nitudes are added to equals, equals arise therefrom, he demon strated tliis other which is considered equally evident: if equals are taken from equal magnitudes, equals remain. It was said he should have assumed both or demonstrated both. But I was not of that opinion, and I believed that it was always so much gained to have diminished the number of the axioms. Addition is no doubt anterior to subtraction and more simple, because the two terms are employed in addition in the same way, which is not the case in subtraction. Arnauld did the opposite of Boberval. He assumed still more than Euclid. As for the maxims, they are sometimes taken as established propositions whether evident or not. That may be well for beginners whom scrupulousness holds back; but when the establishment of science is the question, it is a dif ferent matter. Thus it is that they are often taken in ethics and even among the logicians in their topics, in some of which there is a good supply, but a part of which contain enough of them vague and obscure. For the rest, I said a long time since publicly and privately that it is important to demon strate all our secondary axioms, which we ordinarily use, by reducing them to the primitive or immediate and indemon strable axioms, which I recently and elsewhere called the identicals.
§ 2. Ph. Knowledge is self-evident when the agreement or disagreement of ideas is immediately perceived. § 3. But there are truths, not recognized as axioms, which are none the less self-evident. Let us see if the four species of agree ment of which we spoke not long since (chap. 1, § 3, and chap. 3, § 7), viz.: identity, connection, relation, and real ex istence, furnish us with them. § 4. As for identity or diver sity, we have as many evident propositions, as we have dis tinct ideas, for we can deny both, as in saying man is not a horse, red is not blue. Further the statement what is, is, is as evident as the statement a man is a man.
Tli. It is true, and I have already remarked that it is as evident to say ecthetically in particular A is A, as to say in 1686; Objections centre Descartes; and La Loyique ou I' Art dePenser, or the celebrated Port Royal Logic, 1662, written in conjunction with Nicole (c/. infra, p. 530, note 1), the best specimen of the logic of the Cartesian school. It has been translated into English most admirably by Prof. Thos. Spencer Baynes. — TK.
en. VTI] ON HUMAN UNDERSTANDING 465 general: it is what it is. But to deny the subjects of different ideas one of another is not always certain, as I have already remarked; as if any one wished to say, the trilateral (or that which has three sides) is not a triangle, because, in fact, tri- laterality is not triangularity; again, if any one had said: the pearls of Slusius l (of which I spoke to you not long since) are not the lines of the cubic parabola, he would have been mis taken, and yet that would have appeared evident to many people. The late Mr. Hardy,2 Conseiller au Chatelet de Paris, an excellent geometer and orientalist and well versed in the ancient geometers, who has published the commentary of Marinus on the Data of Euclid, was so prepossessed with the fact that the oblique section of the cone called an ellipse is different from the oblique section of the cylinder, that the demonstration of Serenus 3 appeared to him paralogistic, and I could gain nothing against him by my remonstrances: as he was nearly as old as Roberval, when I saw him, and I was a very young man, a difference which could give me very little persuasive power as regards him, although in other respects I was on very good terms with him. This example may show in passing what prepossession may do even in the case of clever people, for he was truly prepossessed, and Hardy is spoken of 2 Claude Hardy, born near the close of the sixteenth century, died in 1678, was a barrister by profession, and became in 1626 " Conseiller au Chatelet de Paris." He was acquainted with not less than thirty-six ancient and modern languages, and made a profound study of mathematics. Descartes chose him as one of his judges in his controversy with Fermat, in 1698, over Format's De maximis et minimis. Hardy published the Data Euclidis, Greek text, with Latin trans., together with the commentary of Marinus, the Neo-Platonist (cf. Zeller, Philos. d. Griech., III., 2 [Vol. 6], 833, 3d ed., 1881), who lived in the fifth century, and was a disciple and the successor of Proclus (cf. ante, p. 108, note 2), at Paris, 1625, 4to. Leibnitz speaks of Hardy as an "homme de merite, grand geometre, et grand orientaliste," cf. Dutens, 5, 610. — TR.
