The problem of which Leibnitz here speaks is the modern form of the prob lems of the rectification and quadrature of the circle or the calculation and construction of TT and of a square mathematically exactly equal in area to a given circle. In referring to it he distinguishes between an "ordinary," or algebraic, and an "extraordinary," or, as it is now termed, transcendental equation. The problem is to prove that TT cannot be the root of any equation having whole numbers for coefficients, or that TT is not algebraical. The quad rature of the circle has long been known to be an unsolvable problem, — Leib nitz knew this, — but the impossibility of its solution has only recently been demonstrated. Not until mathematicians possessed the methods furnished by the theory of definite integrals and the departments of higher algebra devel oped in the last few decades was this demonstration possible. With the aid of these methods Prof. Lindemann of Konigsberg succeeded, in June, 1882, in demonstrating with exactness the non-algebraic character of TT, and thus proved for the first time that the rectification and the squaring of the circle with ruler and compasses is impossible. For his proof, cf. Uber die Ludolph'sche Zahl, in the " Sitzungsberichte d. kungl. Pr. Akad. d. Wiss., zu Berlin," June 22, 1882, pp. 679-682; the " Comptes Rendus " of the French Academy, Vol. 115, pp. 72-74, " Math. Annalen," Vol. 20, pp. 213-225.
For an historical sketch of the problem of circle-quadrature, cf. Holtzen- dorff and Virchow, Sammlung gemeinverstiindlicher wissenschaftlicher Vor- triif/e, Heft 67; an article from this by H. Schubert, in " The Mouist," January, 1891, Vol. 1, No. 2, pp. 197-228, reprinted in Smithsonian Report, 1890, pp. 97- 120; and the article "Squaring of the Circle," in the Encyclnp. Brit., 9th ed., Vol. 22, pp. 450-454 (American Reprint). On Leibnitz's relation to the prob lem, cf. Kummer, Festrede am Leibniz t age, in the " Monatsberichte d. Pr. Akad. d. Wiss.," July, 1867, p. 387 sq., an article which, while not directly taking into consideration the present interesting passage for the history of mathematics, is admirable in its understanding of the peculiarities of Leib nitz's method of thought. — TR.
426 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv tion, which would introduce the irrational or even the unknown quantity into the exponent/ for which, however, an extended calculation would be reqiiired, and in which the difficulty will not easily be solved unless we some daj7 find a short method for its solution. But to exclude all the finite expressions is impos sible, as I myself know, and to determine exactly the best is an immense task. And all this shows us that the human mind proposes questions so strange, especially when the infinite enters therein, that we must not be astonished if there is some difficulty in making them out, so much the more as all depends often on a short method in these geometrical matters, which cannot always be determined on, just as fractions cannot always be reduced to the lowest terms or the divisors of a number be found. It is true that we may always have these divisors if they are possible, because their number is finite; but when what we have to examine is infinitely variable and ascends by degrees, we are not its master though we wish to be, and it is too laborious to do all that is necessary in order to attempt methodically to reach the short method or the rule of progression exempting us from going farther. And as its usefulness does not correspond to the labor, its success is left to posterity, which will be able to enjoy it when this labor or prolixity is diminished by the new preparations and means which time may furnish. Unless the persons who devote themselves from time to time to these studies determine to do properly what is necessary in order to further progress, we cannot hope to advance much in a short time. And we must not think that all is done, since indeed in ordinary geometry, we still have 110 method for determining the best constructions when the problems are a little complex. A certain progres sion of synthesis should be mixed with our analysis in order the better to succeed. And I remember to have heard it said that the Pensionary De Witt 2 had some thoughts on this subject.]
1 Cf. Letter to Arnauld, Gerhardt, Leibniz, philos. Schrift., 2, 61, 62, Janet, CEuvres philos. de Leibniz, 1, 622. — Tu.
2 John De Witt, 1625-1<>72, an illustrious Dutch statesman, was a steadfast opponent of the House of Orange, whose re-elevation to power in the United Provinces lie labored earnestly and for many years successfully to prevent. His plans for his country were finally defeated by the diplomacy of Louis XIV., the opposition of the Calvinist clergy, and the change in the popular CH. in] ON HUMAN UNDERSTANDING 427 Ph. There is, indeed, another difficulty, to know whether a purely material being thinks or not,1 and perhaps we shall never be capable of knowing this, although we have ideas of matter and of thought, for the reason that it is impossible for us to discover by contemplation of our own ideas without revelation, whether God has not given to some masses of matter, fitly dis posed, the power to perceive and to think, or whether he has not united and joined to matter so disposed an immaterial substance that thinks. For as regards our notions, it is no more difficult for us to conceive that God, if he pleases, can add to our idea of matter the faculty of thinking, than to under stand that he joins to it another substance with the faculty of thinking, since we know not wherein thinking consists, nor to what kind of substance this all powerful Being has been pleased to give that power, which cannot exist in any created being save by virtue of the good pleasure and the bounty of the Creator.
