in being ludicrous: epic is like tragedy a representation of superior men, but by means of narrative and unlimited in time: tragedy is a representation of an action superior and complete, in a day if possible, by means of action, and accomplishing by pity and fear the purgation of such passions (Poetics, 1449 b 24). Music is a part of moral education; and for this end we should use the most moral harmonies. But music has also other ends and uses, and on the whole four; namely amusement, virtue, occupation and purgation of the affections; for some men are liable more than others to pity and fear and enthusiasm, but from sacred melodies we see them, when they have heard those which act orgiastically on the soul, becoming settled by a kind of medicine and purgation (κάθαρσις), and being relieved with pleasure. Finally, art is not morality, because its end is always a work of art, not virtuous action: on the other hand, art is subordinate to morality, because all the ends of art are but means to the end of life, and therefore a work of art which offends against morality is opposed to the happiness and the good of man. Such is Aristotle’s productive science or art, contained in hisRhetoricandPoetics, compared with hisEthicsandPolitics.
I.Speculation(θεωρία).—Since things are individuals, and there is nothing, and nothing universal, beyond them, there are two kinds of knowledge (γνῶσις), sense (αἴσθησις) of individuals, intellect (νοῦς) of universals. Both powers know by being passively receptive of essence propagated by an efficient cause; but, while in sense the efficient cause is an external object (ἔξωθεν), in intelligence it is active intellect (νοῦς τῷ ποιεῖν) propagating its essence in passive intellect (νοῦς παθητικός). Nevertheless, without sense there is no knowledge. Sense receives from the external world an essence,e.g.of white, which is really universal as well as individual, but apprehends it only as individual,e.g.this white substance: intellect thereupon discovers the universal essence but only in the individuals of sense. This intellectual discovery requires sensation and retention of sensation; so that sense (αἴσθησις) receives impressions, imagination (φαντασία) retains them as images, intellect (νοῦς) generalizes the universal, and, when it is intelligence of essence, is always true.
This is the origin of knowledge, psychologically regarded (in theDe Anima). Logically regarded, the origin of all teaching and learning of an intellectual kind is a process of induction (ἐπαγωγή) from particulars to universal, and of syllogism (συλλογισμός) from universal to further particulars; induction, whenever it starts from sense, becomes the origin of scientific knowledge (ἐπιστήμη); while there is also a third process of example (παράδειγμα) from particular to particular, which produces only persuasion. In acquiring scientific knowledge, syllogism cannot start from universals without induction, nor induction acquire universals without sense. At the same time, there are three species of syllogism, scientific, dialectical and eristical or sophistical; and in consequence there are different ways of acquiring premisses. In order to acquire the knowledge of the true and primary principles of scientific knowledge, and especially theintelligence of the universal essence of the subject, which is always true, the process of knowledge consists of (1) sense (αἴσθησις), which receives the essence as individual, (2) memory (μνήμη), which is a retention of sensible impression, (3) experience (ἐμπειρία), which consists of a number of similar memories, (4) induction (ἐπαγωγή), which infers the universal as a fact (τὸ ὅτι), (5) intellect (νοῦς), which apprehends the principle (ἀρχή); because it is a true apprehension that the universal induced is the very essence and formal cause of the subject: thereupon, scientific syllogism (ἐπιστημονικὸς συλλογισμός), making the definition (ὁρισμος) of this essence the middle term (τὸ μέσον), becomes a demonstration (ὁρισμος) of the consequences which follow from the essence in the conclusion. Such then is science. In order to acquire the probabilities (τὰ ἔνδοξα) of opinion (δόξα), which are the premisses of dialectical syllogism, the process is still induction, as in science, but dialectical induction by interrogation from the opinions of the answerers until the universal is conceded: thereupon the dialectical syllogism (διαλεκτικὸς συλλογισμός) deduces consequent opinions in the conclusion. Nor does the process of acquiring the premisses of eristical syllogism, which is fallacious either in its premisses or in its process, differ, except that, when the premisses are fallacious, the dialectical interrogations must be such as to cause this fallacy. Hence, as science and dialectic are different, so scientific induction and syllogism must be distinguished from dialectical induction and syllogism. Dialectic is useful, for exercise, for conversation and for philosophical sciences, where by being critical it has a road to principles. But it is by a different process of sense, memory, experience, induction, intelligence, syllogism, that science becomes knowledge of real causes, of real effects, and especially of real essences from which follow real consequences, not beyond, but belonging to real substances. So can we men, not, as Plato thought, by having in our souls universal principles innate but forgotten, but by acquiring universal principles from sense, which is the origin of knowledge, arrive at judgments which are true, and true because they agree with the things which we know by sense, by inference and by science. Such is Aristotle’s psychological and logical realism, contained in theDe Animaand logical treatises.
