NOTES AND ILLUSTRATIONS TO PART SECOND, FIRST DIVISION. NOTE A, p. 32.-Fundamental Laws of Belief. (2 1.) Or the fault in Euclid's arrangement which I have here remarked, some of the ancient editors were plainly aware, as they removed the two Theorems in question from the class of Axioms, and placed them, with at least an equal impropriety, in that of Postulates. "In quibusdam codicibus," says Dr. Gregory, Axiomata 10 et 11 inter postulata numerantur."-Euclidis quæ supersunt omnia. Ex Recensione Davidis Gregorii. Oxonii, 1703, p. 3. The 8th Axiom too in Euclid's enumeration is evidently out of its proper place. Καὶ τὰ ἐφαρμόζοντα ἐπ' ἄλληλα ἴσα ἄλληλοις ἐστί:—thus translated by Dr. Simson ; Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another." This, in truth, is not an axiom, but a definition. It is the definition of geometrical equality;-the fundamental principle upon which the comparison of all geometrical magnitudes will be found ultimately to depend. For some of these slight logical defects in the arrangement of Euclid's Definitions and Axioms, an ingenious, and, I think, a solid apology, has been offered by M. Prévost, in his Essais de Philosophie. According to this author, (if I rightly understand his meaning,) Euclid was himself fully aware of the objections to which this part of his work is liable; but found it impossible to obviate them, without incurring the still greater inconvenience of either departing from those modes of proof which he had resolved to employ exclusively in the composition of his Elements;1 or of revolting the student, at his first outset, by prolix and circuitous demonstrations of manifest and indisputable truths. I shall distinguish by italics, in the following quotation, the clauses to which I wish more particularly to direct the attention of my reader. "C'est donc l'imperfection (peut-être inévitable) de nos conceptions, qui a 1 By introducing, for example, the idea of Motion, which he has studied to avoid, as much VOL. III. as possible, in delivering the Elements of Plane Geometry. 2 A engagé à faire entrer les axiomes pour quelque chose dans les principes des sciences de raisonnement pur. Et ils y font un double office. Les uns remplacent des définitions. Les autres remplacent des propositions susceptibles d'être démontrées. J'en donnerai des exemples tirés des Elémens d'Euclide. "Les axiomes remplacent quelquefois des définitions très faciles à faire, comme celle du mot tout. (El. Ax. 9.) D'autres suppléent à certaines définitions difficiles et qu'on évite, comme celles de la ligne droite et de l'angle. Quelques axiomes remplacent des théorêmes. J'ignore si (dans les principes d'Euclide) l'axiome 11 peut être démontré (comme l'ont cru Proclus et tant d'autres anciens et modernes.) S'il peut l'être, cet axiome supplée à une démonstration probablement laborieuse. 66 Puisque les axiomes ne font autre office que suppléer à des définitions et à des théorêmes, on demandera peut-être qu'on s'en passe. Observons 1o. Qu'ils évitent souvent des longueurs inutiles. 2°. Qu'ils tranchent les disputes à l'époque même où la science est imparfaite. 3°. Que s'il est un état, auquel la science puisse s'en passer (ce que je n'affirme point) il est du moins sage, et même indispensable, de les employer, tant que quelque insuffisance, dans ce degré de perfection où l'on tend, interdit un ordre absolument irréprochable. Ajoutons 4°. Que dans chaque science il y a ordinairement un principe qu'on pourroit appeler dominant, et qui par cette raison seule (et indépendamment de celles que je viens d'alléguer) a paru devoir être sorti, pour ainsi dire, du champ des définitions pour être mis en vue sous forme d'axiome. Tel me paroit être en géométrie le principe de congruence contenu dans le 8 Axiome d'Euclide."-Essais de Philosophie, tom. ii. pp. 30, 31, 32. These remarks go far, in my opinion, towards a justification of Euclid for the latitude with which he has used the word Axiom in his Elements. As in treating, however, of the fundamental laws of human belief, the utmost possible precision of language is indispensably necessary, I must beg leave once more to remind my readers, that, in denying Axioms to be the first principles of reasoning in mathematics, I restrict the meaning of that word to such as are analogous to the first seven in Euclid's list. Locke, in what he has written on the subject, has plainly understood the word in the same limited sense. NOTE B, p. 54.