3 Serenus of Antissa, in the island of Lesbos, a Greek geometer, who lived in the fourth century, was the author of two treatises, De Sectione Cylindri et Coni, libri duo, which, according to Brunet, appeared, together with the Conies of Apollonius of Perga, the Lemmas of Pappus of Alexandria, and the commentaries of Eutocius of Ascalonita, at Bonn, 15(56, fol., reprinted at Pis- toja, 1696, fol., and afterwards edited and published by Halley, Oxford, 1710, fol. Hardy could not have seen either of the last two editions, since he died in 1678, but possibly lie may have been acquainted with that of 1566; and, if not, then, as Schaarschmidt says, he must refer to Mersenne's Synopsis, Paris, 1644, which contains, pp. 276-312, an abridgment of Apollonius and Serenus. — TR.
2H 466 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv with esteem in the letters of Descartes.1 But I brought him forward only to show how we may be mistaken in denying one idea of another, if we have not thoroughly enough examined them where it is necessary.
§ 5. Ph. As regards connection or coexistence, we have very few self-evident propositions; there are, however, some, and it appears to me that this is a self-evident proposition: two bodies cannot be in the same place.
Th. Many Christians contest the point with you, as I have already remarked, and even Aristotle and those who after him admit real and exact condensations, reducing one and the same entire body into a smaller place than it before filled, and those who, as the late Mr. Comenius 2 has done in a little book writ ten expressly for the purpose, claim to overthrow3 modern natural philosophy by the experiment of the air-gun, cannot be expected to agree therewith. If you take the body as the impenetrable mass, your statement will be true, because it will be identical or nearly so; but that the real body is such will be denied you. At least it will be said that God could 1 Schaarschmid't thinks that Leibnitz here confounded Hardy with Roberval, whom he had mentioned just before. Descartes frequently mentions Hardy in his letters, cf., for example, Pt. I., epist. Ill, Pt. II., epist. 61, 98, 101, 108, Pt. III., epist. 34, GO, 63, etc.; he also corresponded with him, and doubtless valued him highly, as witness his choice of him as arbiter in his controversy with Fermat, but he nowhere in his letters appears expressly to praise him; while he speaks thus of Roberval in Lib. III., epist. 56, ad Fermatium: " qui procul dubio inter primaries seculi nostri geometras censeri debet." — TK.
2 Johu Amos Comenius, 1592-1671, the last bishop of the old Moravian and Bohemian Brethren, devoted himself chiefly to the reform and regulation of public education and instruction, and wrote many works on pedagogy, which he collected and published under the title Opera didactica omnia, Amsterdam, 1657, 4 vols., fol. He also did some work in physical science, publishing his Disquisitio de caloris et friyoris natura, Amsterdam, 1659, 12mo, which the writer in Michaud, Biog. Univ., 9, 345, says is the only one of his physical works deserving to be in demand, and his Physic.se, ad lumen divinum refor- matse synopsis, Leipsig, 1633, and Amsterdam, 1643, Eng. trans., London, 1(551, which is, perhaps, the book to which Leibnitz here refers. For Leibnitz's esti mate of a portion of the writings of Comenius, cf. Dutens, 5, 181. For an account of his life and work, cf. S. S. Laurie, Comenius. His Life and Educa tional Works, 4th ed. (Pitt Press Series), Cambridge, Univ. Press, 1893; also a reprint of the same, with five portraits, a somewhat extended and annotated bibliography, and photographic reproductions of pages from early editions of Comenius' works, published by C. W. Bardeen, Syracuse, N. Y., 1893. — TR.
3 Gerhardt reads "reserver," probably a Ms. or typographical error; Erd- inann, Jacques, and Janet read " renverser," which, as the context requires, the translation follows. — TR.
en. vn] ON HUMAN UNDERSTANDING 467 make it otherwise, so that this impenetrability will be ad mitted only as conformed to the natural order of things which God has established and of which experiment assures us, although elsewhere it would be necessary to admit that it is also very conformable to reason.
§ G. Ph. As for the relations of the modes, mathematicians have formed many axioms upon the one relation of equality, like that of which you have just spoken, that if equals be taken from equals the remainders are equal. But it is not less evident, I think, that one and one are equal to two, and that if from the five fingers of one hand you take away two and then two others from the five fingers of the other hand, the number of the fingers remaining will be equal.