2.Practice(πρᾶξις).—In this natural world of real substances, human good is not an imitation of a supernatural universal form of the good, but is human happiness; and this good is the same both of the individual as a part and of the state as a whole. Ethics then is a kind of Politics. But in Ethics a man’s individual good is his own happiness; and his happiness is no mere state, but an activity of soul according to virtue in a mature life, requiring as conditions moderate bodily and external goods of fortune; his virtue is (1) moral virtue, which is acquired by habituation, and is a purposive habit of performing actions in the mean determined by right reason or prudence; requiring him, not to exclude, but to moderate his desires; and (2) intellectual virtue, which is either prudence of practical, or wisdom of speculative intellect; and his happiness is a kind of ascending scale of virtuous activities, in which moral virtue is limited by prudence, and prudence by wisdom; so that the speculative life of wisdom is the happiest and most divine, and the practical life of prudence and moral virtue secondary and human. Good fortune in moderation is also required as a condition of his happiness. Must we then, on account of misfortunes, look with Solon at the end, and call no man happy till he is dead? Or is this altogether absurd for us who say that happiness is an activity? Virtuous activities determine happiness, and a virtuous man is happy in this life, in spite of misfortunes unless they be too great; while after death he will not feel the misfortunes of the living so much as to change his happiness. Still, for perfect happiness a man should prefer the speculative life of divine intellect, and immortalize (ἀθανατίζειν) as far as possible. For intellect is what mainly makes a man what he is, and is divine and immortal.
To turn from Ethics to Politics, the good of the individual on a small scale becomes on a large scale the good of the citizen and the state, whose end should be no far-off form of good, and no mere guarantee of rights, but the happiness of virtuous action, the life according to virtue, which is the general good of the citizen. Hence, the citizen of the best state is he who has the power and the purpose to be governed and govern for the sake of the life according to virtue.
A right government is one which aims at the general good, whereas any government which aims at its own good is a deviation. Hence governments are to be arranged from best to worst in the following order:—
I. Right governments (ὀρθαὶ πολιτείαι), aiming at the general good:—
i. Monarchy, of one excelling in virtue:ii. Aristocracy, of a class excelling in virtue:iii. Commonwealth, of the majority excelling in virtue.
i. Monarchy, of one excelling in virtue:
ii. Aristocracy, of a class excelling in virtue:
iii. Commonwealth, of the majority excelling in virtue.
II. Deviations (παρεκβάσεις), aiming at the good of the government:—
i. Democracy, aiming at the good of the majority:ii. Oligarchy, aiming at the good of the few:iii. Tyranny, aiming at the good of one.
i. Democracy, aiming at the good of the majority:
ii. Oligarchy, aiming at the good of the few:
iii. Tyranny, aiming at the good of one.
Such is Aristotle’s practical philosophy, contained in his maturedNicomachean Ethics, and his unfinishedPolitics.
3.Production(ποίησις).—Production differs from practice in being an activity (ἐνέργεια;e.g.building) which is always a means to a work (ἔργον;e.g.a house) beyond itself. Productive science, or art, is an intellectual habit of true reasoning from appropriate principles, acquired from experiences, and applied to the production of the work which is the end of the art. All the arts are therefore at once rational and productive. They are either for necessity (e.g.medicine) or for occupation (e.g.poetry), the former being inferior to the latter. Rhetoric is a faculty on any subject of investigating what may be persuasive (πιθανόν), which is the work of no other art; its means are artificial and inartificial evidences (πίστεις), and, among artificial evidences, especially the logical arguments of example and enthymeme. Poetry is the art of producing representations; (1) in words, rhythm and harmony (ἁρμονία, “harmony” in the original sense); (2) of men like ourselves, or better as in tragedy, or worse as in comedy; (3) by means of narrative as in epic, or by action as in the drama. The cause of poetry is man’s instinct of representation and his love of representations caused by the pleasure of learning. Comedy is representation of men inferior in being ludicrous: epic is like tragedy a representation of superior men, but by means of narrative and unlimited in time: tragedy is a representation of an action superior and complete, in a day if possible, by means of action, and accomplishing by pity and fear the purgation of such passions (Poetics, 1449 b 24). Music is a part of moral education; and for this end we should use the most moral harmonies. But music has also other ends and uses, and on the whole four; namely amusement, virtue, occupation and purgation of the affections; for some men are liable more than others to pity and fear and enthusiasm, but from sacred melodies we see them, when they have heard those which act orgiastically on the soul, becoming settled by a kind of medicine and purgation (κάθαρσις), and being relieved with pleasure. Finally, art is not morality, because its end is always a work of art, not virtuous action: on the other hand, art is subordinate to morality, because all the ends of art are but means to the end of life, and therefore a work of art which offends against morality is opposed to the happiness and the good of man. Such is Aristotle’s productive science or art, contained in hisRhetoricandPoetics, compared with hisEthicsandPolitics.
