For abbreviations used in this volume for these and other works, see the Bibliography given at the end.
This appeared in The Monist for October, 1916.
G. 1848, p. 29; see also G. math., III, pp. 71, 72, and Cantor, III, p. 40.
A fair-minded consideration, like everything emanating from the pen of De Morgan, is given of the matter in a recent edition of his Essays on the Life and Work of Newton. The tale is told with the charm characteristic of De Morgan, and the edition is rendered very valuable by the addition of notes, commentary, and a large number of references supplied by the editor, P. E. B. Jourdain (Open Court Publishing Co.). Special attention is directed to De Morgan’s summary of the unfairness of the case in Note 3 at the foot of pages 27-28.
See under 11 below: also cf. the original Latin as given in G. 1846, p. 4, “per amicum conscium.”
The account here given is substantially that given by Gerhardt in an article in Grunert’s Archiv der Mathematik und Physik, 1856; pp. 125-132.
This article is written in contradiction to the view taken by Weissenborn in his Principien der höheren Analysis, Halle, 1856. It is worthy of remark that the partisanship of Gerhardt makes him omit in this article all mention of the review which Leibniz wrote for the Acta Eruditorum on Newton’s work, De Quadratura Curvarum, which really drew upon him the renewal of the attack, by Keill. The passage which was objected to by the English mathematicians as being tantamount to a charge of plagiarism, in addition to the insult implied, according to their thinking, in making Newton the fourth proportional to Cavalieri, Fabri and Leibniz, is however given by Gerhardt in his preface to the Historia (G. 1846, p. vii).
Fatio’s correspondence with Huygens is to be found in Ch. Hugenii aliorumque seculi XVII virorum celebrium exercitationes mathematicae et philosophicae, ed. Uylenbroeck, 1833.
Bernoulli (Jakob), Opera, Vol. I, p. 431.
Ibid., p. 453.
Cantor, III, p. 221.
In the opening paragraph of the “postscript,” page 11.
The account which follows is taken from Williamson’s article, “Infinitesimal Calculus,” in the Times edition of the Encyc. Brit. The memoir referred to contains a passage, of which the following is a translation (G., 1846, p. v) :
“Perhaps the distinguished Leibniz may wish to know how I came to be acquainted with the calculus that I employ. I found out for myself its general principles and most of the rules in the year 1687, about April and the months following, and thereafter in other years; and at the time I thought that nobody besides myself employed that kind of calculus. Nor would I have known any the less of it, if Leibniz had not yet been born. And so let him be lauded by other disciples, for it is certain that I cannot do so. This will be all the more obvious, if ever the letters which have passed between the distinguished Huygens and myself come to be published. However, driven thereto by the very evidence of things, I am bound to acknowledge that Newton was the first, and by many years the first, inventor of this calculus; from whom, whether Leibniz, the second inventor, borrowed anything, I prefer that the decision should lie, not with me, but with others who have had sight of the paper of Newton, and other additions to this same manuscript. Nor does the silence of the more modest Newton, or the forward obtrusiveness of Leibniz....”
Truly another Roland in the field, and one in a vicious mood. What with other claimants to the method, such as Slusius, etc., at least as far as the differentiation of implicit functions of two variables is concerned, it would almost seem that the infinitesimal calculus was not an invention, but a gradual development of the fundamental principles of the ancient mathematicians.
See De Morgan’s Newton, p. 26 and pp. 148, 149, where the Scholium is translated. The original Latin of this Scholium to Lemma II of Book II of the Principia, the altered Scholium that appeared in the second and third editions, with a note remarking on the change, will be found on pp. 48, 49, in Book II of the “Jesuits’ Edition” of Newton (Editio Nova, edited by J. M. F. Wright, Glasgow, 1822; the third and best edition of the work of Le Seur and Jacquier).
Phil. Trans., 1708; see also Cantor, III, p. 299.
For a discussion, see Rosenberger, Isaac Newton und seine physikalischen Principien, Leipsic, 1895.
The manner of the opening of this postscript would seem to indicate that something had been mentioned with regard to the matter of his irritation about imputed obligations to Barrow in the body of the letter; this cannot be ascertained, for Gerhardt does not quote the letter in connection.
Leibniz can hardly with justice call Barrow his contemporary; Barrow anticipated him by half a dozen years at least. For Barrow had published his Lectiones Geometricae in 1670, while the very earliest date at which Leibniz could have obtained his results is the end of 1672; and there is reason to believe, as I have shown in my edition of the Lectiones, that Barrow was in possession of his method many years before publication, and had most probably communicated his secret to Newton in 1664.
It is to be noted that the sole topic of this postscript is geometry, of which Leibniz candidly states that he knew practically nothing in 1672.
Most probably the Institutiones arithmeticae of Johann Lantz, published at Munich in 1616; Cantor, III, p. 40.
Possibly the Geometria practica of Christopher Clavius, better known as an editor of Euclid; he was the professor at Rome under whom Gregory St. Vincent studied. There are repeated references to Clavius in Cantor, II and III, Index, q. v.
It is worth remarking that neither Lanzius nor Clavius is mentioned in the Historia.
It has been stated that, according to Descartes’s own words, the intricacies of his Géométrie were intentional; it certainly has the character of a challenge to his contemporaries. There is no preparation, such as marks a book of the present day on coordinate geometry; Descartes starts straightway on the solution of a problem given up as insoluble by the ancients. No wonder that young Leibniz found some difficulty with his first attempt to read it.
In 1635, Cavalieri published his Geometria indivisibilibus, and thus laid the foundation stone of the integral calculus. It would seem that Roberval was really the first inventor, or at least an independent inventor of the method; but he lost credit for it because he did not publish it, preferring to keep it to himself for his own use. Other examples of this habit are common among the mathematicians of the time.
The book referred to was published in 1654. It appeared as the second volume of a work whose first volume was a critique and refutation of the quadrature of the circle published by Gregory St. Vincent; this second volume was not the work of Leotaud, as the second part of the title showed: “necnon CURVILINEORUM CONTEMPLATIO, olim inita ab ARTUSIO DE LIONNE, Vapincensi Episc.” It therefore appears to have been an edited reprint of the work of De Lionne, the bishop of Gap (ancient name, Vapin-cum). Since part of this treatise is devoted to the “lunules of Hippocrates” (see Cantor, I, pp. 192-194), it may have had some influence with Leibniz in giving him the first idea for his evaluation of π.
Literally, “I was about to swim without corks.”
Leibniz here would appear to assert that he had considered some form of rectangular coordinate geometry, the association with the name of Descartes being fairly conclusive. Vieta’s In Artem Analyticam Isagoge explained how algebra could be applied to the solution of geometrical problems (Rouse Ball) ; for further information see Cantor.
This seems to have been an improvement on the adding machine of Pascal, adapting it to multiplication, division and extraction of roots. Pascal’s machine was produced in 1642, and Leibniz’s in 1671.
Huygens’s Horologium Oscillatorium was published in 1673; we are thus provided with an exact date for the occurrence of the conversation that set Leibniz on to read Pascal and St. Vincent. This was after his first visit to London, from which he returned in March, “having utilized his stay in London to purchase a copy of Barrow’s Lectiones, which Oldenburg had brought to his notice” (Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert; German edition by Mayer, p. 66). Leibniz himself mentions in a letter to Oldenburg, dated April 1673, that he has done so. Gerhardt (G. 1855, p. 48) states that he has seen, in the Royal Library of Hanover the copy of Barrow’s Lectiones Geometricae, so that it must have been the combined edition of the Optics and the Geometry, published in 1670, that Leibniz bought.
Thus, before he is advised to study Pascal by Huygens, he has already in his possession a copy of Barrow. It is idle that any one should suppose that Leibniz bought this book on the recommendation of a friend in order merely to possess it; Leibniz bought books, or borrowed them, for the sole purpose of study. Unless we are to look upon this account of his reading as the result of lack of memory extending back for thirty years, there is only one conclusion to come to, barring of course the obviously brutal one that Leibniz lied; and this conclusion is that at the first reading the only thing that Leibniz could follow in Barrow was the part that he marked Novi dudum (“Knew this before”), and this was the appendix to Lecture XI, which dealt with the Cyclometria of Huygens, as Barrow calls the book entitled De Circuli Magnitudine Inventa. The absence of any more such remarks is almost proof positive that Leibniz knew none of the rest before. Hence he must have read the Barrow before he had filled those “hundreds of sheets” that he speaks of later, with geometrical theorems that he has discovered; for at the end of the postscript we are considering he states that “in Barrow, when his Lectures appeared, I found the greater part of my theorems anticipated.” There is something very wrong somewhere; for this would appear to state that it was the second edition of Barrow, published in 1874, that Leibniz had bought; it is impossible, as the words of Leibniz stand, that they should refer to the 1670 edition, for it had been published before Leibniz arrived in Paris. It is however certain from Leibniz’s letter to Oldenburg that it could not be the 1674 edition, for the date of the letter is 1673. In this letter Leibniz merely makes a remark on the optical portion; but it could not have been the separate edition of the Optics, published in 1669, for Gerhardt states that the copy he has seen contains the Geometry with notes in the margin.
To those who have ever waded through the combined edition of Barrow’s Optics and Geometry, it may be that rather a startling suggestion will occur. It was sheer ill-luck that drove Leibniz, after studying the Optics (perhaps on the journey back from London, for we know that this was a habit of his), to get tired of the five preliminary geometrical lectures in all their dryness, and on reaching home, just to skim over the really important chapters, missing all the important points, and just the name of Huygens catching his eye. This is a new suggestion as far as I am aware; everybody seems to decide between one of two things, either that Leibniz never read the book until the date he himself gives, “Anno Domini 1675 as far as I remember,” or else that he purposely lied. I will return to this point later; meanwhile see Cantor, III, pp. 161-163, and consult the references given in the footnotes to these pages; the pros and cons of the conflict between probability and Leibniz’s word are there summarized.
Pascal’s chief work on centers of gravity is in connection with the cycloid, and solids of revolution formed from it. His method was founded on the indivisibles of Cavalieri. His work was issued as a challenge to contemporaries under the assumed name of Amos Dettonville, and under the same name he published his own solutions, after solutions had been given by Huygens, Wallis, Wren and others.
The method of ductus plani in planum, the leading or multiplication of a plane into a plane, employed by Gregory St. Vincent in the seventh book of his Opus Geometricum (1649) is practically on the same fundamental principle as the present method of finding the volume of a solid by integration. A simple explanation may be given by means of the figure of a quarter of a cone.
Let AOBC be the quarter of a circular cone (Fig. A), of which OA is the axis, and ABC the base, so that all sections, such as abc, are parallel to ABC and perpendicular to the plane AOC. Let ad be the height of a rectangle equal in area to the quadrant abc, so that ad is the average height of the variable plane abc ; then the volume of the figure is found by multiplying the height of the variable plane as it moves from O to the position ABC by the corresponding breadth of the plane OAC, i. e., by bc, and adding the results.
Fig. A.