Tli. [This question is without doubt incomparably more important than the preceding; but I venture to say to you, sir, that I wish it were as easy to touch souls in order to influence them for their good, and to heal bodies of their diseases, as I think it is in our power to determine this ques tion. I hope you will admit at least that I can advance icith- out offending against modesty and without speaking as a master in default of good reasons; for besides speaking only accord ing to received and common opinion, I think I have brought thereto an attention not common. In the first place, I grant you, sir, that when we have only confused ideas of thought and of matter, as is ordinarily the case, we must not be astonished if we do not see the means of solving such questions. As I remarked a little before, one who has only the ideas of the angles of a triangle commonly held will never think of finding feeling towards the Prince of Orange, occasioned by the recollection of their country's obligations to his ancestors. He was massacred, with his brother Cornelius, in the revolution which put the Prince William (afterwards William III. of England) at the head of the United Provinces. He published Elementa linearum curvurum, Leyden, K>50; The Hague, 1709. — Tit.
1 Cf. ante, p. 5<i sq., where Leibnitz shows that Locke afterwards gave up the opinion which he once advanced as possible that matter can think. For Leib nitz, who regards matter as a mere phenomenon and not a reality, the question does not exist. — TR.
428 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv them always equal to two right angles. We must consider that matter taken as a complete being (i.e. secondary matter in distinction from the primary, which is something purely passive and consequently incomplete) is only a mass, or that which results therefrom, and that every real mass supposes simple substances or real unities, and when we further consider what belongs to the nature of these real unities, i.e. perception and its consequences, we are transferred so to speak into another world, that is to say into the intelligible world of substances while before we have been only among the phenomena of the senses. And this knowledge of the interior of matter shows us sufficiently its natural capability, and that whenever God shall give it organs suitable for rational expression, the imma terial substance which reasons will not fail to be given it also, in virtue of that harmony which is also a natural consequence of substances. Matter cannot subsist without immaterial substances, i.e. without the unities; after which the question should no longer be asked whether God is free to give them to it or not; and if these substances had not in themselves this correspondence or harmony of which I have just spoken, God would not act in accordance with the natural order. To speak in an entirely simple manner of giving or according powers is to return to the naked faculties of the schoolmen and to im agine minute self-subsisting entities, which may go in and out like pigeons from a pigeon-house. It is making substances of them without being aware of it. The primitive powers con stitute substances themselves, and the derivative powers, or, if you prefer, the faculties, are only modes of being, which must be derived from substances, and are not derived from matter so long as it is only a machine, i.e. so long as it is abstractly considered only as the incomplete essence of primary matter, or passivity pure and simple. As to which I think you will agree, sir, that it is not within the power of mere mechanism to produce perception, sensation, reason. They must then spring from some other substantial thing. To desire God to act otherwise and to give to things accidents which are not modes of being or modifications derived from substances, is to have recourse to miracles and to what the schoolmen called the obediential power, by a kind of supernatural exaltation, as when certain theologians claim that the fire of hell burns up sepa en. in] ON HUMAN UNDERSTANDING ±29 rated souls. In which case it may indeed be doubted whether it was the fire that acted or whether God did not himself produce the effect, acting in place of the fire.]
