en. xvn] ON HUMAN UNDERSTANDING 571 and sometimes we do not know whether nature has them within its folds for the case in question. For example, what is apparently simpler than the notion of the prime number? i.e. a whole number indivisible by every other except unity and itself. Yet we seek also a positive and easy sign in order to recognize them certainly without trying all the prime divisors less than the square root of the given prime. There are a number of signs which make known without much calculation that a given number is not prime, but we ask for one which is easy and which makes known certainly that it is prime when it is so. This it is which also makes algebra as yet so imper fect, although nothing is better known than the ideas of which it makes use, since they signify only numbers in general; for the public has not yet the means of extracting the irrational roots of any equation beyond the fourth degree 1 (excepting in a very limited case), and the methods which Diophant,2 Scipio Ferreus,8 1 In Leibnitz's day, as the text states, equations of the 2d, 3d, and 4th degrees were reduced to pure equations, but the reduction of equations of higher degrees than the 4th remained an unsolved problem, on which mathe maticians spent much labor, until Niels Henrik Abel, 1802-1829, a Norwegian mathematician of great ability and acuteness, demonstrated (1824) that the quintic equation and a fortiori the general equation of any order higher than live, is incapable of solution by radicals. Cf. Abel, Demonstration de Vim- possibilite de la resolution alyebrique lies Equations generates qui passent le qitatrieme degre, in (Euvres completes, ed. by Holmboe, 2 vols., Christiania, 1839, Vol. 1, pp. 5-24, and in Crelle, "Journ. f. Math.," 182(i, Vol. 1, pp.
2 Diophantus, c. 325-c. 409, a celebrated Greek mathematician of the Alexandrian school, gave, in his Arithmeticorum lib. VI., a method for the solution of equations of the 1st and 2d degrees. The Ms. of his Arithmetic was discovered in 1460 in the Vatican Library by the astronomer Regiomon- tanus, 14'!6-1476, and was published in a Latin trans., without the original, by Xylamler, in 1575. The Greek text, with a more complete trans., and a com mentary by Bachet de Merzeriac, whose skill in indeterminate analysis especially fitted him for the task, appeared in 1621. The best ed., based upon that of Bachet, including the Greek text with Latin trans., is that by Pierre Fermat, 1601-1665, the celebrated French mathematician, who supplemented the commentary of Bachet by valuable notes of his own. It is found in Vol. 1, pp. 65-341 of Fermat, Opera Mathematica, 2 vols., fol., Tolosai, 1670, 1679.
-TR.
-TR.
8 Scipionc del Ferro orFerri, c. 1465-1525, an Italian mathematician, taught arithmetic- and geometry at Bologna from 1496 till his death. About 1505 he discovered the solution of a particular case of cubic equations, which he did not publish, but communicated to his favorite pupil Antonio del Fiore, who in 1635 challenged Tartaglia to a trial of skill in resolving algebraical problems requiring a knowledge of this rule. Tartaglia in 1630 had already solved two cases of cubic equations, and before the time for the contest came solved two 572 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv and Lewis Ferrari l used respectively for the second, third, and fourth degrees in order to reduce them to the first, or in order to reduce an affected equation to a pure, are wholly different from each other, i.e. that which is used for one degree differs a degree from that used for another. For the second degree, or the quadratic equation, is reduced to the first by merely eliminating the second term. The third degree, or the cubic equation, has been solved, because in separating the unknown quantity into parts there happily arises from these an equation of the second degree. And in the fourth degree, or the biquadratics, something is added to the two sides of the equation to render it capable of extraction on both sides, and then it is happily found that to obtain this result an equation of the third degree only is needed. But all this is only a mix ture of good luck or chance with art or method. And in trying it on these last two degrees we knew not whether it would be successful. Further still, another artifice is necessary to suc cess in the fifth or sixth degree, which are the sursolids or bicubes. And although Descartes believed that the method he used in the fourth, conceiving the equation as produced by two other quadratic equations (but which cannot at bottom give more than that of Lewis Ferrari), would succeed also in the sixth, it is not found to be so. This difficulty shoAvs that even the clearest and most distinct ideas do not always give us all we ask for and all that may be drawn from them. And this makes us also judge that algebra is very far from being the art more. He thus easily won the victory, as his problems could be solved only by one or the other of his own three rules which were unknown to Fiore, and not by the remaining rule which was the only one known to Fiore. Tartaglia's discoveries were improved and published by Cardan in connection with his own in 1545, as a supplement to a treatise on arithmetic and algebra pub lished in 1539. Cf. Cardan, Ojwa omnia, Vol. 4, pp. 249-204. On Ferro, c/. Libri, Hist, des Sciences Math, en Italic, Vol. 3, pp. 148-151; Montucla, Hist, des Math., Vol. 1, p. 479, ed. 1758, Vol. 1, p. 591, ed. 1799-1802. — TK.