As we shall see later, Leibniz does not fully appreciate the real meaning of the method; on the other hand Wallis uses the method with good effect in his Arithmetica Infinitorum, and states that he has come to it independently. In the above case he would have stated that the product in each case was proportional to the square on ac, drawn an ordinate ae at right angles to Oa, so that ae represented the product, and so formed the parabola OeEAaO, of which the area is known to him. This area is proportional to the volume of the cone.
Ungulae denote hoof-shaped solids, such as the frusta of cylinders or cones cut off by planes that are not parallel to one another.
Figure 1 (see above) is of extreme interest. First of all it is not Barrow’s “differential triangle,” which is that of Fig. B below ; this of course is only what those who believe Leibniz’s statement that he received no help from Barrow, would expect. By the way, the figure given by Cantor as Barrow’s is not quite accurate. (Cantor, III, p. 135.)
Fig. B (BARROW).
Fig. C (PASCAL).
But neither is it the figure of Pascal, which is that of Fig. C. Of course, I am assuming that Gerhardt has given a correct copy of the figure given by Leibniz in his manuscript; although that which I have given of it, a faithful copy of Gerhardt’s, shows that his curve was not a circle. I also assume that Cantor is correct in the figure that he gives from Pascal; although Cantor says that the figure occurs in a tract on the sines of a quadrant, and not, as Leibniz states, in a problem on the measurement of the sphere. Indeed it seems to me that the figure is more likely to be connected with the area of the zone of a sphere and the proof that this is equal to the corresponding belt on the circumscribing cylinder than anything else. I am bound to assume these things, for I have not had the opportunity of seeing either of the figures in the original for myself. It is strange, in this connection, that Gerhardt in one place (G. 1848, p. 15) gives 1674 as the date of the publication of Barrow, and in another place (G. 1855, p. 45) seven years later, he makes it 1672, and neither of them is correct as the date of the copy that Leibniz could possibly have purchased, namely 1670. This is culpable negligence in the case of a date upon which an argument has to be founded, for one can hardly suspect Gerhardt of deliberate intent to confuse. Nevertheless, like De Morgan, I should have felt more happy if I could have given facsimiles of Barrow’s book, and Leibniz’s manuscript and figure.
Lastly, there is in Barrow (what neither Gerhardt, Cantor, nor any one else, with the possible exception of Weissenborn, seems to have noticed) chapter and verse for Leibniz’s “characteristic triangle.” Fig. D is the diagram that Barrow gives to illustrate the first theorem of Lecture XI. This is of course, as is usual with Barrow, a complicated diagram drawn to do duty for a whole set of allied theorems.
Fig. E.
In the proof of the first of these theorems occur these words:
“Then the triangle HLG is similar to the triangle PDH (for, on account of the infinite section, the small arc HG can be considered as a straight line).
Hence, HL : LG = PD : DH, or HL. DH = LG. PD,
i. e., HL. HO = DC. Dψ.
By similar reasoning, it may be shown that, since the triangle GMF is similar to the triangle PCG,....”
If now the lines in italics are compared with that part of the figure to which they refer, which has been abstracted in Fig. E, the likeness to Leibniz’s figure wants some explaining away, if we consider that Leibniz had the opportunity for seeing this diagram. Such evidence as that would be enough to hang a man, even in an English criminal court. (Further, see Note 46.)
To sum up, I am convinced that Leibniz was indebted to both of Barrow’s diagrams, and also to that of Pascal (for I will call attention to the fact that he uses all three, as I come to them) and I think that after the lapse of thirty years he really could not tell from whom he got his figure. In such a case it would be only natural, if he knew that it was from one of two sources and he was accused of plagiarizing from the one, that he should assert that it was from the other. Hence, by repetition, he would come to believe it. But even this does not explain his letter to d’Hospital, where he says that he has not obtained any assistance from his methods; unless again we remember that this letter is dated 1694, twenty years after the event.
Great importance, in my opinion hardly merited, is attached to the use by Leibniz of the phrase momento ex axe in this place, and in his manuscripts under the heading Analysis Tetragonistica ex Centrobarycis, dated October, 1675.
The Latin word momentum, a contraction of movimentum, has a primary meaning of movement or alteration, and a secondary meaning of a cause producing such movement. The present use of the term to denote the tendency of a force to produce rotation is an example of the use of the word to denote an effect ; from the second idea, we have first of all its interpretation as something just sufficient to cause the alteration in the swing of a balance (where the primary idea still obtains), hence something very small, and especially a very small element of time.
Thus we see that Leibniz uses the term in its primary sense, for he employs it in connection with a method ex Centrobarycis, and in its mechanical sense, and it is thus fairly justifiable to assume that he got the term from Huygens; in just this sense we now speak of the moment of inertia.
Newton’s use of the term is given in Lemma II of Book II of the Principia, in the following way.
“I shall here consider such quantities as undetermined or variable, as it were increasing or decreasing by a continual motion or flow (fluxus) ; and their instantaneous (momentanea) increments or decrements I shall denote (intelligo = understand) by the name “moments”; so that increments stand for moments that are added or positive (affirmativis), and decrements for those that are subtracted or negative.”
This has nothing whatever to do with what Leibniz means by a moment, and it seem ridiculous to bring forward the use of this word as evidence that Leibniz had seen Newton’s work, or even heard of it through Tschirnhaus, before the year 1675.
The fact that in another place, where I will refer to it again, he uses the phrase “instantaneous increment” is quite another matter.
The use of the word moment in this mechanical sense is here perfectly natural. See Cantor, III, p. 165 ; also Cantor, II, p. 569, where the idea is referred back at least to Benedetti (1530-1590) ; but the idea is fundamental in the theorems due to Pappus concerning the connection between the path of the center of gravity of an area and the surfaces and volumes of rings generated by the area, of which the proofs were given by Cavalieri. When, however, and by whom, the word moment was itself first used in this connection, I have been unable to find the slightest trace. (See p. 195.)
With due regard to the statement that Leibniz “had looked through Cavalieri” before he went to Paris, it is not remarkable that he did not notice very much at all in Cavalieri. Cavalieri’s Geometria Indivisibilibus is not a book to be “looked through.” It is a work for weeks of study. I cannot say whether the idea involved in Leibniz’s characteristic triangle is used by Cavalieri as such; but I do not see how else he could have given proofs (as stated by Williamson in his article on “Infinitesimal Calculus” in the Times edition of the Encyc. Brit.) of Pappus’s theorem for the area of a ring; and I should think that it is morally certain that Cavalieri is the source from which Wallis obtained his ideas for the rectification of the arc of the spiral. I had occasion to refer to a copy in the Cambridge University Library, and what I saw of it in the short time at my disposal determined me to make a translation of it, with a commentary, as soon as I had enough time at my disposal. “As one reads tales of romance” !
The moment is proportional to the area of the surface formed by the rotation of the curve C(C) about AP. Barrow does not at first use the method to find the areas of surfaces of revolution; he prefers to straighten out the curve C(C), and erect the ordinates BC, (B) (C) perpendicular to the curve thus straightened; i. e., he works with the product BC.C(C) as it stands. But, after giving the determination of the surface of a right circular cone as an example of the method, and as a means of combating the obiec-tions of Tacquet to the method of indivisibles, he goes on to say: “Evidently in the same manner we can investigate most easily the surfaces of spheres and portions of spheres (nay, provided all necessary things are given or known, any other surfaces that are produced in this way). But I propose to keep, to a great extent, to more general methods” (end of Lecture II). Thus we find that Barrow does not give any further examples of the determination of the areas of surfaces of revolution until Lecture XII. And why? Because he is not writing a work on mensuration, but a calculus. The reference to the method of indivisibles however shows that in Barrow’s opinion, if Cavalieri had not used his method for the determination of the area of the surface of a sphere, then he ought to have done so.
It is difficult to see also how Huygens could have performed his constructions unless he had used the method that Leibniz claims to have discovered.
It is strange that Roberval, as an independent discoverer of the method of indivisibles, did not perceive the method of the constructions of Huygens. Bullialdus is Ismael Bouilleau (Martin’s Biog. Philos.), or Boulliau (Poggen-dorff), author of works on conics, arithmetic of infinites, astronomy, etc. Cf. Seth Ward: In Ismaelis Bullialdi astron. philos. fundainento inquisitiones. Oxford, 1653.
This conversation probably took place late in 1673; see a note on the alteration of the date of a manuscript dated November 11, 1673, where the 3 was originally a 5 (see p. 93).
The method of Slusius (de Sluze, or Sluse) is as follows:
Suppose that the equation of the given curve is
x3−2x2y + bx2−b2x + by2− y3 = 0.
Slusius takes all the terms containing y, multiplies each by the corresponding index of y ; then similarly takes all the terms containing x, multiplies each by the corresponding index of x, and divides each term of the result by x; the quotient of the former by the last expression gives the value of the subtangent. This is practically the content of Newton’s method of analysis per aequationes, and Slusius sent an account of it to the Royal Society in January, 1673. It was printed in the Phil. Trans., as No. 90. This is given by Gerhardt (G. 1848. p. 15) as an example of the method of Slusius. It is rather peculiar that Gerhardt does not mention that this is the example given by Newton in the oft-quoted letter of December 10, 1672, and represents what Newton “guesses the method to be.” As it stands in G. 1848, it would appear to be a quotation from the work of Slusius himself. There is evidence that Leibniz had seen the explanation given in the Phil. Trans., or had been in communication with Slusius; this will be referred to later, but it may be said here that this fact makes Leibniz somewhat independent of any necessity of having seen Newton’s letter.
Some point is made of the question why, if Leibniz had seen the “differential triangle” of Barrow, he should have called it by a different name. If there were any point in it at all, it would go to prove that Barrow’s calculus was published by Barrow as a differential calculus. But there is no point, for Barrow never uses the term! It is a product of later growth, by whom first applied I know not. Leibniz, thus free to follow his logical plan of denominating everything, uses a term borrowed from his other work. He thus defines a character or characteristic. “Characteristics are certain things by means of which the mutual relations of other things can be expressed, the latter being dealt with more easily than are the former.” See Cantor, III, p. 33f.
Gregory’s Geometriae Pars Universalis was published at Padua in 1668. Leibniz had either this book, or the Barrow in which one of Gregory’s theorems is quoted, close at hand in his work. For he gives it as an example of the power of his calculus, referring to a diagram which is not drawn. This diagram I was unable to draw from the meager description of it given by Leibniz, until I looked up Barrow’s figure, in default of being able to obtain a copy of Gregory’s work; thereupon the figure was drawn immediately.
Here indeed it must be admitted that Leibniz is—suffering from a lapse of memory. As has been said before, Barrow’s lectures appeared in 1670 and were in the possession of Leibniz before ever he dreamed of his theorems. But what can one expect when admittedly this account (from which the Historia was in all probability written up) is purely from memory, aided by the few manuscripts that he had kept. Gerhardt does not say that he has found, nor does he publish, any manuscripts that could possibly give the order in which the text-books that Leibniz procured were read. Which of us, at the age of 57, could say in what order we had read books at the age of 27; or, if by then we had worked out a theory, could with accuracy describe the steps by which we climbed, or from a mass of muddle and inaccuracies, say to whom we were indebted for the first elementary ideas that we had improved beyond all recognition? I doubt whether any of us would recognize our own work under such circumstances.