Ph. You surprise me somewhat by your elucidations, and you anticipate me in many of the things I was going to say to you upon the limits of our knowledge. I should have said to you that we are not in a state of vision, as the theologians call it, that faith and probability must suffice us as regards many things, and particularly as regards the immateriality of the soul; that all the great ends of morality and religion are established upon sufficiently good foundations without the aid of the proofs of this immateriality drawn from philosophy; and that it is evident that he who has begun to make us sub sist here as sensible and intelligent beings, and who has pre served us many years in this state, can and will make us enjoy also a similar state of sensibility in the other life, and make us capable of receiving there the retribution he has designed for men according as they shall have conducted themselves in this life; in fine that we may judge by this that the necessity to determine for or against the immateriality of the soul is not so great as some people too zealous for their own views have wished to persuade us. [I was going to say all this to you, and more besides to the same effect, but I see now how differ ent is the statement that we are sensible, thinking, immortal beings by nature and the statement that we are so only by mir acle. It is a miracle, in fact, which I know I must admit if the soul is not immaterial; but this view of miracle, besides being without foundation, will not produce a sufficiently good effect in the minds of most people. I see clearly also from the way you understand the matter, that we can decide rationally as regards •the present question, without finding it needful to depart to the enjoyment of the state of vision and to find ourselves in the company of those superior spirits who penetrate very deeply into the internal constitution of things and whose living and penetrating sight and vast field of knowledge may make us imagine by conjecture what happiness they must enjoy.] I had supposed it entirely beyond our knowledge to combine sen sation ivith extended m>atter, and existence with a thing which has absolutely no extension. I had therefore become convinced that those who took sides here followed the unreasonable method 430 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv of certain persons, who, seeing that things considered from a certain side are incomprehensible, throw themselves headlong upon the opposite side, although it is no less unintelligible; a procedure which arose in my opinion from the fact that some having their mind too deeply buried so to speak in matter, could not accord any existence to that which is not material; and others not finding that thought is included in the natural faculties of matter, concluded that God himself could not give life and perception to a solid substance without putting therein an immaterial substance; while I now see that if He had done so it would be by a miracle, and that this incomprehensibility of the union of the soul and the body or of the union of sensa tion with matter seems to cease through your hypothesis of pre- established harmony between different substances.]
Th. [In fact there is nothing unintelligible in this new hypothesis, since it attributes to the soul and to bodies only the modifications which we experience in ourselves and in them; and only makes them appear more regular and more connected than has been thought hitherto. The difficulty which remains exists only as regards those who wish to imagine what is only intelligible,1 as if they wished to see sounds or hear colors, and these are they who refuse existence to everything which is net extended, a view which will compel them to refuse it to God himself, i.e. to renounce the causes and reasons of changes and of such changes: these reasons being incapable of arising from extension and from natures purely passive, and not indeed wholly from particular and inferior active natures without the pure and universal act of the supreme substance.]
Ph. One objection remains for me with reference to things whose matter is naturally susceptible of feeling. The body so far as we can conceive it, is capable only of striking and effecting a body, and motion can produce nothing but motion: so that when we agree that the body produces pleasure or pain or the idea of a color or sound, we seem compelled to abandon our reason, to go beyond our own ideas, and to at tribute this production solely to the good pleasure of our Creator. What reason have we then to conclude that it is not the same with perception in matter? I almost see what CH. in] ON HUMAN UNDERSTANDING 431 reply you can make, and although you have already said some thing regarding it more than once, I understand you better now, sir, than I have done. But I shall be very glad to hear further what reply you will make regarding it upon this important occasion.
Tli. [You rightly judge, sir, that I shall say that matter cannot produce pleasure, pain, or thought in us. It is the soul itself which produces them in conformity to what takes place in matter. And some clever people among the moderns begin to declare that they understand occasional causes only as I. Now this being posited, there occurs nothing unintelligible, except that we cannot distinguish all that enters into our con fused perceptions, which contain even the infinite, and which are the detailed expression of what occurs in bodies. As for the good pleasure of the Creator, it must be said that he is ruled by the natures of things, so that he produces and con serves therein only what suits them and can be explained, at least in general, by their natures; for the detail often sur passes us as much as the care and power of arranging the grains of a mountain of sand according to the order of the figures, although there is here nothing difficult to understand but the multitude. Otherwise if this knowledge were in itself beyond us, and if we could not indeed conceive the rea son of the relations of the soul and body in general, in fine, if God gave to things accidental poivers detached from their natures, and consequently removed from reason in general, there would be a back door for calling back the too occult quali ties which no mind can understand, and these little goblins of faculties incapable of reason, Et quidquid Scliola finxit otiosa: helpful goblins who proceed to appear like the gods of the theatre, or like the fairies of the Amadis, and who will do at need all that a philosopher wishes, without ceremony and without tools. But to attribute the origin of these powers to the good pleasure of God appears to me a thing not quite congruous with him who is the supreme reason, with whom everything is regular, everything consistent. This good pleasure would not indeed be good, nor pleasure, if there were not a perpetual parallelism between the power and the wisdom of God.]
432 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv § 8. Ph. Our knowledge of identity and diversity goes as far as our ideas, but that of the connection of our ideas, §§9, 10. as regards their coexistence in one and the same subject is very imperfect and almost nothing § 11. especially as regards secondary qualities as colors, sounds, and tastes § 12. because we do not know their connection with the primary qualities, i.e. § 13. how they depend upon size, figure, or motion. § 15. We know a little more of the incompatibility of the secondary qualities; for a subject cannot have, for example, two colors at the same time, and when they seem to be seen in an opal or in an infusion of lignum nephriticum, it is in different parts of the object. § 16. It is the same with the active and passive powers of bodies. Our researches in this direction must depend on experience.