1 Ludovico or Luigi Ferrari, 1522-1502, or 15G5, an Italian mathematician, a pupil of Cardano (cf. ante, p. 5G(>, note 1), and Professor of Mathematics at Milan and at the University of Bologna, discovered the demonstration of the formula for the resolution of equations of the 3d degree, sentbyTartaglia, c. 1500-1557, to Cardan under the form of an enigma, and shortly after this discovery solved equations of the 4th degree. For an account of his demonstration, cf. Cardan, Ars mar/na, 1545, chap. 15. De cubo et quadratis ajqualibus numero, § 3, Opera omnia, 10 vols., Lugdnni, 1G63, Vol. 4, p. 254. On Ferrari, cf. Libri, Hist, des Sciences Math, en Italic, Vol. 3, pp. 180, 181; Montucla, Hist, des Math., Vol. 1, pp. 484, 485, ed. 1758, Vol. 1, pp. 596, 597, ed. 1799-1802. — Tn.
en. xvii] ON HUMAN UNDERSTANDING 573 of invention, since it needs a more general art; and we may say, indeed, that the art of signs (Specieuse) in general, i.e. the art of characters, is a marvellous means of assistance, since it aids the imagination. It will not be doubted in view of the arithmetic of Diophant and the geometrical books of Apollonius and Pappus that the ancients possessed it to a certain extent. Vieta l has given it more extension by expressing not only what is asked for, but also the given numbers, by general characters, doing in calculating what Euclid already did in reasoning, and Descartes has extended the application of this calculus to geometry, indicating lines by equations. Nevertheless, even after the discovery of our modern algebra, Bouillaud 2 (Tsmael Bullialdus), no doubt an excellent geometer, whom, moreover, I knew in Paris, regarded only with wonder the demonstrations of Archimedes upon the spiral, and could not understand how this great man had thought of employing the tangent of this line as the dimension of the circle. Father Gregory of St. Vincent3 appears to have divined it, thinking that it was at tained by the parallelism of the spiral and the parabola. But this method is only a particular one, whilst the new calculus of infinitesimals 4 which proceeds by the method of the differ ences which I have thought of and successfully shared with the public, gives a general one, wherein this discovery concerning 2 Ismael Boulliau, I(i05-16!>4, a French mathematician and astronomer, who was the first to give, in his Ad astro noinos inonita duo, lt>57, a plausible ex planation of the change in the light of some stars by attributing to them an axial revolution which shows successively their obscure and luminous parts. He was the author of several works, among which is the De lineis spiralibus demons tr at iones, 1657, which Leibnitz, perhaps, had in mind here. — TR.