TU. That one and one make two, is not properly a truth, but it is the definition of two; although it is true and evident that it is the definition of a possible thing. As for the axiom of Euclid applied to the fingers of the hand, I willingly admit that it is as easy to conceive what you say of the fingers as to see it in the case of A and B; but in order not to do often the same thing, you observe it generally, and afterwards it is sufficient to make subsumptions. Otherwise, it is as if you preferred calculation by particular numbers to universal rules, which would be obtaining less than is possible. For it is of more account to solve the general problem: to find two num bers whose sum makes a given number, and whose difference also makes a given number, than merely to seek two numbers whose sum makes ten, and whose difference makes six. For if I proceed in this second problem according to the method of numerical algebra, mixed with the literal (specieuse), the calculation will be as follows: a + b = 10, and a — b = 6; of which by adding together the right side to the right, and the left side to the left, I produce the result, a + & + a — b = 10 + 6, i.e. (since +& and —b cancel each other) 2 a = 16 or a = 8. Subtracting the right side from the right, and the left from the left (since to take away a — b is to add — a -(- &) I produce the result a + b — a + b = 10 — 6, i.e. 2 b = 4 or b = 2. Thus, I shall in truth have the a and b I ask for, which are 8 and 2, which satisfy the problem, i.e. whose sum is 10 and whose difference is 6; but I have not thereby the general method for any other numbers, which we might wish 408 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv or be able to put in the place of 10 or 6, a method which I could, however, find with the same facility as these two num bers 8 and 2, by putting x and v in the place of the numbers 10 and 6. For proceeding the same as before we shall have a + & + a — b = x + v, i.e. 2a = x-}-voi'a — ^x + v, and we shall also have a + b — a + b = x — v, i.e. 2b = x — v or 6 = ^ x — v. This calculation gives this theorem or general canon, that when two numbers are required whose sum and differ ence are given, you have only to take as the greater of the required numbers, half of the sum made from the given sum and difference; and for the less of the required numbers, half of the difference between the given sum and difference. You see also that I might have dispensed with the letters, if I had treated the numbers as letters, i.e. if instead of putting 2a = 16 and 2 6 = 4, I had written 2a = 10 + 6 and 2 b = 10 - 6, which would have given me a = •£• (10 + 6) and b = ^ (10 — 6). Thus, in the particular calculation itself 1 should have had the general calculation, taking these symbols 10 and 6 as general numbers, as if they were the letters x and v; in order to have a truth or method more general, and taking these same charac ters 10 and 6 also for the numbers they ordinarily signify, I shall have a sensible example which may serve, indeed, as a proof. And as Vieta1 has substituted letters for numbers for the sake of greater generality, I have desired to reintroduce the numerical characters, since they are more serviceable in algebra (specieuse) even than the letters. I have found this of much use in large calculations for avoiding errors and even 1 Francois Viete, 1540-1603, better known by the Latin form of his name, Vieta, was a distinguished French mathematician, who is often regarded as the founder of modern algebra, because of his introduction of the general use of letters as symbols of undetermined, and therefore general, quantities, thus opening up the way for the higher mathematical analysis, afterwards carried on by Descartes and others. To him is also due the invention of the different simple transformations now used in the solution of equations, such as adding to or subtracting from the members of an equation the same quantity, or mul tiplying or dividing them by the same quantity. He first enounced the princi ple of homogeneity or the principle that all the quantities in an equation should be of one kind, — lines, surfaces, solids, or supersolids, — a principle which, after three centuries of controversy, has now been adopted generally by mathemati cians. His various mathematical writings, which, being a man of wealth, he printed at his own expense and distributed among the scholars of Europe, were collected and edited by F. van Schooten, Professor of Mathematics at Leyden, aided by J. Golius and Mersenne, and published under the title of Opera mathematica, Leyden, 1(>4<>, 1 vol., fol. — TR.
CH. vn] ON HUMAN UNDERSTANDING 469 in the application of proofs, such as the casting away of the nines in the midst of the computation without waiting for the result, when there are only numbers instead of letters; which may often be when you employ skill in the positions, so that the suppositions are found true in the particular case, besides the use there is of seeing the connections and orders which the letters alone cannot always make the mind discern so well, as I have elsewhere shown, after I found that the good char acteristic is one of the greatest aids of the human mind.
§ 7. Ph. As for real existence which I had counted as the fourth species of existence which may be noticed in ideas, it can furnish us no axiom, for we have not indeed a demonstra tive knowledge of beings outside us, God alone excepted.