Aristotle, even in this sketch of his system, shows himself to be the philosopher of facts, who can best of all men bear criticism; and indeed it must be confessed that he retained many errors of Platonism and laid himself open to the following objections. Two substances, being individuals,e.g.Socrates and Callias, are in no way the same, but only similar, even in essence,e.g.Socrates is one rational animal, Callias another. A universal,e.g.the species man, is not predicate of many individuals (ἓν κατὰ πολλῶν,Post. An.i. II), but a whole number of similar individuals,e.g.all men; and not a whole species, but only an individual, is a predicate of such individual,e.g.Socrates is a man, not all men, and one white thing, not all white things. Consequently, a species or genus is not a substance, as Aristotle says it is in theCategories(inconsistently with his own doctrine of substances), but a whole number of substances,e.g.all men, all animals. Similarly, the universal essence of a species is not one and the same as each individual essence, but is the whole number of similar individual essences of the similar individuals of the species,e.g.all rational animals. Consequently, the universal essence of a species of substances is not one and the same eternal essence in all the individuals of a species but only similar, and is not substance as Aristotle calls it in theMetaphysics, inconsistently with his own doctrine of substance, but is a whole number of similar substances,e.g.all rational animals which are what all men are. Hence again, the natural world of species and essences is not eternal, but only endures as long as there are individual substances. Hence, moreover, a natural substance or body as an efficient cause or force causes an effect on another, not by propagating one eternal essence of a species into the matter of the other, but so far as we really understand force, by their reciprocally preventing one another from occupying the same place at the same moment on account of the mutual resistance of any two bodies. The essence of a natural substance,e.g.wood, is not immateriate, but is the whole body as what it is. The matter of a natural substance is not a primary matter which is one indeterminate substratum of all natural substances, but is only one body as able to be changed by a force which is another substance able to change it,e.g.a seed becoming wood, wood becoming coal, &c. A natural substance or body, therefore, is not a heterogeneous compound of essence and matter, but is essence as what it is, matter as able passively to be changed, force as able actively to change. The simple bodies which are the matter of the rest are not terrestrial earth, water, air, fire,and a different celestial aether, but whatever elementary bodies natural science, starting anew from mechanics and chemistry, may determine to be the matter of all other bodies whatever. Nature does not aim at God as end, but God, thinking and willing ends, produces and acts on nature. Soul is not an immateriate essence of an organic body capable, but an immateriate conscious substance within an organic body. Sensation is not the reception of the selfsame essence of an external body, but one’s perception of one’s sentient organism as affected, and especially of its organs resisting one another,e.g.one’s lips, hands, &c., preventing one another from occupying the same place at the same moment within one’s organism. Intelligence does not differ from sense by having no bodily organ, but the nervous system is the bodily organ of both. Intelligence is not active intellect propagating universal essence in passive intellect, but only logical inference starting from sense, and both requiring nervous body and conscious soul. It is not always a true apprehension of essence, but often, especially in physical matter, such as sound or heat or light, takes superficial effects to be the essence of the thing. Aristotle did not altogether solve the question, What is, and scarcely solved at all the question, How do we know the external world?
We might continue to object. But at bottom there remains the fundamental position of Aristotelianism, that all things are substances, individuals separate though related; that some things are attributes, real only as being some individual substance somehow affected, or, as we should say, modified or determined; and that without individual substances there is nothing, and nothing universal apart from individuals. There remains too the consequence that there are different substances, separate from but related to one another; and these substances of three irreducible kinds, natural, supernatural, human. Aristotelianism has to be considered against the philosophy which preceded it and against the philosophy which has since followed it. Platonism preceded it, and was the metaphysical doctrine that all things are supernatural—forms, gods, souls. Idealism has since followed it, and is the metaphysical doctrine that all things are mind and states of mind. Aristotelianism intervenes between ancient Platonism and modern Idealism, and is the metaphysical doctrine that all things are substances, natural and supernatural and human. It is a philosophy of substantial things, standing as avia mediabetween a philosophy of the supernatural and a philosophy of mind. There are three alternatives, which may be put as questions which every thinker must ask himself. Are the things which surround me in what I call the environment,—the men, the animals, the plants, the ground, the stones, the water, the air, the moon, the sun, the stars and God—are they shadows, unsubstantial things, as formerly Platonism made all things to be except the supernatural world of forms, gods and souls? Or are they, as modern Idealism says, mind and states of mind? Or are they really substances separate from, though related to, myself, who am also a substance? The Aristotelian answer is—“Yes, all things are substances, but not all supernatural, nor all mental; for some are natural substances, or bodies”; and by that answer Aristotelianism stands or falls.