Th. [The ideas of sensible qualities are confused, and the powers which should produce them furnish in consequence only ideas into which some confusion enters: thus the connections of these ideas can be known otherwise than by experience only as they are reduced to the distinct ideas which accom pany them, as has been done (for example) in regard to the colors of the rainbow and of prisms. And this method pre sents a beginning in analysis which is of great use in physics; and by following it I doubt not that medicine in time will find itself considerably more advanced, especially if the public is a little better interested than hitherto.]
§ 18. Ph. As for the knowledge of relations it is the largest field of our knowledge and it is difficult to determine how far it may extend. Progress depends on our sagacity in finding intermediate ideas. Those who are ignorant of algebra cannot imagine the wonderful things that may be done in this field by means of this science. And I do not see that it is easy to determine what new means of perfecting other parts of our knowledge may yet be found out by a penetrating mind. At least the ideas regarding quantity are not the only ones capable of demonstration; there are others, perhaps the most important part of our contemplation, from which we might deduce certain knowledge, if vices, passions, and domineering did not directly oppose the execution of such enterprise.
Th. [Nothing is truer than what you, sir, here say. What is there more important, supposing it is true, than what I be- CH. in] ON HUMAN UNDERSTANDING 433 lieve we have determined upon the nature of substance, upon unity and multiplicity, upon identity and diversity, upon the constitution of individuals, upon the impossibilities of void and atoms, upon the origin of cohesion, upon the law of con tinuity, and the other laws of nature; but chiefly upon the harmony of things, the immateriality of souls, the union of the soul and the body, the conservation of souls, and even of the animal after death. And there is nothing in all this which I do not think demonstrated or demonstrable.]
Ph. [It is true that your hypothesis appears extremely con sistent and of great simplicity: a clever man in France who desired to refute it, admits publicly that he was impressed by it. And it is a simplicity very different from that which I see. It will be well to show this doctrine more and more in its true light. But in speaking of things which are of most importance to us, I thought of morality to which I admit your metaphysic gives wonderful foundations: but without digging so deep, it has sufficiently firm foundations, although perhaps they do not extend as far (as I remember that you remarked) when a nat ural theology like yours is not their base. Yet the considera tion of the goods of this life alone already serves to establish important consequences for regulating human society. We can estimate justice and injustice as incontestably as in math ematics; for example this proposition: There cannot be injustice ivhere there is no property, is as certain as any demonstration which is in Euclid; property being the right to a certain thing, and injustice the violation of a right. It is the same with this proposition: No government allows absolute liberty. For gov ernment is the establishment of certain laws, whose execution it demands, and absolute liberty is the power each one has of doing whatever he pleases.
Th. [You use the word property a little differently from its ordinary use, for you mean by it the right of one person to a thing to the exclusion of the right of another. Thus if there were no property, as if all were common, there nevertheless might be injustice. By thing in the definition of property you must also further understand action; for otherwise, if there were therein no right to things, it would be always an injustice to prevent men from acting where they find it needful. But according to this explanation it is impossible that there be 110 434 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv property. As for the proposition concerning the incompati bility of government with absolute liberty, it belongs to the number of the corollaries, i.e. the propositions, which it is sufficient to point out. There are some in jurisprudence which are more complex, as for example, those concerning what is called jus aocrescendi,1 concerning the conditions and many other matters; and I indicated them when I published in my youth some theses upon the conditions, in which I demon strated some of them. And if I had leisure, I would retouch them.2] Ph. [That would afford pleasure to the curious, and serve to anticipate any one who might reprint them without re vision.]
Tli. [That is what happened to my " Ars Combinatoria," 3 as I have already complained. It was a fruit of my early youth, and yet it was reprinted a long time after without consulting me and without indicating even that it was a second edition, and this made some think to my prejudice that I was capable 1 Cf. Poste, Gains, Elements of Roman Law, Bk. II., 199 (p. 262, 3d ed., Oxford, 1890): " Illud constat si duobus pluribusve per vindicationem eadem res legata sit, sive conjunctim, sive disjunctim, si omiies veniant ad legatum, partes ad singulos pertinere, et deficientis portionem collegatario adcrescere." Also Sandars, Inst. of Justinian, Lib. II., Tit. XX., 8 (p. 226, 8th ed., London, 1888): "Si eadem res duobus legata sit, sive conjunctim sive disjunctim: si ambo perveniant ad legatum, scinditur inter eos legatum; si alter deficiat, quia aut spreverit legatum, aut vivo testatore decesserit, vel alio quolibet modo defecerit, totum ad collegatarium pertinet."