All the syllogisms of the first thirteen numbers fall under the first mood of the First Figure.

This syllogistic concatenation is the spinal cord of mathematical demonstration. The mathematician shortens the form

is fixed at one of the points hitherto remaining moveable, all motion is taken from the body.

In the treatise by Tiberghien, a follower of Krause (Logique, la Science de la Connaissance, Paris, 1865), the anti-Kantian opinion expressed in the above-quoted tractate, that the certainty of mathematical axioms is compatible with an empirical origin of the conception of space, is combated. I there said, that Kant's demonstration for the à priority of the intuition of space is only an indirect one, which is grounded on the disjunction given by experience or independent of all experience (empirical or à priori); and the demonstration is fallacious because of the incompleteness of the disjunction, for there is a third possibility, viz. the intellectual working up the empirical data according to logical laws without the à priori (i.e. independent of all experience) elements of knowledge. If we do not reach mathematical knowledge by direct observation, it does not follow that it is absolutely independent of all observation. The mathematical fundamental axioms are partly analytical judgments (§ 83), but are partly, so far as they are synthetical judgments, like physical judgments, eg. the law of gravitation based immediately on observation, the geometrical on the observation of the relations of space, and the arithmetical on the observation of objects of the same kind leading to the notion of number. From these fundamental axioms the theorems are derived by means of a syllogistic deduction, which does not follow purely subjective laws, but is founded on the presupposition of an objective arrangement, which our thinking only reproduces, and this presupposition itself rests on the combined external and internal experience (cf. §§ 28, 41 ff., 73, 81, and several of the remarks to §§ 137, 138 ff.). Tiberghien answers (p. 244 f.) with the question, Why does Kant leave the third possibility unnoticed? and answers it thus: It is because the critique of pure reason had shown that there is no knowledge without elements à priori, and that thus the elaboration proposed is manifestly absurd.' But this answer involves an error in reference to Kant's actual course of demonstration. It is only necessary to read Kant's work to be convinced that Kant proceeds upon this disjunction in his reasoning, that he employs it as premise, and not, as Tiberghien asserts, as a result or conclusion of a demonstration independent of it. The third possibility is to be called

of expression, but the syllogistic form of thought cannot be removed without destroying the force of the demonstration itself.

Physics also can explain particular phenomena from syllogistic rules only in a syllogistic form of thought. Every application of a mathematical formula to a given case is made by means of a syllogistic subsumption of the special under a universal relation of magnitude or situation. The province of the syllogism, and more especially of the mood Barbara, in Physics extends beyond that of the mathematical formula. The law that the warm body must radiate part of its warmth out through the atmosphere towards a colder surrounding it, if it is not separated from it by protecting media, and so must grow cold, adds to our meteorological knowledge by means of syllogistic subsumption, without being brought to a mathematical form. an evident absurdity,' if the subjective presupposition is thought to be an unalterable truth, that all orderly arrangement has its origin in ourselves only. But this itself is first deduced from this disjunction, whose completeness is what is questioned, and therefore the argument undeniably proceeds in a vicious circle. If Tiberghien on his side believes this hypothesis to be controvertible, there is not even the appearance of justice for that designation. Kant's rejection of the weightiest of all objections which have been raised against his doctrine, by a mere jest ('Ex pumice aquam!' Kr. d. pr. V., pref.), whose application involves the Kantian hypotheses, may be explained and excused from Kant's subjective isolation at his own stand-point, but not accepted. Lastly, so far as the infinitude of space goes, which Tiberghien lays so much stress on, this can only be understood by us in the negative sense, that the possibility of advance to any other place is not taken away, and only this notion is mathematical. Reimann and H. Helmholtz, in the above-quoted articles, recognise the empirical basis of Geometry as settled. Cf. Beneke, Syst. d. Log. ii. 51 ff.

[Another side of the Logic of Geometry stated in this paragraph, that the major premise so defines the geometrical notion that the intuitions from the construction may be dispensed with in the proof, is combated by Prof. W. R. Smith in two papers read before the Royal Society of Edinburgh (Transactions, vol. xxv.), 'On Mr. Mill's Theory of Geometrical Reasoning,' and 'Hegel and the Metaphysics of the Fluxional Calculus.' Cf. Translator's Preface.]