3 Gregoire de Saint-Vincent, 1584-16<;7, a Flemish geometer who was much occupied with the problem of the quadrature of the circle, and whose prin cipal work is the Opus geometricum quadrature circuit et section/at n com', 4 For Leibnitz's account of his discovery of the " calculus of infinitesimals," cf. his Historia et orif/o calculi differentialis, Gerhardt, Leibniz, math. Sch'rift., II., 1 [Vol. 5],:592-410; also his letter, April 18, 171(i, to the Countess Kiel- mannsegge, Dutens, Lribitit. op. orn., 3, 45(i-4u'l. For Leibnitz's various writ ings on the subject, cf. Gerhardt, op. cit., II., 1 [Vol. 5], 141^18; Dutens, up. cit., Vol. 3, passim. Dutens, Vol. 3, contains also much material concerning the controversy between Leibnitz and Newton regarding the discovery of the calculus. Further accounts are given in Guhrauer, Leibnitz, Eine Biographic, 1, 280-320, Jaucourt, Historia vitse Leibnilii, and Montncla, Hist. d. Math., Vol. 2 (both in Dutens, op. cit. Vol. 3, pp. xii-xl, xli-lv), and in Encyclop. Brit., 9th ed., Vol. 13, Article, " Infinitesimal Calculus." — TK.
574 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv the spiral is mere play and a sample of the easiest, like nearly all we have before discovered in the matter of the dimensions of curves. The reason of the advantage of this new calculus is, moreover, that it relieves the imagination in the problems, which Descartes excluded from his geometry under the pre text that they most frequently lead to mechanics, but at bottom because they did not agree with his calculus. As for errors arising from ambiguous terms, it is our business to avoid them.
Ph. There is also a case in which reasoning cannot be applied, but in which also there is no need of it and in which sight is worth more than reasoning. It is in intuitive knoivl- edye, where the connection of ideas and truths is immediately seen. Such is the knowledge of indubitable maxims, and I am tempted to believe that this is the degree of evidence which the angels have at present, and which the spirits of just men made perfect will have in a future state regarding a thousand things which now escape our knowledge. § 15. But demon stration, based upon mediate ideas, gives a reasoned knowledge. This is because the connection of the mediate idea with the extremes is necessary and is seen by a juxtaposition of evidence, similar to that of a yard-stick applied now to one cloth and now to another to show that they are equal. § 16. But if the con nection is only probable, the judgment gives only an opinion.
Th. God alone has the advantage of having only intuitive knowledge. But very happy souls, however detached they are from these material bodies, and the genii themselves, however exalted they are, although they have a knowledge without com parison more intuitive than ours, and often see at a glance what we discover only by the force of consequences after having em ployed time and labor, must likewise find difficulties in their path without which they would not have the pleasure of making dis coveries, which pleasure is one of the greatest. And we must always admit that there will be an infinite number of truths concealed from them either wholly or for a time, whereto they must attain by force of consequences and by demonstration or even frequently by conjecture.
Ph. [These genii then are only animals more perfect than we; it is just as if you said with the emperor of the moon: it is all as here.~\ CH. xvn] ON HUMAN UNDERSTANDING 575 Tli. I will say so, not entirely, but in regard to the ground of things, for the modes and degrees of perfection vary infinitely. Meanwhile the ground is everywhere the same, a maxim which is fundamental with me and reigns in all my philosophy. I con ceive things unknown or confusedly knownonly after the manner of those which are distinctly known to us; a procedure which makes philosophy very easy, and I believe indeed that it must do so. But if this philosophy is simplest in its ground, it is also the richest in its modes, because nature may vary them infinitely, as indeed she has done with as much abundance, order, and ornateness as it is possible to imagine. This is the reason why I believe there is no genius however sublime who has not an infinite number of them above him. Yet although we are very inferior to so many intelligent beings, we have the advantage of not being visibly controlled upon this globe where we hold indisputably the first rank; and with all the ignorance in which we are immersed we have always the pleasure of seeing nothing which surpasses us. And if we were vain we might judge as Caesar, who preferred to be first in a country town rather than second in Rome. For the rest, I speak here only of the natural knowledge of these spirits and not of the beatific vision,1 or of the supernatural light that God is pleased to give them.
1 The term " beatific vision" (visio beatified) denotes in theological and religious thought the direct and immediate or intuitive vision of God enjoyed by the saints and angels in heaven and supposed to constitute their essential bliss. The philosophical significance of the idea as historically developed, with respect both to its speculative and practical uses, lies in the fact that the visio beatific<i was regarded as the sole means of obtaining absolute truth and of realizing absolute blessedness. The idea thus involved more or less ex plicitly a species of knowledge supernaturally mediated in some unknown and unexplained way, but considered, because of the method of its mediation, as far superior in certainty and completeness to any knowledge that finite beings could attain through the unaided action of their own intellectual powers — in brief, as the very perfection of knowledge attainable by such beings.