Th. We can always say that this proposition I exist, is of the highest evidence, being a proposition which cannot be proved by any other, or rather an immediate truth. And to say / think, therefore I am, is not properly to prove existence by thought, since to think and to be thinking is the same thing; and to say, I am thinking, is already to say, / am. You can, however, exclude this proposition from the number of the axi oms with some reason, for it is a proposition of fact, based upon an immediate experience, and it is not a necessary prop osition, whose necessity is seen in the immediate agreement of ideas. On the contrary, it is only God who sees how these two terms, /and existence are united, i.e. why I exist. But if the axiom is taken more generally as an immediate or non- provable truth, we may say that this proposition, / am, is an axiom, and in every case we may be assured that it is a primi tive truth, or rather unum ex primis cognitis inter terminos com- plexos, i.e. that it is one of the first known statements which is understood in the natural order of our knowledge, for it may be that a man has never thought expressly of forming this proposition, which, however, is innate to him.
§ 8. Ph. [I have always believed that the axioms have little influence upon the other parts of our knowledge. But you have disabused me, since you have indeed shown an important use of identical propositions. Suffer me, however, sir, to set before you still what I have in mind upon this article, for your explanations may also serve to make others return from their error.] § 8. It is a celebrated rule in the schools that 470 LEIBNITZ'S CRITIQUE OF LOCKE [UK. iv all reasoning comes from things already known and admitted, ex praecognitis et praeconcessis. This rule seems to cause these maxims to be regarded as truths known to the mind before the others, and the other parts of our knowledge as truths dependent upon the axioms. § 9. [I think I have shown (Book I., chap. 1) that these axioms are not the first known, the child knowing much sooner that the rod which I show him is not the sugar he has tasted, than all the axioms you please. But you have distinguished between particular knowledge or experiences of facts and the principles of a universal and necessary knowledge (and herein I admit that it is necessary to recur to axioms) as also between the natural and accidental order].
Tli. I have also added that in the natural order the state ment, that a thing is what it is, is prior to the statement that it is not another; for the question here does not concern the history of our discoveries, which is different in different men, but the connection and natural order of truths, which is always the same.1 But your remark, viz.: that what the child sees is only a fact, deserves still more reflection; for the expe riences of the senses do not give truths absolutely certain (as you have often yourself, sir, observed not long since), nor are they exempt from all danger of illusion. For if it is allow able to make fictions metaphysically possible, sugar might imperceptibly be changed into a rod, in order to punish the child who has been naughty, as water is changed into wine with us on Christmas eve, if it has been well prepared (mori- gene).2 But in all cases the pain (you will say) that the rod inflicts will never be the pleasure the sugar gives. I reply that the child will take it into his head as late to make an express proposition about this, as to notice this axiom, that you caii- 1 Leibnitz here calls attention to a very important distinction, viz.: the dis tinction between the historical and the natural or logical order of our knowl edge. The genesis of our knowledge, its gradual rise in the course of our lives, is always a matter of individual experience, the experience of no two indi viduals being precisely alike; while the natural or logical order and connec tion of truths, being grounded in reason, is always the same for all. Leibnitz's remark further suggests that the origin of a principle or truth is not its justification, a common fallacy in much of the investigation of the present day, and that the ultimate criteria of truth are philosophical, not historical. Cf. Bowne, Metaphysics, pp. 13 sq., New York: Harper and Bros., 1882. — TK.
2 Duncan, Philos. Wks. of Leibnitz, p. 354, translates: "rectified." — TB.
CH. vn] ON HUMAN UNDERSTANDING 471 not truly say that what is is not at the same time, although he can very well perceive the difference of the pleasure and the pain, as well as the difference between perceiving and not perceiving.
§ 10. Ph. There are, however, a number of other truths as self-evident as these maxims. For example, that one and ttvo are equal to three, is as evident a proposition as that axiom which states: that the tvhole is equal to the sum of all its parts.