The superficies of the earth by night, under a clear sky, is warmer than the surrounding space, and is not separated from it by a cloud-covering protecting it from cooling. It therefore must radiate a part of its warmth, and become cooled (until the warmth of the sun makes reparation). The explanation of the formation of dew rests on the syllogism: Every cooling object whose temperature is below that of the so-called point of dew, attracts to itself out of the atmosphere a part of the watery vapour contained in it, and causes it to precipitate itself on it; the superficies of the earth, and especially of plants, are colder in clear nights than the atmosphere, in consequence of the radiation of the heat to the space around; and, therefore, when the cooling exceeds a certain limit, they attract a portion of the watery vapour contained in the atmosphere, and make it precipitate itself on them.

The application of grammatical laws to individual cases is a syllogistic process of thought. The verbs (verba sentiendi et declarandi) which denote an intellectual activity (the recognition an existence of what is) require in Latin the construction of the accusative with the infinitive; persuadere with the meaning to convince (that something is) denotes an intellectual activity, and requires this construction. Verbs which refer to a striving (after something which is to be) take the construction of ut; persuadere in the meaning of to persuade (to do something) belongs to this class, and, with this meaning, takes the construction of ut.

The like holds good of the application of the laws of justice. Theft is the crime in which a moveable thing belonging to another is taken from his possession or custody. The deed which this accused person has done is a crime of this kind. Therefore it is theft. Theft requires a severer punishment than the appropriation of something found (which was not in the possession or custody of another, if the former possessor had lost or abandoned it). The deed done by this accused person is theft. Therefore it requires severer punishment. In the application of a law to a given case the major premise is established by laying down the law; the minor premise has to

do with facts, and is found out by actual sight, avowal, testimony or circumstantial evidence. If an authentic interpretation lies between the law and its enforcement, then in this case the law is the major premise, and a delivery of the court, by which the meaning of an expression used in the law is stated (e.g. whether the erroneous subjective view, that something may have happened which has not happened, be an opinion' in the sense of law or not), is the minor premise, and a rule directly applicable to one of the individual cases present (or directly excluding this applicability) is the conclusion.

In the province of Ethics the particular is known from the general syllogistically, however much the expression may despise syllogistic prolixity. This prolixity indeed is not required, because the ethical relations lie so near the universal human consciousness. But the course of ethical thinking is syllogistic, if we, e.g. affirm of a definite person, whom we have known to be faithful to duty, that he is worthy of esteem. For we subsume the individual case under the universal law that fidelity to duty establishes the ethical claim to esteem.

The like holds good of the understanding of historical phenomena. Besides the explanation given by Schiller of the vehemence and length of the Thirty Years' War-that in religious wars, and more especially in later times, the individual takes his side from personal conviction-the following example may bear witness to the force of this form of thought. Those individuals who have freed the elements of culture, separately acquired by the noblest and best endowed peoples of antiquity, from their national limitations, and made them extend over all peoples of the earth capable of civilisation, are among the men of antiquity who are of most world-wide importance. Men who recognised the elements universally true for man in the rich treasury of Grecian Art and Science, of Roman law and policy, acquired by the labour of centuries, and, in a still greater degree in religious ideas jealously guarded by the Jewish people, who have freed their eternal truths from the temporal and transient

Already quoted, § 101.

veil of national restriction, who have advanced them to a new and purer position, and have prepared the way for their universal diffusion,—they are, each in his province, men who have freed elements of culture, &c. Hence, they are the men of antiquity who are of the most world-wide importance. If this conclusion is referred to individual persons, in whose effects on the history of the world that character has shown itself, this reference in its logical form takes the same mode of inference. If the major premise of the first syllogism requires verification, this is to be done only in a like syllogistic form of thought, viz. by presenting a universal law of development, which mankind, as one whole ethical organism, must obey.

§ 111. The three remaining moods of the First Figure in the stricter sense have the forms e a e, a ii, e i o, and take the names Celarent, Darii, Ferio. In these names the place in the alphabet of the initial consonant signifies the order of the moods, and the vowels in their succession signify the characteristic logical form of the major and minor premises and of the conclusion.

In the mood Celarent a universally negative conclusion (No S is P) is derived from a universally negative major premise (No M is P) and a universally affirmative minor premise (Every S is M), according to the following scheme :

The proof of the validity lies in the relation of spheres. If M is quite separated from P, and S contained in M, then S must be quite separated from P.