Again Leibniz makes a bad mistake in affecting to despise the work of his rivals—for that is what the words, “these things were perfectly easy to the veriest beginner who had been trained to use them,” makes us believe. It is also bad taste, for, besides Barrow, Huygens also remained true to the method of geometry till his death. The sentence which follows savors of conceit; as a matter of fact it was left to others, such as the Bernoullis, to make the best use of the method of Leibniz. The great thing we have to thank Leibniz for is the notation; it is a mistake to call this the invention of a notation for the infinitesimal calculus. As we shall see, Leibniz invented this notation for finite differences, and only applied it to the case in which the differences were infinitely small. Barrow’s method, of a and e, also survives to the present day, under the disguise of h and k, in the method by which the elements of the calculus are taught in nine cases out of ten. For higher differential coefficients the suffix notation is preferable, and later on the operator D is the method par excellence.
Here Leibniz seems to be unable to keep from harking back to the charge made by Fatio, suggesting that by the publication of his letters by Wallis this charge has been proved to be absolutely groundless.
It is possible that this may mean “has received high commendation”; for elogiis may be the equivalent of eulogy, in which case celebratus est must be translated as “has been renowned.”
This is untrue. As has been said, the attack was first made publicly in 1699; at this time, although Huygens had indeed been dead for four years, Tschirnhaus was still alive, and Wallis was appealed to by Leibniz. It is strange that Leibniz did not also appeal to Tschirnhaus, through whom it is suggested by Weissenborn that Leibniz may have had information of Newton’s discoveries. Perhaps this is the reason why he did not do so, since Tschirnhaus might not have turned out to be a suitable witness for the defense. Leibniz must have had this attack by Fatio in his mind, for he could hardly have referred to Keill as a novus homo, while we know that he did not think much of Fatio as a mathematician. To say that there never existed any uncertainty as to the name of the true inventor until 1712 is therefore sheer nonsense; for if by that he means to dismiss with contempt the attack of Fatio, whom can he mean by the phrase novus homo ? The sneering allusion to “the hope of gaining notoriety by the discussion” can hardly allude to any one but Fatio. Finally if Fatio is dismissed as contemptible, the second attack by Keill was made in 1708. If it was early in the year, Tschirnhaus was even then alive, though Wallis was dead.
Gerhardt says in a note (G. 1846, p. 22) that his real name was probably Kramer; for what reason I am unable to gather. Cantor says distinctly that his name was Kaufmann, and this is the usually accepted name of the man who was one of the first members of the Royal Society and contributed to its Transactions. It seems to me that Gerhardt is guessing; the German word Kramer means a small shopkeeper, while Kaufmann means a merchant. To Mercator is due the logarithmic series obtained by dividing unity by (1 + x) and integrating the resulting series term by term; the connection with the logarithm of (1 + x) is through the area of the rectangular hyperbola y(1 + x) =0. See Reiff, Geschichte der unendlichen Reihen.
Newton obtained the general form of the binomial expansion after the method of Wallis, i. e., by interpolation. See Reiff.
We now see what was Leibniz’s point; the differential calculus was not the employment of an infinitesimal and a summation of such quantities; it was the use of the idea of these infinitesimals being differences, and the employment of the notation invented by himself, the rules that governed the notation, and the fact that differentiation was the inverse of a summation; and perhaps the greatest point of all was that the work had not to be referred to a diagram. This is on an inestimably higher plane than the mere differentiation of an algebraic expression whose terms are simple powers and roots of the independent variable.
Why is Barrow omitted from this list? As I have suggested in the case of Barrow’s omission of all mention of Fermat, was Leibniz afraid to awake afresh the sleeping suggestion as to his indebtedness to Barrow? I have suggested that Leibniz read his Barrow on his journey back from London, and perhaps, tiring at having read the Optics first and then the preliminary five lectures, just glanced at the remainder and missed the main important theorems. I also make another suggestion, namely, that perhaps, or probably, in his then ignorance of geometry he did not understand Barrow. If this is the case it would have been gall and wormwood for Leibniz to have ever owned to it. Then let us suppose that in 1674 with a fairly competent knowledge of higher geometry he reads Barrow again, skipping the Optics of which he had already formed a good opinion, and the wearisome preliminary lectures of which he had already seen more than enough. He notes the theorems as those he has himself already obtained, and the few that are strange to him he translates into his own symbolism. I suggest that this is a feasible supposition, which would account for the marks that Gerhardt states are made in the margin. It would account for the words “in which latter I found the greater part of my theorems anticipated” (this occasion in future times ranking as the first time that he had really read Barrow, and lapse of memory at the end of thirty years making him forget the date of purchase, possibly confusing his two journeys to London) ; it would account for his using Barrow’s differential triangle instead of his own “characteristic triangle.” As Barrow tells his readers in his preface that “what these lectures bring forth, or to what they may lead you may easily learn from the beginnings of each,” let us suppose that Leibniz took his advice. What do we find? The first four theorems of Lecture VIII give the geometrical equivalent of the differentiation of a power of a dependent variable; the first five of Lecture IX lead to a proof that, expressed in the differential notation,
(ds/dx)2 = 1 + (dy/dx)2;
the appendix to this lecture contains the differential triangle, and five examples on the a and e method, fully worked out; the first theorem in Lecture XI has a diagram such that, when that part of it is dissected out (and Barrow’s diagrams want this in most cases) which applies to a particular paragraph in the proof of the theorem, this portion of the figure is a mirror image of the figure drawn by Leibniz when describing the characteristic triangle (turn back to note 30). I shall have occasion to refer to this diagram again. The appendix to this lecture opens with the reference to the work of Huygens; and the second theorem of Lecture XII is the strangest coincidence of all. This theorem in Barrow’s words is:
“Hence, if the curve AMB is rotated about the axis AD, the ratio of the surface produced to the space ADLK is that of the circumference of a circle to its diameter; whence, if the space ADLK is known, the said surface is known.”
The diagram given by Barrow is as usual very complicated, serving for a group of nine propositions. Fig. F is that part of the figure which refers to the theorem given above, dissected out from Barrow’s figure. Now remember that Leibniz always as far as possible kept his axis clear on the left-hand side of his diagram, while Barrow put his datum figure on the left of his axis, and his constructed figures on the right; then you have Leibniz’s diagram and the proof is by the similarity of the triangles MNR, PMF, where FZ = PM ; and the theorem itself is only another way of enunciating the theorem that Leibniz states he generalized from Pascal’s particular case! Lastly, the next theorem starts with the words: “Hence the surfaces of the sphere, both the spheroids and the conoids receive measurement.” What a coincidence!
Fig. F.
As this note is getting rather long, I have given the full proof of the first two theorems of Barrow’s Lecture XII as a supplement, at the end of this section.
The sixth theorem of this lecture is the theorem of Gregory which Leibniz also gives later; I will speak of this when I come to it. As also, when we discuss Leibniz’s proof of the rules for a product, etc., I will point out where they are to be found in Barrow ready to his hand.
Yet if all this were so, he could still say with perfect truth that, in the matter of the invention of the differential calculus (as he conceived the matter to consist, that is, the differential and integral notations and the method of analysis), he derived no assistance from Barrow. In fact, once he had absorbed his fundamental ideas, Barrow would be less of a help than a hindrance.
Apollonian geometry comprised the conic sections or curves of the second degree according to Cartesian geometry; curves of a higher degree and of a transcendent nature, like the spiral of Archimedes, were included under the term “mechanical.”
The great discovery of Descartes was not simply the application of geometry; that had been done in simple cases ages before. Descartes recognized the principle that every property of the curve was included in its equation, if only it could be brought out. Thus Leibniz’s greatest achievement was the recognition that the differential coefficients were also functions of the abscissa. The word function was applied to certain straight lines dependent on the curve, such as the abscissa itself, the ordinate, the chord, the tangent, the perpendicular, and a number of others (Cantor. III, preface, p. v). This definition is from a letter to Huygens in 1694. There is therefore a great advance made by 1714, the date of the Historia, since here it is at least strongly hinted that Leibniz has the algebraical idea of a function.
With regard to Newton, at least, this is untrue. Without a direct reference to the original manuscript of Newton it is quite impossible to state whether even Newton wrote 0 or 0 ; even then there may be a difficulty in deciding, for Gerhardt and Weissenborn have an argument over the matter, while Reiff prints it as 0. However this may be there is no doubt that Newton considered it as an infinitely small unit of time, only to be put equal to zero when it occurred as a factor of terms in an expression in which there also occurred terms that did not contain an infinitesimally small factor. This was bound to be the case, since Newton’s x and 
were velocities. In short, expressing Newton’s notation in that of Leibniz, we have
0 or 0 = (dx/dt). dt
and therefore 0 is an infinitesimal or a differential equal to Leibniz’s dx.
This is in a restricted sense true. No one seems to have felt the need of a second differentiation of an original function; those, who did, differentiated once, and then worked upon the function thus obtained a second time in the same manner as in the first case. Barrow indeed considered only curves of continuous curvature, and the tangents to these curves; but Newton has the notation 
, etc. But the idea had been used by Slusius in his Meso-labum (1659), where a general method of determining points of inflection is made to depend on finding the maximum and minimum values of the subtangent. Lastly, it can hardly be said that Leibniz’s interpretation of ∫∫ ever attained to the dignity of a double integral in his hands.
David Gregory is not the only sinner! Leibniz, using his calculus, makes a blunder over osculations, and will not stand being told about it; he simply repeats in answer that he is right (Rouse Ball’s Short History).
The names of the committee were not even published with their report. In fact the complete list was not made public until De Morgan investigated the matter in 1852 ! For their names see De Morgan’s Newton, p. 27.
What then made Leibniz change his mind?
It is established that this was Johann (John) Bernoulli; see Cantor, III, p. 313f ; Gerhardt gives a reference to Bossut’s Geschichte, Part II, p.219.
This seems to be an intentional misquotation from Bernoulli’s letter, which stated that Newton did not understand the meaning of higher differentiations. At least, that is what Cantor says was given in the pamphlet.
It is established that the pamphlet referred to was also an anonymous contribution by Leibniz himself! Is it strange that hard things are both thought and said of such a man?