Literature.—The Aristotelian philosophy is to be studied first in Aristotle’s works, which are the best commentaries on one another; the best complete edition is the Berlin edition (1831-1870), by Bekker and Brandis, in which also are the fragments collected by V. Rose, the scholia collected by Brandis, and the index compiled by Bonitz. After reading the remains of the Peripatetic school, the Greek commentators should be further studied in this edition. The Latin commentators, the Arabians and the schoolmen show how Aristotle has been the chief author of modern culture; while the vindication of modern independence comes out in his critics, the greatest of whom were Roger and Francis Bacon. Since the modern discovery of the science of motion by Galileo which changed natural science, and the modern revolution of philosophy by Descartes which changed metaphysics, the study of Aristotle has become less universal; but it did not die out, and received a fresh stimulus especially from Julius Pacius, who going back through G. Zabarella to the Arabians, and himself gifted with great logical powers, always deserves study in his editions of theOrganonand thePhysicsand in hisDoctrinae Peripateticae. In more recent times, as part of the growing conviction of the essentiality of everything Greek, Aristotle has received marked attention. In France there are the works of Cousin (1835), Félix Ravaisson, who wrote on theMetaphysics(1837-1846), and Barthélemy St Hilaire, who translated theOrganonand other works (1844 seq.). In Germany there has been a host of commentaries, among which we may mention theOrganonedited (1844-1846) by F. Th. Waitz (not so well as by Pacius), theDe Animaedited (1833) by F.A. Trendelenburg and later by A. Torstrik, theHistoria Animaliumby H. Aubert and F. Wimmer (1868), theEthicsby K.L. Michelet (1827), theMetaphysicsby A. Schwegler (1847) and (best of all) by H. Bonitz (1848), who is the most faithful of all commentators, because to great industry and acumen he adds the rare gift of confessing when he does not understand, and when he does not know what Aristotle might have thought. With Aristotle’s works before one, with theIndex Aristotelicus, and the edition and translation of theMetaphysicsby Bonitz on one side, and Zeller’sDie Philosophie der Griechen, ii. 2, “Aristoteles” (trans. by Costelloe and Muirhead), on the other side, one can go a considerable way towards understanding the foundations of Aristotelianism.In England scholars tend to take up certain parts of Aristotle’s philosophy. Grote indeed intended to write a general account of Aristotle like that of Plato; but hisAristotlewent little further than the logical writings. From Cambridge we have J.W. Blakesley’sLife of Aristotle, E.M. Cope’sRhetoric, Dr Henry Jackson’sNicomachean Ethics, v., S.H. Butcher’sPoetics, Hicks’sDe Anima, J.E. Sandys’sAthenian Constitution, Jebb’sRhetoric(ed. Sandys). Oxford in particular, since the beginning of the 19th century, has kept alive the study of Aristotle. E. Cardwell in his edition of theNicomachean Ethics(1828) had the wisdom to found his text on the Laurentian Manuscript (Kb); E. Poste wrote translations of thePosterior AnalyticsandSophistici Elenchi; R. Congreve edited thePolitics; A. Grant edited theNicomachean Ethics; E. Wallace translated and annotated theDe Anima; B. Jowett translated thePolitics; W.L. Newman has edited thePoliticsin four volumes; Dr Ogle has translated theDe Partibus Animalium, with notes; R. Shute wrote aHistory of the Aristotelian Writings; Professor J.A. Stewart has writtenNotes on the Nicomachean Ethics; Professor J. Burnet has issued an annotated edition of theNicomachean Ethics, and W.D. Ross has translated theMetaphysics. All these are, or were, Oxford men; and it remains to mention two others: I. Bywater, who as an Aristotelian scholar has done much for the improvement of Bekker’s text, especially of theNicomachean Ethicsand thePoetics; and F.G. Kenyon, who has the proud distinction of having been the first modern editor of theἈθηναίων πολιτεία.
Literature.—The Aristotelian philosophy is to be studied first in Aristotle’s works, which are the best commentaries on one another; the best complete edition is the Berlin edition (1831-1870), by Bekker and Brandis, in which also are the fragments collected by V. Rose, the scholia collected by Brandis, and the index compiled by Bonitz. After reading the remains of the Peripatetic school, the Greek commentators should be further studied in this edition. The Latin commentators, the Arabians and the schoolmen show how Aristotle has been the chief author of modern culture; while the vindication of modern independence comes out in his critics, the greatest of whom were Roger and Francis Bacon. Since the modern discovery of the science of motion by Galileo which changed natural science, and the modern revolution of philosophy by Descartes which changed metaphysics, the study of Aristotle has become less universal; but it did not die out, and received a fresh stimulus especially from Julius Pacius, who going back through G. Zabarella to the Arabians, and himself gifted with great logical powers, always deserves study in his editions of theOrganonand thePhysicsand in hisDoctrinae Peripateticae. In more recent times, as part of the growing conviction of the essentiality of everything Greek, Aristotle has received marked attention. In France there are the works of Cousin (1835), Félix Ravaisson, who wrote on theMetaphysics(1837-1846), and Barthélemy St Hilaire, who translated theOrganonand other works (1844 seq.). In Germany there has been a host of commentaries, among which we may mention theOrganonedited (1844-1846) by F. Th. Waitz (not so well as by Pacius), theDe Animaedited (1833) by F.A. Trendelenburg and later by A. Torstrik, theHistoria Animaliumby H. Aubert and F. Wimmer (1868), theEthicsby K.L. Michelet (1827), theMetaphysicsby A. Schwegler (1847) and (best of all) by H. Bonitz (1848), who is the most faithful of all commentators, because to great industry and acumen he adds the rare gift of confessing when he does not understand, and when he does not know what Aristotle might have thought. With Aristotle’s works before one, with theIndex Aristotelicus, and the edition and translation of theMetaphysicsby Bonitz on one side, and Zeller’sDie Philosophie der Griechen, ii. 2, “Aristoteles” (trans. by Costelloe and Muirhead), on the other side, one can go a considerable way towards understanding the foundations of Aristotelianism.