The idea originated in Plato's conception of an immediate intuition, going beyond rational thought, of the pure forms of reality or the Ideas. Trans formed by 1'hilo and Plotinus into their ecstatic intuition, or that identifica tion of the human with the Divine in which all consciousness of individual personality is lost; combined by Clement and Origen, in view of certain ex pressions in the Pauline epistles, with the thought of a personality in union with whom the self-consciousness of the individual is preserved; and still further developed by Augustine, as the principle of the absolute and imme diate certainty of inner experience or consciousness involving within itself the idea of God as the absolute personality and the sum and essence of all 570 LEIBNITZ'S CKITIQUE OF LOCKE [BK. iv § 19. Ph. As each one makes use of reasoning either with regard to himself or with reference to another, it will not be useless to make some reflections xipon four sorts of arguments which men are wont to use in order to draw others to their opinions or at least so to keep them in awe as to prevent them from contradicting. The first argument may be called argu- mentum ad vercundiam, when we cite the opinion of those who have acquired authority by their knowledge, rank, power, or otherwise; for when another does not yield to it promptly, he is liable to be censured as full of vanity and even to be charged with insolence. § 20. There is also 2) argumentum ad ignorantiam, i.e. to demand that the opponent admit the proof or assign a better. § 21. There is 3) argumentum ad kominem, when we press a man by what he has himself said. § 22. Finally, there is 4) argumentum adjudicium, which con sists in employing proofs drawn from some one of the sources of knowledge or probability. This is the only one of all which advances and instructs us; for if from respect I dare not con tradict, or if I have nothing better to say, or if I contradict myself, it does not follow that you are right. I may be modest, ignorant, deceived, and you prove yourself to be mistaken also.
Th. It is doubtless necessary to make a difference between what is proper to be said and what is truly to be believed. Yet as the majority of truths may be boldly maintained, there is some prejudice against an opinion that it is necessary to con ceal. The argument ad ignorantiam is valid in cases of pre sumption where it is reasonable to hold to an opinion till the contrary is proved. The argument ad hominem has this effect, that it shows that one or the other assertion is false and that truth, this conception passed into the philosophical and religious thinking and life of the Christian Church, and became especially prominent in the teachings of the Mediaeval Mystics. On this historical development cf. "\Vin- delband, Hist, of Phllos., trans, by Tufts, pp. 119 sq., 227 sq., '249 sq., '27()tiq.; Zeller, Philos. d. Griech., III., 2 [Vol. 6], 413 sq., 611 sq., 854, note 4, 3d ed., 1881; Bonn, Greek P/iilosophers, 2, 311 sq.
In a modified form this intuition of divine things became what the Church fathers and the theological and philosophical writers of the Middle Age termed the lumen f/ratise, " the light of grace," the supernatural light given through divine inspiration, in opposition to the lumen naturule or " natural light," the rational knowledge given by nature to all men as such. Cf. Neiv Essays, Bk. I., chap. 1, § 21, Th., ante, p. 71; Gerhardt, Leibniz, philos. Schrift.,6, 4!>4s7., nOSttq.; Hamilton's Reid, Note A, § V., IV.. 1, note f, Vol. 2, p. 7(>3, § VI., 20-22, 25-20, pp. 77(>-778, 54, p. 785, 8tii ed., 1880. — TK.