Again this is Leibniz himself! Had he then no friends at all to speak for him and dare subscribe their signatures to the opinion? Unfortunately Tschirnhaus was dead at the time of the publication of the Commercium Epistolicum, but he could have spoken with overwhelming authority, as Leibniz’s co-worker in Paris, at any time between the date of Leibniz’s review of Newton’s De Quadratura in the Acta Eruditorum until his death in 1708, even if he had died before the publication of Keill’s attack in the Phil. Trans. of that year was made known to him. Does not this silence on the part of Tschirnhaus, the personal friend of Leibniz, rather tend to make Leibniz’s plea, that his opponents had had the shrewdness to wait till Tschirnhaus, among others, was dead, recoil on his own head, in that he has done the very same thing? Leibniz must have known the feeling that this review aroused in England, and, Huygens being dead, Tschirnhaus was his only reliable witness. Of course I am not arguing that Leibniz did found his calculus on that of Newton. I am fully convinced that they both were indebted to Barrow, Newton being so even more than Leibniz, and that they were perfectly independent of one another in the development of the analytical calculus. Newton. with his great knowledge of and inclination toward geometrical reasoning, backed with his personal intercourse with Barrow, could appreciate the finality of Barrow’s proofs of the differentiation of a product, quotient, power, root, logarithm and exponential, and the trigonometrical functions, in a way that Leibniz could not. But Newton never seems to have been accused of plagiarism from Barrow; even if he had been so accused, he probably had ready as an answer, that Barrow had given him permission to make any use he liked of the instruction that he obtained from him. Leibniz, when so accused. replied by asserting, through confusion of memory I suggest, that he got his first idea from the works of Pascal. Each developed the germ so obtained in his own peculiar way; Newton only so far as he required it for what he considered his main work, using a notation that was of greatest convenience to him, and finally falling back on geometry to provide himself with what appealed to him as rigorous proof; Leibniz, more fortunate in his philosophical training and his lifelong effort after symbolism, has ready to hand a notation, almost developed and perfected when applied to finite quantities, which he saw with the eye of genius could be employed as usefully for infinitesimals. De Morgan justly remarks that one dare not accuse either of these great men of deliberate untruth with regard to specific facts; but it must be admitted that neither of them can be considered as perfectly straightforward; and the political similitude, which Cantor speaks of, in which nothing is too bad to be said of an opponent, seems to have applied just as much to the mathematician of the day as to the politician.
This was given in more detail in the first draught of this essay (G. 1846, p. 26) : Hitherto, while still a pupil, he kept trying to reduce logic itself to the same state of certainty as arithmetic. He perceived that occasionally from the first figure there could be derived a second and even a third, without employing conversions (which themselves seemed to him to be in need of demonstration), but by the sole use of the principle of contradiction. Moreover, these very conversions could be proved by the help of the second and third figures, by employing theorems of identity; and then now that the conversion had been proved, it was possible to prove a fourth figure also by its help, and this latter was thus more indirect than the former figures. He marveled very much at the power of identical truths, for they were generally considered to be useless and nugatory. But later he considered that the whole of arithmetic and geometry arose from identical truths, and in general that all undemonstrable truths depending on reasoning were identical, and that these combined with definitions yield identical truths. He gave as an elegant example of this analysis a proof of the theorem, The whole is greater than its part.
It is fairly certain that Leibniz could not possibly at this time have perceived that in this theorem he has the germ of an integral. The path to the higher calculus lay through geometry. As soon as Leibniz attained to a sufficient knowledge of this subject he would recognize the area, under a curve between a fixed ordinate and a variable one as a set of magnitudes of the kind considered, the ordinates themselves being the differences of the set; he would see that there was no restriction on the number of steps by which the area attained its final size. Hence, in this theorem he has a proof to hand that integration as a determination of an area is the inverse of a difference. This does not mean the inverse of a differentiation, i. e., the determination of a rate, or the drawing of a tangent. As far as I can see, Leibniz was far behind Newton in this, since Newton’s fluxions were founded on the idea of a rate; also Leibniz apparently does not demonstrate the rigor of a method of infinitely narrow rectangles.
It is a pity that we are not told the date at which Leibniz read his Wallis; it is a greater pity that Gerhardt did not look for a Wallis in the Hanover Library and see whether it had the date of purchase on it (for I have handled lately several of the books of this time, and in nearly every case I found inserted on the title page the name of the purchaser and the date of purchase). I make this remark, because there arises a rather interesting point. Wallis, in his Arithmetica Infinitorum, takes as the first term of all his series the number 0, and in one case he mentions that the differences of the differences of the cubes is an arithmetical series. He also works out fully the sums of the figurate numbers (or as Leibniz calls them the combinatory numbers) ; the general formulas for these sums he calls their characteristics. He also remarks on the fact that any number (see table, p. 32) can be obtained by the addition of the one before it and the one above it (which is itself the sum of all the numbers in the preceding column above the one to the left of that which he wishes to obtain). Thus, in the fourth column 4 is the sum of 3 (to the left) and I (above), i. e., the sum of the two first numbers in column three; 10 is the sum of 6 (to the left) and 4 (above, which has been shown to be the sum of the first two numbers of column three), and therefore 10 is the sum of the first three numbers in column three. Now my point is, assuming it to have been impossible that Leibniz had read Wallis at the time that he was compiling his De Arte, we have here another example, free from all suspicion, of that series of instances of independent contemporary discoveries that seems to have dogged Leibniz’s career.
The name surdesolid to denote the fifth power is used by Oughtred, according to Wallis. By Cantor the invention of the term seems to be credited to Dechales, who says, “The fifth number from unity is called by some people the quadrato-cubus, but this is ill-done, since it is neither a square nor a cube and cannot thus be called the square of a cube nor the cube of a square: we shall call it supersolidus or surde solidus” (Cantor, III, p. 16). Wallis himself uses “sursolid.”
This theorem is one of the fundamental theorems in the theory of the summation of series by finite differences, namely,
which is usually called the direct fundamental theorem; for although Leibniz could not have expressed his results in this form since he did not know the sums of the figurate numbers as generalized formulas (or I suppose not, if he had not read Wallis), and apparently his is only a special case, yet it must be remembered that any term of the first series can be chosen as the first term. It is interesting to note that the second fundamental theorem, the inverse fundamental theorem, was given by Newton in the Principia, Book III, lemma V, as a preliminary to the discussion on comets at the end of this book. Here he states the result, without proof, as an interpolation formula; (it is frequently referred to as Newton’s Interpolation Formula) ; it may however be used as an extrapolation formula, in which case we have
um+n = um + nC1 
Δum + nC2 Δ2 um + etc.
In the two formulas as given here, the series are
What are we to understand by the inclusion of this series in this connection? Does Leibniz intend to claim this as his? I have always understood that this is due to Johann Bernoulli, who gave it in the Acta Eruditorum for 1694, in a slightly different form, and proved by direct differentiation; and that Brook Taylor obtained it as a particular case of a general theorem in and by finite differences. If Leibniz intended to claim it, he has clearly anticipated Taylor. It is quite possible that Leibniz had done so, even in his early days; and as soon as in 1675, or thereabouts, he had got his signs for differentiation and integration, it is possible that he returned to this result and expressed it in the new notation; for the theorem follows so perfectly naturally from the last expression given for a − ω. But it is hardly probable, for Leibniz would almost certainly have shown it to Huygens and mentioned it.
The other alternative is that here he is showing how easily Bernoulli’s series could have been found in a much more general form, i. e., as a theorem that is true (as he indeed states) for finite differences as well as for infinitesimals; the inclusion of this statement makes it very probable that this supposition is a correct one. This leads to a pertinent, or impertinent, question. Brook Taylor’s Methodus Incrementorum was published in 1715; the Historia was written some time between 1714 and 1716; Gerhardt states that there were two draughts of the latter, and that he is giving the second of these. In justice to Leibniz there should be made a fresh examination of the two draughts, for if this theorem is not given in the original draught it lays Leibniz open to further charge of plagiarism. I fully believe that the theorem will be found in the first draught as well and that my alternative suggestion is the correct one.
In any case, the tale of the Historia is confused by the interpolation of the symbolism invented later (as Leibniz is careful to point out). The question is whether this was not intentional. And this query is not impertinent, considering the manner in which Leibniz refrains from giving dates, or when we compare the essay in the Acta Eruditorum, in which he gives to the world the description of his method. Weissenborn considers that “this is not adapted to give an insight into his methods, and it certainly looks as if Leibniz wished deliberately to prevent this.” Cf. Newton’s “anagram” (sic), and the Geometry of Descartes, for parallels.
In reference to the employment of the calculus to diagrammatic geometry, as will be seen later, Leibniz says:
“But our young friend quickly observed that the differential calculus could be employed with figures in an even more wonderfully simple manner than it was with numbers, because with figures the differences were not comparable with the things which differed; and as often as they were connected together by addition or subtraction, being incomparable with one another, the less vanished in comparison with the greater.”
This makes what has just gone before date from the time previous to his reading of the work of Cavalieri. See note following.
This is about the first place in which it is possible to deduce an exact date, or one more or less exact. According to Leibniz’s words that immediately follow it may be deduced that it was somewhere about twelve months before the publication of the Hypothesis of Physics—if we allow for a slight interval between the dropping of the geometry and the consideration of the principles of physics and mechanics, and .a somewhat longer interval in which to get together the ideas and materials for his essay—that he had finished his “slight consideration” of Leotaud and Cavalieri. This would make the date 1670, and his age 24.
This essay founded the explanation of all natural phenomena on motion, which in turn was to be explained by the presence of an all-pervading ether; this ether constituted light.
The dedication of the Nova methodus in 1667 to the Elector of Mainz (ancient name Moguntiacum) procured for Leibniz his appointment in the service of the latter, first as an assistant in the revision of the statute-book, and later on the more personal service of maintaining the policy of the Elector, that of defending the integrity of the German Empire against the intrigues of France, Turkey and Russia, by his pen.
This probably refers to the time when his work on the statute-book was concluded, and Leibniz was preparing to look for employment elsewhere.
This is worthy of remark, seeing that Leibniz had attempted to explain gravity in the Hypothesis physica nova by means of his concept of an ether. The conversation with Huygens had results that will be seen later in a manuscript (see § 4, p.65) where Leibniz obtains quadratures “ex Centrobarycis.” It also probably had a great deal to do with Leibniz’s concept of a “moment.”
The use of the word veterno—which I have translated “lethargy” as being the nearest equivalent to the fundamental meaning, the sluggishness of old age—coupled with his remark that he was in no mind to enter fully into these more profound parts of mathematics, sheds a light upon the reason why he had so far done no geometry. Also the last words of the sentence give the stimulus that made him cast off this lethargy; namely, shame that he should appear to be ignorant of the matter. This would seem to be one of the great characteristics of Leibniz, and might account for much, when we come to consider the charges that are made against him.
We have here a parallel (or a precedent) for my suggestion that Leibniz was mentally confusing Barrow and Pascal as the source of his inspiration for the characteristic triangle. For here, without any doubt whatever, is a like confusion. What Pell told him was that his theorems on numbers occurred in a book by Mouton entitled De diametris apparentibus Solis et Lunae (published in 1670). Leibniz, to defend himself from a charge of plagiarism, made haste to borrow a copy from Oldenburg and found to his relief that not only had Mouton got his results by a different method, but that his own were more general. The words in italics are interesting.
Of course these words are not italicized by Gerhardt, from whom this account has been taken (G. 1848, p. 19) ; nor does he remark on Leibniz’s lapse of memory in this instance. Further there is no mention made of it in connection with the Historia, i. e., in G. 1846. Is it that Gerhardt, as counsel for the defense, is afraid of spoiling the credibility of his witness by proving that part of his evidence is unreliable? Or did he not become aware of the error till afterward? See Cantor, III, p. 76.