In England scholars tend to take up certain parts of Aristotle’s philosophy. Grote indeed intended to write a general account of Aristotle like that of Plato; but hisAristotlewent little further than the logical writings. From Cambridge we have J.W. Blakesley’sLife of Aristotle, E.M. Cope’sRhetoric, Dr Henry Jackson’sNicomachean Ethics, v., S.H. Butcher’sPoetics, Hicks’sDe Anima, J.E. Sandys’sAthenian Constitution, Jebb’sRhetoric(ed. Sandys). Oxford in particular, since the beginning of the 19th century, has kept alive the study of Aristotle. E. Cardwell in his edition of theNicomachean Ethics(1828) had the wisdom to found his text on the Laurentian Manuscript (Kb); E. Poste wrote translations of thePosterior AnalyticsandSophistici Elenchi; R. Congreve edited thePolitics; A. Grant edited theNicomachean Ethics; E. Wallace translated and annotated theDe Anima; B. Jowett translated thePolitics; W.L. Newman has edited thePoliticsin four volumes; Dr Ogle has translated theDe Partibus Animalium, with notes; R. Shute wrote aHistory of the Aristotelian Writings; Professor J.A. Stewart has writtenNotes on the Nicomachean Ethics; Professor J. Burnet has issued an annotated edition of theNicomachean Ethics, and W.D. Ross has translated theMetaphysics. All these are, or were, Oxford men; and it remains to mention two others: I. Bywater, who as an Aristotelian scholar has done much for the improvement of Bekker’s text, especially of theNicomachean Ethicsand thePoetics; and F.G. Kenyon, who has the proud distinction of having been the first modern editor of theἈθηναίων πολιτεία.
(T. Ca.)
ARISTOXENUS,of Tarentum (4th centuryB.C.), a Greek peripatetic philosopher, and writer on music and rhythm. He was taught first by his father Spintharus, a pupil of Socrates, and later by the Pythagoreans, Lamprus of Erythrae and Xenophilus, from whom he learned the theory of music. Finally he studied under Aristotle at Athens, and was deeply annoyed, it is said, when Theophrastus was appointed head of the school on Aristotle’s death. His writings, said to have numbered four hundred and fifty-three, were in the style of Aristotle, and dealt with philosophy, ethics and music. The empirical tendency of his thought is shown in his theory that the soul is related to the body as harmony to the parts of a musical instrument. We have no evidence as to the method by which he deduced this theory (cf. T. Gomperz,Greek Thinkers, Eng. trans. 1905, vol. iii. p. 43). In music he held that the notes of the scale are to be judged, not as the Pythagoreans held, by mathematical ratio, but by the ear. The only work of his that has come down to us is the three books of theElements of Harmony(ῥυθμικὰ στοιχεῖα), an incomplete musical treatise. Grenfell and Hunt’sOxyrhynchus Papyri(vol. i., 1898) contains a five-column fragment of a treatise on metre, probably this treatise of Aristoxenus.
The best edition is by Paul Marquard, with German translation and full commentary,Die harmonischen Fragmente des Aristoxenus(Berlin, 1868). The fragments are also given in C.W. Müller,Frag. Hist. Graec., ii. 269 sqq.; and R. Westphal,Melik und Rhythmik d. klass. Hellenenthums(2nd vol. edited by F. Saran, Leipzig, 1893). Eng. trans. by H.S. Macran (Oxford, 1902). See also W.L. Mahne,Diatribe de Aristoxeno(Amsterdam, 1793); B. Brill,Aristoxenus’ rhythmische und metrische Messungen(1871); R. Westphal,Griechische Rhythmik und Harmonik(Leipzig, 1867); L. Laloy,Aristoxène de Tarente et la musique de l’antiquité(Paris, 1904); SeePeripatetics,Pythagoras(Music) and art. “Greek Music” in Grove’sDict. of Music(1904). For the Oxyrhynchus fragment seeClassical Review(January 1898), and C. van Jan in Bursian’sJahresbericht, civ. (1901).