-Fundamental Laws of Belief. (8 3.) The prevalence in India of an opinion bearing some resemblance to the Berkeleian Theory, may be urged as an objection to the reasoning in the text; but, on examination, this resemblance will be found much slighter than has been generally apprehended. (See Philosophical Essays, pp. 81, 82, et seq. [Works, vol. v., Essay ii., ch. 2].) On this point the following passage from Sir William Jones is decisive; and the more so, as he himself has fallen into the cominon mistake of identifying the Hindu belief with the conclusions of Berkeley and Hume. 'The fundamental tenet of the Védánti school consisted, not in denying the existence of matter, that is, of solidity, impenetrability, and extended figure, (to deny which would be lunacy,) but in correcting the popular notion of it, and in contending, that it has no essence independent of mental perception, that existence and perceptibility are convertible terms, that external appearances and sensations are illusory, and would vanish into nothing, if the divine energy, which alone sustains them, were suspended but for a moment;1 an opinion which Epicharmus and Plato seem to have adopted, and which has been maintained in the present century with great eloquence, but with little public applause; partly because it has been misunderstood, and partly because it has been misapplied by the false reasoning of some unpopular writers, who are said to have disbelieved in the moral attributes of God, whose omnipresence, wisdom, and goodness, are the basis of the Indian philosophy. I have not sufficient evidence on the subject to profess a belief in the doctrine of the Védánta, which human reason alone could, perhaps, neither fully demonstrate nor fully disprove; but it is manifest that nothing can be farther removed from impiety than a system wholly built on the purest devotion."- Works of Sir William Jones, vol. i. pp. 165, 166. From these observations, (in some of which, I must be permitted to say, there is a good deal of indistinctness, and even of contradiction,) it may on the whole be inferred—1. That in the tenets of the Védánti school, however different from the first apprehensions of the unreflecting mind, there was nothing inconsistent with the fundamental laws of human belief, any more than in the doctrine of Copernicus concerning the earth's motion. 2. That these tenets were rather articles of a theological creed than of a philosophical system; or, at least, that the two were so blended together, as sufficiently to account for the hold which, independently of any refined reasoning, they had taken of the popular belief. In this last conclusion I am strongly confirmed, by a letter which I had the pleasure of receiving, a few years ago, from my friend Sir James Mackintosh, then Recorder of Bombay. His good nature will, I trust, pardon the liberty I take in mentioning his name upon the present occasion, as I wish to add to the following very curious extract, the authority of so enlightened and philosophical an observer. Amidst the variety of his other important engagements, it is to be hoped that the results of his literary researches and speculations, while in the East, will not be lost to the world. ". . . . I had yesterday a conversation with a young Bramin of no great learning, the son of the Pundit (or assessor for Hindu law) of my Court. He told me that, besides the myriads of gods whom their creed admits, there was one whom they know by the name of BRIM, or the great one, without form or limits, whom no created intellect could make any approach towards conceiving; that, in reality, there were no trees, no houses, no land, no sea, but all without was Maia, or illusion, the act of BRIM; that whatever we saw or felt was only a dream, or, as he expressed it in his imperfect English, thinking in one's sleep; and that the reunion of the soul to BRIM, from whom it originally sprung, was the awakening from the long sleep of finite existence. All this you have heard and read before as Hindu speculation. What struck me was, that speculations so refined and abstruse should, 1 Sir William Jones here evidently confounds the system which represents the material universe as not only at first created, but as every moment upheld by the agency of Divine Power, with that of Berkeley and Hume, which, denying the distinction between primary and secondary qualities, asserts that extension, figure, and impenetrability, are not less inconceivable without a percipient mind, than our sensations of heat and cold, sounds and odours. According to both systems, it may undoubtedly be said that the material universe has no existence independent of mind; but it ought not to be overlooked, that in the one, this word refers to the Creator, and in the other, to the created percipient. |