An instance is referred to on p. 85 of De Morgan’s Newton, showing the sort of thing that was done by the committee. This however is not connected with a letter to Oldenburg, but to Collins. It may be taken as a straw that shows the way the wind blew.
Observe that nothing has been said of the fact that Leibniz had purchased a copy of Barrow and took it back with him to Paris.
Cf. the remark in the postscript to Bernoulli’s letter, where Leibniz says that the work of Descartes, looked at at about the same time as Clavius, that is, while he was still a youth, “seemed to be more intricate.”
The libellus referred to would seem to be the work on the cycloid, written by Pascal in the form of letters, from one Amos Dettonville, to M. de Carcavi.
This theorem is given, and proved by the method of indivisibles, as Theorem I, of Lecture XII in Barrow’s Lectiones Geometricae; and Theorem II is simply a corollary, in which it is remarked:
“Hence the surfaces of the sphere, both the spheroids, and the conoids receive measurement....”
The proof of these two theorems is given at the end of this section as a supplement. See also Note 46, for its significance.
The whole context here affords suggestive corroboration in favor of the remarks made in Note 31 on the use of the word “moment,” though the connection with the determination of the center of gravity is here overshadowed by its connection with the surface formed by the rotation of an arc about an axis.
The figure given is exactly that given by Gerhardt, with the unimportant exception that, for convenience in printing, I have used U instead of Gerhardt’s θ, a V instead of his 
(a Hebrew T), and a Q for his II. I take it, of course, that Gerhardt’s diagram is an exact transcript of Leibniz’s, and it is interesting to remark that Leibniz seems to be endeavoring to use T’s for all points on the tangent, and P’s for points on the normal, or perpendicular, as it is rendered in the Latin.
This diagram should be compared with that in the “postscript” written nine or ten years before. Note the complicated diagram that is given here.
and the introduction of the secant that is ultimately the tangent, which does not appear in the first figure. From what follows, this is evidently done in order to introduce the further remarks on the similar triangles. It adds to the confusion when an effort is made to determine the dates at which the several parts were made out. For instance, the remark that finite triangles can be found similar to the characteristic triangle probably belongs approximately to the date of his reply to the assertions of Nieuwentijt, which will be referred to later.
The notation introduced in the lettering should be remarked. His early manuscripts follow the usual method of the time in denoting different positions of a variable line by the same letter, as in Wallis and Barrow, though even then he is more consistent than either of the latter. He soon perceives the inconvenience of this method, though as a means of generalizing theorems it has certain advantages. We therefore find the notation C, (C), ((C)), for three consecutive points on a curve, as occurs in a manuscript dated (or it should be) 1675. This notation he is still using in 1703; but in 1714, he employs a subscript prefix. This is all part and parcel with his usual desire to standardize and simplify notations.
This sentence conclusively proves that Leibniz’s use of the moment was for the purposes of quadrature of surfaces of rotation.
“From these results”—which I have suggested he got from Barrow—“our young friend wrote down a large collection of theorems.” These theorems Leibniz probably refers to when he says that he found them all to have been anticipated by Barrow, “when his Lectures appeared.” I suggest that the “results” were all that he got from Barrow on his first reading, and that the “collection of theorems” were found to have been given in Barrow when Leibniz referred to the book again, after his geometrical knowledge was improved so far that he could appreciate it.
The use of the first person is due to me. The original is impersonal, but is evidently intended by Leibniz to be taken as a remark of the writer, “the friend who knew all about it.” The distinction is marked better by the use of the first personal pronoun than in any other way.
Query, all except Leibniz, the Bernoullis, and one or two others.
Tetragonism = quadrature; the arithmetical tetragonism is therefore Leibniz’s value for π as an infinite series, namely,
“The area of a circle, of which the square on the diameter is equal to unity, is given by the series
This is clearly original as far as Leibniz is concerned; but the consideration of a polar diagram is to be found in many places in Barrow. Barrow however forms the polar differential triangle, as at the present time, and does not use the rectangular coordinate differential triangle with a polar figure; nor does Wallis. We see therefore that Leibniz, as soon as ever he follows his own original line of thinking, immediately produces something good.
This is evidently a misprint; it is however curious that it is repeated in the second line of the next paragraph. Probably, therefore, it is a misreading due to Gerhardt, who mistakes AZ for the letters XZ, as they ought to be; and has either not verified them from the diagram, or has refrained from making any alteration.
The symbol 
is here to be read as “and then along the arc to.”
Probably refers to Leibniz’s work on curvature, osculating circles, and evolutes, as given in the Acta Eruditorum for 1686, 1692, 1694. It is to be noted that with Leibniz and his followers the term evolute has its present meaning, and as such was first considered by Huygens in connection with the cycloid and the pendulum. It signified something totally different in the work of Barrow, Wallis and Gregory. With them, if the feet of the ordinates of a curve are, as it were, all bunched together in a point, so as to become the radii vectores of another curve, without rupturing the curve more than to alter its curvature (the area being thus halved), then the first curve was called the evolute of the second and the second the involute of the first. See Barrow’s Lectiones Geometricae, Lecture XII, App. III, Prob. 9, and Wallis’s Arithmetica Infititorum, where it is shown that the evolute, in this sense, of a parabola is a spiral of Archimedes.
The colon is used as a sign of division, and the comma has the significance of a bracket for all that follows. It is curious to notice that Leibniz still adheres to the use of xx for x2, while he uses the index notation for all the higher powers, just as Barrow did; also, that the bracket is used under the sign for a square root, and that too in addition to the vinculum. For an easy geometrical proof of the relation x = 2z2/ (1 + z2), see Note 94.
See Cantor, III, pp. 78-81. Also note the introduction of what is now a standard substitution in integration for the purpose of rationalization.
This term represents what is now generally known as the method of inversion of series. Thus, if we are given
x = y + ay2 + by3 + cy4 + etc.,
where x and y are small, then y = x is a first approximation; hence since y = x − ay2 − by3 − cy4 − etc., we have as a second approximation
y = x − ax2;
substituting this in the term containing y2, and the first approximation, y = x, in the term containing y3, we have
y = x−a(x−ax2)2−bx3 = x−ax2+ (2a2−b)x3,
as a third approximation; and so on.
The relation x = 2z2/(1+z2) can be easily proved geometrically for the circle; hence, by using the orthogonal projection theorem, Leibniz’s result for the central conic can be immediately derived.
Thus suppose that, in the diagrams below, AC is taken to be unity, then AU = z and AX = x.
Then, in either figure, since the Δs BYX, CUA are similar,
AX : XB = AX. XB : XB2 = XY2: XB2 = AU2 : CA2;
hence, for the circle, we have
AX: AB = AU2 = : AC2 + AU2, or x = 2z2/(1+z2);
and similarly for the rectangular hyperbola
AX: AB = AU2: AC2−AU2, or x = 2z2/(1−z2).
Applying all the x’s to the tangent at A, we have (by division and integration of the right-hand side, term by term, in the same way as Mercator)
area AUMA = 2 (z3/3 
z5/5 + z7/7 etc.)
Now, since the triangles UAC, YXB are similar, UA.XB=AC.XY; hence 2ΔAYC = 2UA.AC UA.AX = 2UA.AC AUMA 2 seg. AYA, for Leibniz has shown that AXMA = 2 seg. AYA; hence it follows immediately that
sector ACYA = z z3/3 + z5/5 etc.
Fig. G.
Fig. H.
If now, keeping the vertical axis equal to unity, the transverse axis is made equal to a, Leibniz’s general theorem follows at once from the orthogonal projection relation.
Note that z is, from the nature of the diagrams, less than 1.
Wallis’s expression for π as an infinite product, given in the Arithmetica (or Brouncker’s derived expression in the form of an infinite continued fraction), or the argument used by Wallis in his work, could not possibly be taken as a proof that π could not be expressed in recognized numbers.
The letter that is missing would no doubt have been given, in the event of the Historia being published. According to Gerhardt it is to be found in Ch. Hugenii. . . .exercitationes, ed. Uylenbroeck, Vol. I, p. 6, under date Nov. 7, 1674.
Collins wrote to Gregory in Dec. 1670, telling him of Newton’s series for a sine, etc.; Gregory replied to Collins in Feb. 1671, giving him three series for the arc, tangent and secant; these were probably the outcome of his work on Vera Circuli (1667).
By Mercator; query, also an allusion to Brouncker’s article in the Phil. Trans., 1668.
Quite conclusive; no other argument seems required.
This date, April 12, 1675, is important; it marks the time when Leibniz first began to speak of geometry in his correspondence with Oldenburg, as he says below.
Newton obtained the series for arcsin x from the relation 
: = 1: 
, by expansion and integration, and then the series for the sine by the “extraction of roots.” See Note 93, and, for Newton’s own modification, Cantor, III, p. 73.
It would appear from this that Leibniz could differentiate the trigonometrical functions. Professor Love, on the authority of Cantor, ascribes them to Cotes; but I have shown in an article in The Monist for April, 1916, that Barrow had explicitly differentiated the tangent and that his figures could be used for all the other ratios. Note the word “later” in the next sentence.
Probably only to test Leibniz’s knowledge.
Gerhardt states that in the first draft of the Historia, Leibniz had bordered the Harmonic Triangle, as given here, with a set of fractions, each equal to 1/1, so as to correspond more exactly with the Arithmetical Triangle.
The sign here used appears to be an invention of Leibniz to denote an identity, such as is denoted by ≡ at present.
This, and other formulas of the same kind, had been given by Wallis in connection with the formulas for the sums of the figurate numbers. Wallis called these latter sums the “characters” of the series.
This sentence, in that it breaks the sense from the preceding sentence to the one that follows, would appear to be an interpolated note.
There is an unimportant error here. The first value of x evidently should be 0, and not 1.
Why not? Newton’s dotted letters still form the best notation for a certain type of problem, those which involve equations of motion in which the independent variable is the time, such as central orbits. Probably Leibniz would class the suffix notation as a variation of his own, but the D-operator eclipses them all. For beginners, whether scholastic or historically such (like the mathematicians that Barrow, Leibniz and Newton were endeavoring to teach), the separate letter notation has most to recommend it on the score of ease of comprehension; we find it even now used in partial differential equations.
Leibniz does not give us an opportunity of seeing how he would have written the equivalent of dxdxdx; whether as dx3 or 
or (dx)3.
Ductus and ungulae have already been explained in Notes 28, 29; cuneus denotes a wedge-shaped solid; cf. “cuneiform.”
This is peculiar. The demonstration that follows was beyond the powers of Leibniz in June, 1676 (see pp. 121, 122), probably so until Nov., 1676, when he was in Holland, and possibly later still. Hence the result would have been communicated to Huygens by letter, and there would be an answer from Huygens. I have been so far unable to find such a letter.