The best edition is by Paul Marquard, with German translation and full commentary,Die harmonischen Fragmente des Aristoxenus(Berlin, 1868). The fragments are also given in C.W. Müller,Frag. Hist. Graec., ii. 269 sqq.; and R. Westphal,Melik und Rhythmik d. klass. Hellenenthums(2nd vol. edited by F. Saran, Leipzig, 1893). Eng. trans. by H.S. Macran (Oxford, 1902). See also W.L. Mahne,Diatribe de Aristoxeno(Amsterdam, 1793); B. Brill,Aristoxenus’ rhythmische und metrische Messungen(1871); R. Westphal,Griechische Rhythmik und Harmonik(Leipzig, 1867); L. Laloy,Aristoxène de Tarente et la musique de l’antiquité(Paris, 1904); SeePeripatetics,Pythagoras(Music) and art. “Greek Music” in Grove’sDict. of Music(1904). For the Oxyrhynchus fragment seeClassical Review(January 1898), and C. van Jan in Bursian’sJahresbericht, civ. (1901).
ARISUGAWA,the name of one of the royal families of Japan, going back to the seventh son of the mikado Go-Yozei (d. 1638). After the revolution of 1868, when the mikado Mutsu-hito was restored, his uncle, Prince Taruhito Arisugawa (1835-1895), became commander-in-chief, and in 1875 president of the senate.After his suppression of the Satsuma rebellion he was made a field-marshal, and he was chief of the staff in the war with China (1894-95). His younger brother, Prince Takehito Arisugawa (b. 1862), was from 1879 to 1882 in the British navy, serving in the Channel Squadron, and studied at the Naval College, Greenwich. In the Chino-Japanese War of 1894-95 he was in command of a cruiser, and subsequently became admiral-superintendent at Yokosuka. Prince Arisugawa represented Japan in England together with Marquis Ito at the Diamond Jubilee (1897), and in 1905 was again received there as the king’s guest.
ARITHMETIC(Gr.ἀριθμητική, sc.τέχνη, the art of counting, fromἀριθμός, number), the art of dealing with numerical quantities in their numerical relations.
1. Arithmetic is usually divided intoAbstract ArithmeticandConcrete Arithmetic, the former dealing with numbers and the latter with concrete objects. This distinction, however, might be misleading. In stating that the sum of 11d. and 9d. is 1s. 8d. we do not mean that nine pennies when added to eleven pennies produce a shilling and eight pennies. The sum of money corresponding to 11d. may in fact be made up of coins in several different ways, so that the symbol “11d.” cannot be taken as denoting any definite concrete objects. The arithmetical fact is that 11 and 9 may be regrouped as 12 and 8, and the statement “11d. + 9d. = 1s. 8d.” is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (§ 11).
2. The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view. There are three main purposes, the practical, the educational, and the scientific;i.e.the subject may be studied with a view to technical skill in dealing with the arithmetical problems that arise in actual life, or for the sake of its general influence on mental development, or as an elementary stage in mathematical study.
3. The practical aspect is an important one. The daily activities of the great mass of the adult population, in countries where commodities are sold at definite prices for definite quantities, include calculations which have often to be performed rapidly, on data orally given, and leading in general to results which can only be approximate; and almost every branch of manufacture or commerce has its own range of applications of arithmetic. Arithmetic as a school subject has been largely regarded from this point of view.
4. From the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy. This aspect of the matter, however, belongs mainly to the period when arithmetic was studied almost entirely for commercial purposes; and even then accuracy was not found always to harmonize with actuality. The development of physical science has tended to emphasize an exactly opposite aspect, viz. the impossibility, outside a certain limited range of subjects, of ever obtaining absolute accuracy, and the consequent importance of not wasting time in attempting to obtain results beyond a certain degree of approximation.
5. As a branch of mathematics, arithmetic may be treated logically, psychologically, or historically. All these aspects are of importance to the teacher: the logical, in order that he may know the end which he seeks to attain; the psychological, that he may know how best to attain this end; and the historical, for the light that history throws on psychology,
The logical arrangement of the subject is not the best for elementary study. The division into abstract and concrete, for instance, is logical, if the former is taken as relating to number and the latter to numerical quantity (§ 11). But the result of a rigid application of this principle would be that the calculation of the cost of 3 ℔ of tea at 2s. a ℔ would be deferred until after the study of logarithms. The psychological treatment recognizes the fact that the concrete precedes the abstract and that the abstract is based on the concrete; and it also recognizes the futility of attempting a strictly continuous development of the subject.
On the other hand, logical analysis is necessary if the subject is to be understood. As an illustration, we may take the elementary processes of addition, subtraction, multiplication and division. These are still called in text-books the “four simple rules”; but this name ignores certain essential differences. (i) If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz. addition and subtraction. (iii) The inclusion of the four processes under one general head fails to indicate the essential difference between addition and multiplication, as direct processes, on the one hand, and subtraction and division, as inverse processes, on the other (§ 59).
6. The present article deals mainly with the principles of the subject, for which a logical arrangement is on the whole the more convenient. It is not suggested that this is the proper order to be adopted by the teacher.
I. Number
7.Ordinal and Cardinal Numbers.—One of the primary distinctions in the use of number is between ordinal and cardinal numbers, or rather between the ordinal and the cardinal aspects of number. The usual statement is thatone, two, three, ...are cardinal numbers, andfirst, second, third, ...are ordinal numbers. This, however, is an incomplete statement; the words one, two, three, ... and the corresponding symbols 1, 2, 3, ... or I, II, III, ... are used sometimes as ordinals,i.e.to denote the place of an individual in a series, and sometimes as cardinals,i.e.to denote the total number since the commencement of the series.