Th. You appear to have forgotten, sir, that I have shown you more than once that the statement one and two is three is only the definition of the term three, so that to say that one and two is equal to three, is to say that a thing is equal to itself. As for this axiom, that the whole is equal to the sum of all its parts, Euclid makes no express use of it. This axiom also needs limitation, for it must be added that these parts must not themselves have a common part, for seven and eight are parts of twelve, but they make more than twelve. The bust and the trunk taken together are more than the man, in that the thorax is common to them both. But Euclid says, that the vchole is greater than its part, a statement which is wholly trustworthy. And the statement that the body is greater than the trunk, differs from the axiom of Euclid only in this, that this axiom is limited to what is exactly necessary: but in exemplifying it and clothing the body you make the intelligible become also sensible, for the statement that a given whole is larger than a given part, is in fact the proposition that a whole is larger than its part, but the features of which are embellished with some coloring or addition: it is as if he who says A B says A. Thus it is not necessary here to oppose the axiom and the example as different truths in this regard, but to consider the axiom as embodied in the example and rendering it true. It is a different matter, if the evidence is not observed in the example itself, and when the affirmation of the example is a consequence, and not merely a subsump- tion of the universal proposition, as may occur indeed in the case of the axioms.
Ph. Our clever author says here: I should like to ask these gentlemen who maintain that all other knowledge (not of fact) depends upon general principles innate and self-evident, what principle they need to prove that tivo and two are four? for 472 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv we know (according to him) the truth of this kind of proposi tions without recourse to any proof. What do you say about it, sir?
Tli. I say that I was awaiting you there well prepared. That two and two are four is not a truth at once immediate, supposing that four signifies three and one. We can then demonstrate it, and in this way: — Definitions. — (1) Two is one and one.
(2) Three is two and one.
(3) Four is three and one.
Axiom. — Putting equal things in their place, the equality remains. Demonstration. — 2 and 2 is 2 and 1 and 1 (by def. 1) 2 + 2 4 Then (by the axiom) 2 and 2 is 4. Which was to be demonstrated.
I might, instead of saying that 2 and 2 is 2 and 1 and 1, say that 2 and 2 is equal to 2 and 1 and 1, and thus with the others. But it may be understood throughout in order to shorten the process; and that, in virtue of another axiom which states that a thing is equal to itself, or that what is the same, is equal.
Ph. [This demonstration, as little necessary as it is in rela tion to its too well known conclusion, serves to show how truths depend on definitions and axioms. Thus I foresee what reply you will make to many objections that are made against the use of axioms. You object that there will be an innu merable multitude of principles; but this is when you reckon among the principles the corollaries which follow from the definitions by the aid of some axiom. And since the defini tions or ideas are innumerable, so also will the principles be in this sense, supposing also with you that the undemonstra- ble principles are the identical axioms. They become innu merable also by exemplification, but at bottom you can reckon A is A and B is B as one and the same principle differently clothed.
Th. Further, this difference of degrees in the evidence makes me disagree with your distinguished author that all en. vn] ON HUMAN UNDERSTANDING 473 these truths called principles and which pass as self-evident, because they are so near the indemonstrable primitive ax ioms, are entirely independent and incapable of receiving the one from the other any light or proof. For they may always be reduced either to axioms, themselves, or to other truths nearer the axioms, as this truth, that two and two make four, has shown you. And I just told you how Roberval diminished the number of Euclid's axioms, by sometimes reducing one to another.
§ 11. Ph. This judicious writer, who has furnished an occa sion for our conferences, agrees that maxims have their use, but he believes that it is rather that of closing the mouth of the obstinate, than of establishing the sciences. I should be very glad, said he, if you would show me some one of these sciences built upon these general axioms which cannot be shown to be sustained as well without axioms.
Tli. Geometry is, without doubt, one of these sciences. Euclid expressly employs axioms in demonstration, and this axiom: that two homogeneous magnitudes are equal when one is neither larger nor smaller than the other, is the basis of the demonstrations of Euclid and Archimedes respecting the size of curvilinears. Archimedes employed axioms of which Euclid had no need; for example, of two lines, each of Avliich is con cave always on the same side, that which encloses the other is the greater. We cannot also dispense with the identical ax ioms in geometry, as, for example, the principle of contradic tion, or the demonstrations which lead to the impossible. And as for the other axioms, which are demonstrable, we may dispense with them, absolutely speaking, and draw conclu sions immediately from the identicals and from the defini tions; but the prolixity of the demonstrations, and the end less repetitions into which you Avould then fall, would cause a horrible confusion, if it were always necessary to begin ab ovo; Avhile by assuming the mean propositions, already proved, we easily pass much farther. This assumption of truths already known is useful, especially as regards the ax ioms, for they recur so often that geometers are compelled to