This only proves the proportionality, enabling Leibniz to convert the equation 2∫dy/y = 3∫dx/x into 2 log y = 3log x. It will hardly suffice as it stands to enable him to deal with such an equation as 2∫dy/y = 3∫x dx; and it is to be noted that Leibniz does not notice at all the constant of integration. Although Barrow has in effect differentiated (and therefore also has the inverse integral theorems corresponding thereto) both a logarithm and an exponential in Lecture XII, App. III, Prob. 3, 4, yet these problems are in such an ambiguous form that it may be doubted whether Barrow was himself quite clear on what he had obtained. Hence this clear statement of Leibniz must be considered as a great advance on Barrow.
Almost seems to read as a counter-charge against Newton of stealing Leibniz’s calculus. Note the tardy acknowledgement that Barrow has previously done all that Newton had given.
The whole effect that this Historia produces in my mind is that the entire thing is calculated to the same end as the Commercium Epistolicum. The pity of it is that Leibniz could have told such a straightforward tale, if events had been related in strict chronological order, without any interpolations of results that were derived, or notation that was perfected, later. A tale so told would have proved once and for all how baseless were the accusations of the Commercium, and largely explained his denial of any obligations to Barrow.
Let MP be perpendicular to the curve AB, and the lines KZL, αφδ such that FZ = MP, µφ = MF. Then the spaces αβδ, ADLK are equal.
For the triangles MRN, PFM are similar, MN : NR = PM : MF,
MN.MF = NR.PM;
that is, on substituting the equal quantities,
µν.µφ = FG. FZ, or rect. µθ = rect. FH.
But the space αβδ only differs in the slightest degree from an infinite number of rectangles such as µθ, and the space ADLK is equivalent to an equal number of rectangles such as FH. Hence the proposition follows.
Hence, if the curve AMB is rotated about the axis AD, the ratio of the surface produced to the space ADLK is that of the circumference of a circle to its diameter; whence if the space ADLK is known, the said surface is known.
Some time ago I assigned the reason why this was so.
Hence, the surfaces of the sphere, both the spheroids, and the conoids receive measurement. For if AD is the axis of the conic section, etc.
§§ 3-10 inclusive appeared in The Monist for April, 1917.
It is impossible to see, without a fuller knowledge of the context,whether this refers to “compensation of errors,” or whether Leibniz is alluding to the possibility of all the finite terms cancelling one another.
Leibniz comes back to this point later; see § 5.
This, without either proof or figure, is a hopeless muddle; and yet it is repeated word for word, without any addition or remark, in Gerhardt’s 1855 publication. Goodness knows what the use of it was supposed to be in this form! Unless Leibniz has omitted some length, which he has supposed to be unity, the dimensions are all wrong.
The sign 
signifies multiplication.
Observe that as yet nothing has been said about the area of surfaces of revolution or moments about the axis, although we should expect them to be mentioned in connection with the figure that is given; for the next manuscript shows that in October 1675, Leibniz has already done a considerable amount of work on moments.
Gerhardt has a footnote to the effect that, as nearly as possible he has retained the exact form of this and the manuscripts that immediately follow; except in the matter of this one sign I have adhered to the form given by Leibniz.
Weissenborn, Principien der höheren Analysis, Halle, 1856.
This a should be x.
Here, in the Latin, “ac m omn.x” should be “a c in omn.x.”
In view of this accurate bit of algebra, the faulty work in subsequent manuscripts seems very unaccountable.
This proves the fundamental theorem given lower down, with regard to a pair of parallel straight lines; and he now goes on to discuss the case of non-parallel straight lines.
The passage in Gerhardt reads:
Datis ergo duobus momentis figurae ex duabus rectis non parallelis, dabitur figurae momentis tribus axibus librationis, qui non sint omnes paralleli inter se, dabitur figurae area, et centrum gravitatis.
For this I suggest:
Datis ergo tribus momentis figurae ex tribus rectis non parallelis, aliter figurae momentis tribus axibus librationis, qui non sunt omnes paralleli inter se. . . .
The passage would then read:
Given three moments of a figure about three straight lines that are not parallel, in other words, the moments of the figure about three axes of libration, which are not all parallel to one another, then the area of the figure will be given and also the center of gravity.
If the alternative words are written down, one under the other, and not too carefully, I think the suggested corrections will appear to be reasonable.
Apparently, here Leibniz is referring back to the theorem at the beginning of the section.
I have given this equation, and those that immediately follow it, in facsimile, in order to bring out the necessity that drove Leibniz to simplify the notation.
We have here a very important bit of work. Arguing in the first instance from a single figure, Leibniz gives two general theorems in the form of moment theorems. The first is obvious on completing the rectangle in his diagram, and this is the one to which the given equation applies. In the other the whole, of which the two parts are the complements, is the moment of the completed rectangle; its equivalent is the equation
omn.xy = ult.x omn.y − omn. omn.y.
Now, although Leibniz does not give this equation, it is evident that he recognized the analogy between this and the one that is given; for he immediately accepts the relation as a general analytical theorem that he can use without any reference to any figure whatever, and proceeds to develop it further. This would therefore seem to be the point of departure that led to the Leibnizian calculus.
Having freed the matter from any reference to figures, he is able to take any value he pleases for the letters. He supposes that z = 1, and thus obtains the last pair of equations. He then considers x and w as the abscissa and ordinate of the rectangular hyperbola xw = a (constant) ; hence omn.a/x or omn. w is the area under the hyperbola between two given ordinates, and therefore a logarithm; and thus omn. omn.a/x is the sum of logarithms, as he states. See Note 60, p. 122.
There only seem to be two possible sources for this paragraph, (1) original work on the part of Leibniz, and (2) from Barrow. For we know that Neil’s method was that of Wallis, and the method of Van Huraet used an ordinate that was proportional to the quotient of the normal by the ordinate in the original curve.
Now Barrow, in Lect. XII, § 20, has the following: “Take as you may any right-angled trapezial area (of which you have sufficient knowledge), bounded by two parallel straight lines AK, DL, a straight line AD, and any line KL whatever; to this let another such area be so related that when any straight line FH is drawn parallel to DL, cutting the lines AD, CE, KL in the points F, G, H, and some determinate line Z is taken, the square on FH is equal to the squares on FG and Z. Moreover, let the curve AIB be such that, if the straight line GFI is produced to meet it, the rectangle contained by Z and FI is equal to the space AFGC; then the rectangle contained by Z and the curve AB is equal to the space ADLK. The method is just the same, even if the straight line AK is supposed to be infinite.”
This striking resemblance, backed by the fact that there seems to be no connection between this theorem and the rest of the paper, that Leibniz gives no attempt at a proof, (indeed I very much doubt whether I could have made out his meaning from the original unless I had recognized Barrow’s theorem) and that Leibniz gives 1675 as the date of his reading Barrow, almost forces one to conclude that this is a note on a theorem (together with an original deduction therefrom by himself) which Leibniz has come across in a book that is lying before him, and that that book is Barrow’s. Against it, we have the facts of the use of the word “quadratrix,” not in the sense that Barrow uses it, namely as a special curve connected with the circle; that the quadratrix is one of the special curves that Barrow considers in the five examples he gives of the Differential Triangle method; and that another example of this method is the differentiation of a trigonometrical function which seems to be unknown to Leibniz.
This is either a misprint, v instead of O, or else Leibniz is in error. For Slusius’s method there must be only two variables in the equation. In the Phil. Trans. for 1672 (No. 90), Sluse gives his method thus:
If y5 + by4 = 2qqν3 − yyν3 then the equation must be written y5 + by4 + yyν3 = 2qqν3 − yyν3; then multiply each term on the left-hand side by the number of y’s in the term, and substitute t in place of one y in each; similarly multiply each term on the left-hand side by the exponent of ν; the equation obtained will give the value of t.
The use of the letters v and y is to be noted in connection with Leibniz’s use of the same letters; it does not seem at all necessary that Leibniz should have seen Newton’s work, with this ready to the former’s hand, as a member of the Royal Society. I suggest that Sluse obtained his rule by the use of a and e, as given in Barrow. Can Barrow’s words usitatum a nobis (in the midst of a passage written in the first person singular) have meant that the method was common property to himself and several other mathematicians that were contemporary with him? This would explain a great deal.
There is evidently a slip here; l should be x.
This is an instance of the care which Leibniz takes; in the work above l has been the difference for y, and a the difference for x; he is now integrating an algebraical expression, and not considering a figure at all; hence l = a, and a is equal to unity, and therefore ∫l3 = l3x = a3x = x! Thus what is generally considered to be a muddle turns out to be quite correct. The muddle is not with Leibniz, it is with the transcriber. It is certain that these manuscripts want careful republishing from the originals; won’t some millionaire pay to have them reproduced photographically in an edition de luxe?
This is, as I am going to show later, on p. 180, palpably a mere analytical translation of Barrow’s geometry.
Since the triangles QLI, WL(L) are similar, QI.B (B) = QL.Q(Q), hence omn.QI (applied to AB) = omn. QL (to AQ) = figure AQLA, hence omn. (QI + QA) = rect. ABLQ = 2ΔABL.
Since l is the difference for y, therefore 2l is the difference for 
υ; this is shown to be (υ + tυ)/t or x(v/t) + υ; and this is the equivalent to (since v/t = dυ/d = dυ)
d(υ) = xdv + υ = xdv + υdx.
The meaning of this is probably a series such as that considered by Wallis. If a, a + d, a + 2d, etc. is the arithmetical progression, and l, l−d, l−2d, etc. is the series reversed, then the series refolded reciprocally is al, (a + d)(l−d), (a + 2d)(l−2d), etc. It may however mean the sum of the squares of the arithmetical progression. But the point is not very important.
The accuracy of the algebra is noteworthy in comparison with the inaccuracies that occur later. There is however a slip: e2 = f2 + d2 and not f2 − d2; this must be a slip and not a misprint, because it persists throughout. It should be noted that the figure given by Gerhardt is careless in that LM is made to pass through A.
Such theorems are also considered in Wallis, where it is shown that the products for two equal parabolas are the squares on the ordinates of a semicircle; the axes of the parabolas being coincident, but set in opposite sense.
This is obviously wrong; the base of the cylinder is the area made up of FL, GM, HN, etc. The whole of this last passage proved to be difficult to make out; Leibniz has not completed his figure, by showing the surface formed by placing the ordinates FL, GM, HN with their middle points at C, D, E, and the ordinates themselves perpendicular to the plane of the curve BCDE, which figure I have added on the right-hand side of Leibniz’s figure. Even when this is given, there is another difficulty added because as given by Gerhardt, CS is the tangent at D instead of the proper line, namely, the perpendicular from C to TS; in addition through a misprint, this line is afterward referred to as TC. Lastly, “the rectangle FLG” is a misprint for FLC, which with Leibniz stands for FL.LC; this notation for a rectangle is, as far as I can remember, used by Wallis and Cavalieri.
When all these errors are revised, what at first sight seemed to be rather a muddle turns out to be an exceedingly neat idea in connection with the moments of a figure, and their use to find an area, although mostly impracticable; it is evidently taken from Pascal (cf. onglets).