On the whole, the ordinal use is perhaps the more common. Thus “100” on a page of a book does not mean that the page is 100 times the page numbered 1, but merely that it is the page after 99. Even in commercial transactions, in dealing with sums of money, the statement of an amount often has reference to the last item added rather than to a total; and geometrical measurements are practically ordinal (§ 26).
For ordinal purposes we use, as symbols, not only figures, such as 1, 2, 3, ... but also letters, as a, b, c, ... Thus the pages of a book may be numbered 1, 2, 3, ... and the chapters I, II, III, ... but the sheets are lettered A, B, C, ... Figures and letters may even be used in combination; thus 16 may be followed by 16a and 16b, and these by 17, and in such a case the ordinal 100 does not correspond with the total (cardinal) number up to this point.
Arithmetic is supposed to deal with cardinal, not with ordinal numbers; but it will be found that actual numeration, beyond about three or four, is based on the ordinal aspect of number, and that a scientific treatment of the subject usually requires a return to this fundamental basis.
One difference between the treatment of ordinal and of cardinal numbers may be noted. Where a number is expressed in terms of various denominations, a cardinal number usually begins with the largest denomination, and an ordinal number with the smallest. Thus we speak of one thousand eight hundred and seventy-six, and represent it by MDCCCLXXVI or 1876; but we should speak of the third day of August 1876, and represent it by 3. 8. 1876. It might appear as if the writing of 1876 was an exception to this rule; but in reality 1876, when used in this way, is partly cardinal and partly ordinal, the first three figures being cardinal and the last ordinal. To make the year completely ordinal, we should have to describe it as the 6th year of the 8th decade of the 8th century of the 2nd millennium;i.e.we should represent the date by 3. 8. 6. 8. 9. 2, the total number of years, months and days completed being 1875. 7. 2.
In using an ordinal we direct our attention to a term of a series, while in using a cardinal we direct our attention to the interval between two terms. The total number in the series is the sum of the two cardinal numbers obtained by counting up to any interval from the beginning and from the end respectively; but if we take the ordinal numbers from the beginning and from the end we count one term twice over. Hence, if there are 365 days in a year, the 100th day from the beginning is the 266th, not the 265th, from the end.
8.Meaning of Names of Numbers.—What do we mean by any particular number,e.g.byseven, or bytwo hundred and fifty-three? We can definetwoasone and one, andthreeasone and one and one; but we obviously cannot continue this method for ever. For the definition of large numbers we may employ either of two methods, which will be called thegroupingmethod and thecountingmethod.
(i)Method of Grouping.—The first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition. Thus, on the denary system (§16) we can give independent definitions to the numbers up to ten, and then regard (e.g.) fifty-three as a composite number made up of five tens and three ones. Or, on the quinary-binary system, we need only give independent definitions to the numbers up to five; the numberssix, seven, ... can then be regarded asfive and one, five and two, ..., a fresh series being started when we get tofive and fiveorten.The grouping method introduces multiplication into the definition of large numbers; but this, from the teacher’s point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary arithmetical processes.
(ii)Method of Counting.—The second method consists in taking a series of names or symbols for the first few numbers, and then repeating these according to a regular system for successive numbers, so that each number is defined by reference to the number immediately preceding it in the series. Thustwostill meansone and one, butthreemeanstwo and one, notone and one and one.Similarlytwo hundred and fifty-threedoes not mean two hundreds, five tens and three ones, butonemore thantwo hundred and fifty-two; and the number which is called one hundred is not defined as ten tens, but as one more than ninety-nine.
9.Concrete and Abstract Numbers.—Number is concrete or abstract according as it does or does not relate to particular objects. On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name themthree.If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. This, however, is a description, not a definition, and we still want a definition for “number” in the phrase “number of times.”
10.Definition of “Number.”—Suppose we fix on a certain sequence of names “one,” “two,” “three,” ..., or symbols such as 1, 2, 3, ...; this sequence being always the same. If we take a set of concrete objects, and name them in succession “one,” “two,” “three,” ..., naming each once and once only, we shall not get beyond a certain name,e.g.“six.” Then, in saying that the number of objects is six, what we mean is that the name of the last object named is six. We therefore only require a definite law for the formation of the successive names or symbols. The symbols 1, 2, ... 9, 10, ..., for instance, are formed according to a definite law; and in giving 253 as thenumberof a set of objects we mean that if we attach to them the symbols 1, 2, 3, ... in succession, according to this law, the symbol attached to the last object will be 253. If we say that this act of attaching a symbol has been performed 253 times, then 253 is anabstract(orpure)number.