en. xvii] ON HUMAN UNDERSTANDING 577 the opponent is deceived whatever way he takes it. We might bring yet other arguments which are used, for example the one we might call ad vertiginem, when we reason thus: if this proof is not received we have no means of attaining certainty upon the point in question, which we take as an absurdity. This argument is valid in certain cases, as if any one wished to deny primitive and immediate truths, for example, that any thing can be and not be at the same time, or that we ourselves exist, for if he were right there would be no means of knowing anything whatever. But when certain principles are produced and we wish to maintain them because otherwise the entire system of some received doctrine would fall, the argument is not decisive; for we must distinguish between what is neces sary to maintain our knowledge and between what serves as a foundation for our received doctrines or practices. Use was sometimes made among jurisconsults of probable reasoning in order to justify the condemnation or torture of pretended sor cerers upon the deposition of others accused of the same crime, for it was said: if this argument falls, how shall we convict them? And sometimes in a criminal case certain authors maintain that in the facts where conviction is more difficult, more slender proofs may pass as sufficient. But this is not a reason. It proves only that we must employ more care, and not that we must believe more thoughtlessly, except in the case of extremely dangerous crimes, as, for example, in the matter of high treason, where this consideration has weight, not to condemn a man, but to prevent him from doing harm; so that there may be a mean, not between guilty and not guilty, but between condemnation and banishment in the jxulgments, where law and custom admit it. Use has been made of a similar argument in Germany for some time in order to give color to the coining of bad money; for (they say) if we must keep to the prescribed rules, we cannot coin it without loss. We must be allowed then to debase its alloy. But besides the fact that we must diminish the weight only and not the alloy or super scription the better to obviate frauds, we suppose a practice necessary which is not so; for no command of heaven nor any human law exists obliging those who have no mine nor occa sion to have silver in bars to coin money; and to make money out of money is a bad practice which naturally carries deterio- 578 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv ration with it. But how (they say) shall we exercise our regale in coining it? The reply is easy. Content yourselves with coining a little from good silver, even at a small loss, if you think its coinage a matter of importance to yourselves, since you have no need nor right to flood the world with debased small coin.
§ 23. Ph. After having said a word concerning the relation of our reason to other men, let us add something about its relation to God, which makes us distinguish between what is contrary to reason and what is above reason. Of the first class is everything which is incompatible with our clear and dis tinct ideas; of the second is every thought whose truth or probability evidently cannot be deduced from sensation or reflection by the aid of reason.1 Thus the existence of more than one God is contrary to reason, and the resurrection of the dead is above reason.
Th. I find something to say regarding your definition of that which is above reason, at least if you connect it with the received use of this phrase; for it seems to me that from the manner in which this definition is couched, it goes too far in one direction and not far enough in the other; and if we fol low it, all that of which we are ignorant and which in our present condition we are unable to know, would be above reason, for example, that a given fixed star is greater or less than the sun; again, that Vesuvius will send out fire in such a year; these are facts the knowledge of which is beyond us, not because they are above reason, but because they are beyond our senses; for we could very well judge of them, if we had more perfect organs or more information about the circumstances. There are also difficulties which are beyond our present faculty, but not beyond reason as a whole; for example, there is n.o astronomer here below who can calculate the detail of an eclipse in the space of a pater and without taking the pen in hand, yet there are perhaps genii to whom that would be mere play. Thus all tilings might be made known or practicable by the aid of reason, by supposing more information concerning the facts, more perfect organs, and a more elevated mind.
1 Cf. Locke, Exam, of Malebranche, § 53, Philos. Wks., Vol. 2, p. 455 (Bohn's CH. xvn] ON HUMAN UNDERSTANDING 579 Ph. This objection ceases if I understand my definition not only of our sensation or reflection, but also of that of every other possible created spirit.
Th. If you take it so, you are right. But the other diffi culty remains, viz.: that there will be nothing above reason according to our definition, because God will always be able to give the means of apprehending by sensation and reflection any truth whatever; as in reality the greatest mysteries be come known to vis by the testimony of God which we recognize by the motives of credibility, upon which our religion is based. And these motives undoubtedly depend upon sensation and reflection. The question then seems to be not whether the existence of a fact or the truth of a proposition can be deduced from the principles which reason uses, i.e. from sensation and reflection, or rather the external and internal sense, but whether a created spirit is capable of knowing the how of this fact, or the a priori reason of this truth; so that we may say that what is beyond reason may indeed be apprehended, but it can not be comprehended by the means and forces of created reason, however great and exalted it be. It is reserved to God alone to understand it, as it belongs to him alone to assert it.1