Note. The values f, g, a, h, are the lengths of TQ, QP, PT, and the perpendicular from Q on PT.
See Cantor, III, p. 183; but neither Cantor nor Gerhardt appears to offer any suggestion as to why this date should have been altered.
See foot of next page.
27 This was obtained in the form omn.p = y2/2, previous to October, 1674, from the Pascal form of the characteristic triangle; it is quoted as a known theorem in the essay dated 29 October, 1675. See §§ 3, 6.
It is probably at this date that he began to revise his ideas as to d diminishing the dimensions; being forced to reconsider them by the occurrence of such equations as wz = y. It is seen in the next paragraph how careful he is to keep his dimensions equal; for he introduces an apparently irrelevant a(= 1) for this purpose. It gradually dawns on him that neither ∫ nor d alters the dimensions, but that a “sum of lines” is really a sum of rectangles, on account of the fact that they are applied in a certain fixed way to an axis; he is not quite certain of this however until well on in the next year, when we find him using ∫dx y.
It is difficult to see exactly what Leibniz means by this statement; I can only guess at substitution by means of the theorem wz = y, the equivalent to the recognition of the fact that y dy/dx.dx = ydy. The wording is however impersonal, and may mean that he himself had never thought of the idea before. Barrow has many such theorems for changing the variable.
Required y = f(x), such that y dy/dx = a2/x ; the solution is y2 = 2a2. logeAx. Weissenborn remarks on the omission of the a as being incorrect; from Leibniz’s standpoint I cannot agree with him. Leibniz, from Mercator’s s work, connects a2/x with the ordinate of the equilateral hyperbola xy = a2, and its integral with the quadrature of this curve. The omission of the a2 only alters the base of the logarithm, and Leibniz merely states that the solution is of a logarithmic nature without attempting to give it exactly.
How does he know until he has tried it? This rather combats the idea that these were mere exercises; it gives this essay the appearance of being a fair copy intended either for publication or for one of his correspondents. If this were the case, the errors later in algebraical work are all the more unintelligible. The idea that Leibniz was a man who was accustomed to writing down his thoughts as he went along does not appeal to me at all; this is the method of the slow-working mind, rather than that of genius.
This seems to be the root of the error into which he falls; he has not yet perceived that the e’s have to be applied to some axis, before he can sum them; and this is to a great extent due to the omission of the dx, taken as constant and equal to unity. He is thus bound to fall back on the algebraical summation of a series.
From the characteristic triangle, AS: AP = dx: dy.
This is of course nonsense. The error seems to arise from the dx being placed outside the integral sign; thus he assumes that dx is constant, while, for the integration, he also assumes that the dy is constant.
We cannot argue from this equation that Leibniz did not at this date appreciate what an infinitesimal was, on account of the infinitesimal being equated to a finite ratio; for since he is assuming that dy is an infinitely small unit, dx really stands for dx/dy.
Note the advance in ideas suggested by the words “infinitely small compared with the former.” Here, of course, the notation BFC is the usual notation of the period for BF.FC, the rectangle contained by BF and FC.
Note in general that this is Leibniz’s equivalent of the modern phrase, “integrate with respect to x.” (For the rest, see fig., p. 93.)
This I think is more likely to be a slip on the part of Leibniz, than a misprint; for in the next line he has AD, which is the correct equivalent of y. Further, AP varies inversely as x, hence the AP’s have to be in harmonical progression, not arithmetical, otherwise x is not equal to x2/2. If on the other hand, we assume three errors of transcription, and replace x for y, AB for AD, AB for AP, the whole thing is correct with an arbitrary base.
It is hardly necessary to point out the error in the arithmetical solution of the quadratic; nor is it important. It is however to be noted that if AC = υ, the equation reduces to υ2 = x(x + υ), and the solution is a pair of straight lines.
This is strongly reminiscent of Barrow, Lect. I (near the beginning) and Lect. III (near the end).
Leibniz, as a logician, should have known better than to trust a single example as a verification of an affirmative rule.
With regard to infinitesimals note the equation dx dy = x!
If Leibniz can see that this equality is “obviously incorrect,” what is the use of the argument that has preceded this sentence; for the final result must also be obviously incorrect.
Leibniz here justifiably verifies the falsity of his supposition being a general rule by a single breach of it. He uses υ = ψ = x, and changes x into x + β; thus,
Here we see the first idea of the method that is the same as that used by Fermat and, afterward by Newton and Barrow; this consideration, whatever the source, is that which leads him later to the substitution x + dx, y + dy in those cases in which Barrow uses a and e.
“ordinando et accommodando,” literally setting in order and adapting. It is to be remembered that Sluse gave only a rule, and not a demonstration of the rule. Part of the rule was that, if the equation in two variables contained terms containing both the variables, these terms had to be set down on each side of the equation. Thus, for the equation y3 = bυυ – yυυ would first of all be written
y3 + yυυ = bυυ − yυυ ..... ordinando (?)
then each term on the left is multiplied by the exponent of y, and each term on the right by that of v, thus,
3y3 + yυυ = 2bυυ − 2yυυ ..... accommodando (?)
and finally one y on the left, in each term, is changed into a t, where t is the subtangent measured along the y axis.
This is hopelessly inaccurate; all except one error, namely, ƒ + 2e, which should be βƒ + 2eω, may be put down to bad transcription. Even if Leibniz’s writing were execrable, the correct version of an ambiguous sign (through bad writing) could easily have been settled, by working through the algebra. Thus the first of the last pair of values, in Leibnizian symbols should be
with a similar correction in the second value.
Even if Leibniz had worked out the correct result, and obtained what he was trying for, namely, ω/β in terms of x, he would have got a very lengthy quadratic, and the roots would be quite beyond his power to use at any time. But he convinces himself that he can thus find the quadrature of any conic, or figures that can be reduced to them.
There is a mistake in sign; a2 − y2 should be y2−a2; hence the work that follows is also wrong.
Although the variables are separable, Leibniz does not recognize the fact that he can make use of this. For later he states that the solution of a problem cannot be obtained from a single equation. In this case we have
Supposing this substitution to have been effected, Leibniz would have concluded that x = υ, and would have stated that he had solved the problem.
But here again he has made an unfortunate choice, for the origin (A) cannot fall on any of the curves Cx = υ or Cx2 ± y2 = ± a2, which is the general solution of the equation. Hence the problem is impossible.
This is quite unintelligible to me as it stands; query, is it an accurate transcription ?
This is tantamount to a confession by Leibniz that he cannot explicitly integrate ∫a2/y, although he knows that it is logarithmic or reduces to the area under the hyperbola; for he has given this in the MS. for Nov. 11.
There are several errors in the letters in this paragraph, which are probably due to transcription; thus, an E for a ( ? badly written) B, an H for an A, etc., would be quite an easily-imagined error, provided the work was not verified during transcription.
The method of Hudde appears to be similar in principle to that of Sluse, while that of Descartes was the construction of the derived function by assuming roots, forming the sum of the quotients of the function divided by each of the assumed root-factors in turn, and comparison with the original function. Both therefore reduce to finding the common measure of the equation to the curve (where the right-hand side is zero) and the differential of it.
Leibniz, however, strange to say, does not note that by taking one of his arbitrary constants, q, equal to f, the equation has its degree lowered in the particular case he has chosen.
In this and the following line I have corrected two obvious misprints; they are evidently not the fault of Leibniz, for the lines that follow from them are correct.
There is some doubt here as to whether Leibniz could have given an example; but it must be remembered that these are practically only notes, mostly for future consideration.
Leibniz has a footnote to this manuscript: “I solved in one day two problems on the inverse methods of tangents, one of which Descartes alone solved, and the other even he owned that he was unable to do.”
This problem is one of them, the first mentioned in the footnote given by Leibniz. But it requires a stretch of imagination to consider Leibniz’s result as a solution. For he ends up with a geometrical construction, that is at least as hard as the construction that can be made by the use of the original data. There is of course the usual misprint that one is becoming accustomed to; but there is also the unusual, for Leibniz, mistake of using his data incorrectly. Starting with the hypothesis that BC : CL = N : BJ, he writes CL = N.BC/BJ (correcting the omission of the factor N), instead of CL= BC. BJ/N.
The solution of the problem is y+n log(y − x + n) = 0, as originally stated, or x = n log(n − y + x), if we continue from Leibniz’s erroneous result dx/dy = n/(y − x).
The point to be noted, however, is that Leibniz does not remark that “this curve appertains to a logarithm.”
Leibniz does not see that this result immediately gives him the equation that he requires. Thus x = c Log y, as he would have written it; the usual omission of the arbitrary constant does not matter in this case, so long as BA is taken as unity, which is possible with Leibniz’s data.
Here he seems to recognize that he has the solution. The next sentence is, however, very strange. As long ago as Nov. 1675 he has written ∫a2/y as Log y, and recognized the connection between the integral and the quadrature of the hyperbola; and yet he says “unless I am mistaken, ∫dy/y is always in our power.” Now notice that in the date there is no day of the month given, contrary to the usual custom with these manuscripts so far; can it be possible that this date was afterward added from memory, and that the manuscript should bear an earlier date? If not we must conclude that Leibniz has not yet attained to a correct idea of the meaning of his integral sign, and is still worried by the necessity (as it appears to him) of taking the y’s in arithmetical progression.
The passage in the original Latin is very ambiguous, and it may be that it is not quite correctly given ; I think, however, that I have given the correct idea of what Leibniz intended. One has to draw an auxiliary curve, in which y = dy/dx, and then find its area; in that case it should be “divided by the differences of the abscissae” instead of “divided by the abscissae.”
An interpolated note, marking a sudden thought or guess; for the next sentence carries on the train of thought that has gone before. Query, some interval of time, either short (such as for a meal) or long (continued the next day), may have occurred here.
This cannot be referred back to the present problem, since Leibniz has already assumed in it that dz and dx are constant. This may account for the fact that he has hesitated to say that the integral represents a logarithm.
This working is intended to apply to the auxiliary curve mentioned above, w standing for dx, and β for dy; hence the curve is not a hyperbola; Leibniz seems to have been misled by the appearance of the equation suggesting xy = constant.
Here apparently he leaves the muddle, in which he has entangled himself, and returns to his original equation; he then remembers that he has found before that the integral in question leads to a logarithm. (See p. 71.)
He has not solved either of them; nor can it be said from this that “Leibniz in 1676 sought and found the curve whose subtangent is constant.” Of all the work that Leibniz has done hitherto, there is none that is so inconclusive as this in comparison.
AT LAST! The recognition of the fact that neither dx nor dy need necessarily be constant, and the use of another letter to stand for the function that is being differentiated, mark the beginning, the true beginning, of Leibniz’s development of differentiation. Later in this manuscript we find him using the third great idea, probably suggested by the second of those given above, namely, the idea of substitution, by means of which he finally attains to the differentiation of a quotient, and a root of a function.
It is very suggestive that this remarkable advance occurs after his second visit to London, while he is staying in Holland. Did some one tell him then of the work of Newton, or of Barrow’s method (which is geometrically an exact equivalent of substitution), pointing out those things of which he had not perceived the drift, or is it the result of his intercourse with Hudde? For the date is that of his stay at The Hague. See Chap. VI, “Leibniz in London.”