Underlying this definition is a certain assumption, viz. that if we take the objects in a different order, the last symbol attached will still be 253. This, in an elementary treatment of the subject, must be regarded as axiomatic; but it is really a simple case of mathematical induction. (SeeAlgebra.) If we take two objects A and B, it is obvious that whether we take them as A, B, or as B, A, we shall in each case get the sequence 1, 2. Suppose this were true for, say, eight objects, marked 1 to 8. Then, if we introduce another object anywhere in the series, all those coming after it will be displaced so that each will have the mark formerly attached to the next following; and the last will therefore be 9 instead of 8. This is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the theorem is true for 8, it is true for 9. But it is true for 2; therefore it is true for 3; therefore for 4, and so on.
11.Numerical Quantities.—If the termnumberis confined to number in the abstract, then number in the concrete may be described asnumerical quantity. Thus £3 denotes £1 taken 3 times. The £1 is termed theunit.A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s.,i.e.6 times 1s. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced.
II. Notation, Numeration and Number-Ideation
12.Terms used.—The representation of numbers by spoken sounds is callednumeration; their representation by written signs is callednotation. The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. This latter presentation may, in the absence of any accepted term, be callednumber-ideation; this word covering not only the perception or recognition of particular numbers, but also the formation of a number-concept.
13.Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system. The essential features which distinguish this from other systems are (1) the limitation of the number of different symbols, only ten being used, however large the number to be represented may be; (2) the use of thezeroto indicate the absence of number; and (3) the principle of local value, by which a symbol in effect represents different numbers, according to its position. The symbols denoting a number are called itsdigits.
A brief account of the development of the system will be found underNumeral. Here we are concerned with the principle, the explanation of which is different according as we proceed on the grouping or the counting system.
(i) On the grouping system we may in the first instance consider that we have separate symbols for numbers from “one” to “nine,” but that when we reach ten objects we put them in a group and denote this group by the symbol used for “one,” but printed in a different type or written of a different size or (in teaching) of a different colour. Similarly when we get to ten tens we denote them by a new representation of the figure denoting one. Thus we may have:
On this principle 24 would represent twenty-four,24two hundred and forty, and24 two hundred and four. To prevent confusion thezeroor “nought” is introduced, so that the successive figures, beginning from the right, may represent ones, tens, hundreds, ... We then have,e.g.,240 to denote two hundreds and four tens; and we may now adopt a uniform type for all the figures, writing this 240.
(ii) On the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, repeating this series indefinitely. There is as yet no distinction between the first object marked 1 and the second object marked 1. We can, however, attach to the 0’s the same symbols, 1, 2, ... 0 in succession, in a separate column, repeating the series indefinitely; then do the same with every 0 of this new series; and so on. Any particular object is then defined completely by the combination of the symbols last written down in each series; and this combination of symbols can equally be used to denote the number of objects up to and including the last one (§ 10).
In writing down a number in excess of 1000 it is (except where the number represents a particular year) usual in England and America to group the figures in sets of three, starting from the right, and to mark off the sets by commas. On the continent of Europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal.
The zero, called “nought,” is of course a different thing from the letter O of the alphabet, but there may be a historical connexion between them (§ 79). It is perhaps interesting to note that the latter-day telephone operator calls 1907 “nineteen O seven” instead of “nineteen nought seven.”
14.Direction of the Number-Series.—There is no settled convention as to the direction in which the series of symbols denoting the successive numbers one, two, three, ... is to be written.
(i) If the numbers were written down in succession, they would naturally proceed from left to right, thus:—1, 2, 3, ... This system, however, would require that in passing to “double figures” the figure denoting tens should be written either above or below the figure denoting ones,e.g.
The placing of the tens-figure to the left of the ones-figure will not seem natural unless the number-series runs either up or down.
(ii) In writing down any particular number, the successive powers of ten are written from right to left,e.g.5,462,198 is
the small figures in brackets indicating the successive powers. On the other hand, in writing decimals, the sequence (of negative powers) is from left to right.
(iii) In making out lists, schedules, mathematical tables (e.g.a multiplication-table), statistical tables, &c., the numbers are written vertically downwards. In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals, though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93).
(iv) In graphic representation measurements are usually made upwards; the adoption of this direction resting on certain deeply rooted ideas (§ 23).
This question of direction is of importance in reference to the development of useful number-forms (§ 23); and the existence of the two methods mentioned under (iii) and (iv) above produces confusion in comparing numerical tabulation with graphical representation. It is generally accepted that the horizontal direction of increase, where a horizontal direction is necessary, should be from left to right; but uniformity as regards vertical direction could only be attained either by printing mathematical tables upwards or by taking “downwards,” instead of “upwards,” as the “positive” direction for graphical purposes. The downwards direction will be taken in this article as the normal one for succession of numbers (e.g.in multiplication), and, where the arrangement is horizontal, it is to be understood that this is for convenience of printing. It should be noticed that, in writing the components of a number 253 as 200, 50 and 3, each component beneath the next larger one, we are really adopting the downwards principle, since the figures which make up 253 will on this principle be successively 2, 5 and 3 (§ 13 (ii)).