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not merely quanta, they are quanta of a particular kind, and as such are subject to laws peculiarly spatial, such as “two points determine a right line." What has been said enables us to see the exact state of the case in reference to fractions. A fraction, as a portion of a permanent quantum, cannot be cogitated without an original reference to space; nevertheless the laws of such a quantum are laws not of space but of time; and when by means of an intuition in space time has been cogitated as an objective reality, we recognise time itself as the primitive quantum, and fractions can be applied to definite portions of time, marked out by the help of external intuitions, and not only to portions of time, but to all objects capable of being. generated by a synthesis in time.
The only passage in Kant referred to by Mr. Mahaffy in the note to pp. 15, 16, which I have not considered, is that in p. 180. I am surprised that Mr. Mahaffy should lay any stress on this passage. If his mode of arguing from it were valid, it would prove that the whole science of Mathematics, according to Kant, rests merely on the evidence of the senses, and is not a priori at all.
If it be true that children learn arithmetic through the intuition of space, it is in no way opposed to the theory which I have been endeavouring to establish; but I am very sure that arithmetic never was learned by means of lightning glances as far as 6, and beyond that number by a cogitatio caeca sive symbolica.
In the note to p. 95 of Fischer's Commentary, Mr. Mahaffy endeavours to account for what he calls the mistake of Mr. Mansel and Dr. Fischer. I have already, in a former part of this essay, considered fully the passages in Kant on number as the schema of quantity, and have adopted Mr. Mahaffy's definition of number with a slight alteration. I have only to add here, that Kant's implied statement that number, as the pure schema of quantity, cannot be reduced to any image, is so far from being favourable to Mr. Mahaffy, that it shows, perhaps, more clearly than any other
passage, the absolute impossibility, in Kant's opinion, of cogitating number as number, except by a successive apprehension and synthesis of its units in time.
Mr. Mahaffy seems to think that the apprehension of a set of objects as a number need not occupy time, and that when it does occupy time there is something peculiar in the mode of apprehension. There is no passage in Kant which I can find supporting this opinion. There is, indeed, one in p. 33,
which as rendered by Mr. Meiklejohn, is opposed to my theory. The following is the passage to which I refer : “Time and space are, therefore, two sources of knowledge from which, a priori, various synthetical cognitions can be drawn. Of this we find a striking example in the cognitions of space and its relations which form the foundation of pure mathematics." I find, on referring to the original, that his passage has been mistranslated, which is, perhaps, the reason why it has not been quoted by Mr. Mahaffy. The literal translation is as follows:
“Time and space are, therefore, two sources of cognition from which, a priori, various synthetical cognitions can be drawn; as especially pure Mathematic, in respect of the cognitions of space, and its relations, gives a brilliant example."
Kant plainly regarded geometry as the portion of Mathematical science, which exhibited most easily and strikingly the truth of his philosophical system. That he did not confine Mathematics to space and its relations is plain from what has been said already, and is explicitly stated in p. 441, “To determine, a priori, an intuition in space (its figure), to divide time into periods, or merely to cognize the quantity of an intuition in space and time, and to determine it by number, all this is an operation of reason by means of the construction of conceptions, and is called mathematical."
The whole theory of Kant, in reference to the foundation of the science of quantity, may, I believe, be thus summed up.
It is only in external intuition that we learn the objective reality of things as quanta ; through it only can we be assured of the objective reality of time itself as the permanent substratum of events, or cogitate time in general as an objective quantum, and definite portions of time as definite quanta ; but if we consider why quantity is attributed to certain objects, we find it is because they are capable of being generated by a synthesis in time, and that through such a synthesis only can they be cogitated as quanta. Thus considered logically, as Cousin would say, time is the primitive quantum, and its laws must be binding on all quanta. Number is altogether incogitable as number, except by the successive apprehension and synthesis of its units in space. It is the pure schema of quantity, for the concept of quantity cannot be explained except by saying it is the determination of a thing whereby it can be cogitated how many times one is placed in it. Numerical truths are based on intuitions in time, and must be exhibited by every collection of objects capable of being regarded as distinct, as well as by all quanta whose quantity is discoverable in apprehension. They are valid in reference to all quanta without exception.
In conclusion, I have only to add, that Kant's remarks, pp. 437, 447, on the method of Algebra, and Mr. Mahaffy's observations on them (note, p. 279, Fischer's Commentary), are most valuable, and tend to confirm the theory put forward in this essay.
By using letters to indicate what we regard as simple quanta, we can by means of the signs + &c., express symbolically the mode in which complex quanta may be constructed in time. By applying operations according to laws, already discovered by analysing intuitions in time, we discover the modes in which new quanta may be generated, and are able to prove that the same quantum may be generated in different ways. Results far too complicated to be discoverable by direct intuition may thus be reached, and, as the whole process is kept steadily before us, and submitted to ocular inspection by means of the symbols, we are secured from error in our deductions. The absolute universality of laws of time, as binding on all phenomena whatever, enables us to discover results, as Mr. Mill has observed, by direct intuition of the symbols themselves. Thus we ascertain how often a particular term occurs in a result by direct counting. By the power which we have of regarding a quantum as simple or complex at will, and of expressing this by the symbols used, we are enabled to concentrate our attention on some particular mode of generating a complex quantum, and to show that it leads to the same result as some other mode.
It is worthy of remark, as confirming to some extent the theory put forward in this essay, that in Algebra, the science of Quantity, there are no signs indicating operations or results of a purely spatial character. If the science of Quantity were altogether dependent on space this would be remarkable. It does not result from any inherent impossibility of expressing spatial relations symbolically, for, as is well known, this has been done in the Quaternion system of Sir W. R. Hamilton, where a + b means not the sum of two quanta, but the diagonal of the parallelogram whose sides are a and b,
TACITUS AND VIRGIL. By THOMAS J. B. BRADY, A. M.,
Trinity College, Dublin.
THE commentators on Tacitus and Virgil have noticed several passages in which the historian has evidently availed himself of the phraseology and thoughts of Virgil, but it does not seem to have occurred to any one to collect the parallel expressions, and present them at one view to the reader. The passages quoted in the present paper, though not forming an exhaustive list, nevertheless show that these echoes of Virgil are a marked and interesting feature in the writings of Tacitus, and prove an intimate acquaintance with, and appreciation of, the poet by the historian. The enthusiastic manner in which Virgil is spoken of in the Dialogus de Oratoribus would (if this work be really one of the writings of Tacitus) prepare us to find traces of the influence of the poet on the mind of the historian (Dial. de Oratoribus, chaps. 12, 13, and 20). Although the interest attaching to these parallelisms is chiefly literary, yet it would appear that they sometimes possess a critical value. In the case of disputed readings, if it be found that one reading recalls a Virgilian expression, it may fairly be preferred, even if less supported by MS. authority than others. Two passages may be here referred to. In Annals, XIII. 55, where the Medicean MS. has the corrupt reading “quotam partem campi iacere," the emendation of Lipsius (adopted by Orelli, Halm, and others), “quo tantam partem campi iacere?” appears greatly supported by Virgil, Georgics, III. 343, “tantum campi iacet.” In the Histories, II. 21, “nox parandis operibus assumpta,” the MS. is well supported against the emendation “absumpta” by the Virgilian parallel, Aen.