This is Barrow all over; even to the words omissis omittendis instead of Barrow’s rejectis rejiciendis. Lect. X, Ex. 1 on the differential triangle at the end of the lecture.
Here we have the idea of substitutions, which made the Leibnizian calculus so superior to anything that had gone before. Note that he still has the.erroneous sign that he obtained for the differentiation of √x at the beginning of this manuscript. Also that the dz is wrongly placed in the denominator of the result.
This line represents the “etc.” of the original equation, and is set down for the purpose of getting the derived terms; the complete derived equation therefore consists of the two lines above and the two below. Note the omission of the negative sign, when changing from the equation to the proportion.
Leibniz, at the beginning, first wrote, “Hudde, Sluse, and others”; but later he struck out all but Sluse. (Gerhardt.)
The complete statement of the method of substitutions.
Leibniz has evidently seen Newton’s work at the time of this composition; also the use of the word “descends” in the next line again suggests Barrow, while the figure is exactly like the top half of the diagram given by Barrow for Lect. XI, 10, which is the theorem of Gregory that is quoted by Leibniz also. For this figure, see Note 71, p. 140.
Leibniz does not give a diagram, but it is not difficult to construct his figure from the enunciation that he gives for it. The whole of this paragraph should be compared with the following extract from Barrow (Lect. XI, 19), piece by piece.
“Again, let AMB be a curve of which the axis is AD and let BD be perpendicular to AD; also let KZL be another line such that, when any point M is taken in the curve AB, and through it are drawn MT a tangent to the curve AB, and MFZ parallel to DB, cutting KZ in Z and AD in F, and R is a line of given length, TF: FM = R : FZ. Then the space ADLK is equal to the rectangle contained by R and DB.
For, if DH = R and the rectangle BDHI is completed, and MN is taken to be an indefinitely small arc of the curve AB, and MEX, NOS are drawn parallel to AD; then we have NO : MO = TF : FM = R : FZ;
NO.FZ = MO.R and FG.FZ = ES.EX.
Hence, since the sum of such rectangles as FG.FZ differs only in the least degree from the space ADLK, and the rectangles ES.EX form the rectangle DHIB, the theorem is quite obvious.”
All the things given are to be found in Barrow, but his name is not even mentioned.
This is the strangest coincidence of all! For, Barrow also quotes this very same theorem of Gregory, and no other theorem; also it occurs in this very same Lect. XI that has been referred to already! Leibniz does not give a diagram; nor from his enunciation could I complete the figure required, until I had referred to the figure given by Barrow!!! The two diagrams are given below for comparison, Barrow’s figure being the one referred to in the note above. Query, is Leibniz’s figure taken from Gregory’s original, which I have not been able to see, or is it the Leibnizian variation of Barrow’s?
The Latin here is rather ambiguous; query, a misprint. But I think I have correctly rendered the argument. It is to be noted that the parabola was at this period always thought of in the form we should now denote by the equation y = x2, and the figure referred to by Leibniz is that which Wallis calls the complement of the semiparabola.
The term is here used with the idea of “vanishing into the far distance.”
This makes (d)dx an inassignable. It may be a misprint due to a slip of Leibniz, or of Gerhardt in transcription; for there is no similarity between it and the statement in the next line. I cannot however offer any feasible suggestion for correction.
This is quite wrong. Leibniz has evidently substituted x + dx for x, etc.; which is not legitimate unless 3X3Y is taken as y + dy + d(y + dy), and so on; even then fresh difficulties would be introduced. As it stands, this line should read
a dy + a ddy = x(dv + ddv) + v(dx + ddx) + (dx + ddx) (dv + ddv).
On account of this error and that noted above, there is not much profit in considering the remainder of this passage.
Translated from an article by Dr. Gerhardt in the Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1891, pp. 157-165, and published in The Monist for Oct., 1917. The notes are mine.
These highly important documents ought to be photographed and published in facsimile.
It seems a pity that Gerhardt has not given the contents of the section labeled “Mechanica,” unless indeed this is all non-mathematical; there may be in it some intimation that would lead to a clue as to the origin of Leibniz’s use of the word moment, meaning thereby, not Newton’s use of the word, but the idea now familiar to us in the determination of the center of gravity of an area, expressed by the equation
x = Σax/Σa,
where a is the element of the area distant x from the axis, x the distance of the center of gravity from that axis, and Σax is the sum of the ‘first moments of the elements’ or ‘the first moment of the whole area.’ See Note 18, below.
“Taugentes omnium figurarum. Figurarum geometricarum explicatio per motum puncti in moto lati.”
In a footnote, Gerhardt asserts that “Barrow’s Lectiones Geometricae appeared in 1672.” This is incorrect; for they were published, combined with the second edition of the Lectiones Opticce, in 1670; nor can Gerhardt be referring to the second edition, for that appeared in 1674 and then as a separate volume. Also, I have, in the little book on The Geometrical Lectures of Isaac Barrow, published by the Open Court Publishing Co., given reasons for supposing that these lectures were never delivered as Lucasian Lectures, though they may have formed the subject-matter for college lectures at Gresham and Trinity. Again, it is not true, although “well known,” that “the method of Barrow was only applicable to such curves as can be expressed by rational functions”; this remark is even only partially true about the differential triangle method; for, as I have shown in the above-mentioned book, Barrow had a complete calculus, which included, among other things, the important idea of substitu tion, which is all that is necessary to complete the “a-and-e” method and make it applicable to surds and fractions, and probably was thus applied by Barrow in working out his constructions; but the whole thing was geometrical, which apparently hid the inner meaning until recently.
To my mind, the mention of but “tangents and local motion” points out that, on Leibniz’s first reading of Barrow, he only perused at all carefully the first five lectures, which are relatively unimportant; or rather it confirms an opinion I had already expressed to Mr. P. E. B. Jourdain : see Note 43, p. 218.
“Locuti sunt mihi de phaenomeno quodam quod Barrovius fatetur se solvere non posse. Newtoni diflicultas soluta hactenus non est, P. Pardies manus dante.”
It seems however that Leibniz attended the meetings of the Royal Society ; at any rate once, when he exhibited the model of his calculating machine. It would be interesting if the roll of members present on all occasions during this period could be obtained, as doubtless they were kept. For such men as Ward were members at the time and attended the meetings, and Ward was, if not in the same class as the three whose names are given, an excellent mathematician ; and, Leibniz, being somewhat of a notable, on account of his connection with the Embassy from Mainz, would surely be introduced to all eminent members present.
The account given by Leibniz himself in the Historia (see above, Chapter III p. 36) reads thus: “He” [for Leibniz wrote in the third person, under the guise of “a friend who knew all about the matter”] “also came across Pell accidentally, and described to him certain of his own observations on numbers, and the latter stated that they were not new, but it had been recently made known by Nikolaus Mercator.... This made Leibniz get the work of Nikolaus Mercator.” As a matter of fact the suggested plagiarism, or what Leibniz took for such a suggestion, was from Mouton and not from Mercator. This is an instance of the lack of memory from which Leibniz suffered; such lack as caused him to make notes of all important points.
See Note 32, p. 171, on the introduction of the Gregorian calendar.
I cannot see what reason Gerhardt has for this statement, considering the contents of Barrow’s book, which we know that Leibniz had purchased; that is, unless we assume either that Leibniz, as I have suggested, did not at that time read the whole of Barrow, or failed to grasp what Barrow had given owing to his (Leibniz’s) incomplete knowledge of geometry.
Leibniz’s own date for the discovery of this result, usually alluded to by him as the “Arithmetical Tetragonism,” is 1674; “But in the year 1674 (so much it is possible to state definitely) he came upon the well-known Arithmetical Tetragonism;....” (see above, Chap. III, p. 42).
See the first critical note, pp. 172ff.
See the first critical note, pp. 172ff.
Observe that Leibniz (or Gerhardt) employs this word in a different sense from that of Barrow, with whom it means the special curve whose equation is y= (r − x) tan πx/2r, a curve that is particularly connected with the circle.
This contradicts both Gerhardt and Leibniz himself, who said that he got it from a consideration of a figure used by Pascal in finding the content of the sphere. See also the first critical note, pp. 172ff.
I will consider this influence in connection with an essay by Gerhardt on this very point in the following chapter, when I shall endeavor to substantiate an opinion I have formed with regard to the earlier manuscripts of Leibniz, which were discovered by Gerhardt, and of which translations are given above, on pp.59-114. I suggest that these do not represent so much the record of his original investigations as notes made while using the works of his predecessors as text-books.
I fail to see how this statement can be completely reconciled with the following well-known quotation from the “Lettre de A. Dettonville a Carcavy” (1658) :
“J‘ay voulu faire cét advertissement pour monstrer que tout ce qui est demonstré par les veritables regles des indivisibles se demonstrera aussi à la rigueur et à la maniere des anciens; et qu’ainsi l’une de ces Methodes ne differe de l’autre qu‘en la maniere de parler; ce qui ne peut blesser les personnes raissonnables quand on les a une fois avertyes de ce qu’on entend par là” (Vol. VIII, p. 352).
Pascal also says on p. 350: “. . . . la doctrine des indivisibles, laquelle ne peut estre rejettée par ceux qui pretendent avoir rang entre les Geometres.”
That is, the method of indivisibles does not differ from the method of exhaustions, except in the way the argument is put; and that the former must be accepted by any mathematician with pretensions to rank among geometers.
The page reference is to the edition of Pascal’s Works in 14 volumes, in the series, Les Grands Ecrivains de la France (pub. Hachette et Cie., Paris, 1914).
Pascal calls it “la balance.” It is worth noting in this connection that Pascal uses the ward “force” and not “moment” for the product of one of his weights and its lever-arm; so that we must look elsewhere for the clue to the use of the word “moment” in this sense by Leibniz.
Several of the problems proposed were solved by Huygens, de Sluse, and Wren; but by special methods, which did not satisfy Pascal, who called for a general method. Later (1670) Barrow gives the rectification of the arc, as a special case of a general theorem (Lect. XII, App. 3, Ex. 2, see my Barrow, p. 177).
See pp. 65ff.
See the second critical note, p. 179.
Leibniz, in the Acta Eruditorum for the year 1700, says, “I can affirm that, when in 1684 I published the elements of my Calculus, I did not know any thing more of Mr. Newton’s inventions in this kind, than what he formerly signified to me by his letters, viz., that he could find tangents without taking away surds;. . . .” As Newton says in the article in Phil. Trans., Vol. XXIX, No. 342, Anno 1714 (usually called the “Recensio”) this “is very extraordinary, and wants an explanation.”
This is feasible, but there is another alternative given by Dr. H. Sloman (The Claim of Leibniz to the Invention of the Differential Calculus, English edition, pub. Macmillan, 1860), which strikes me as even more probable. Sloman’s points are as follows: (1) It is highly probable that Leibniz’s week in London was the last week of that month. (2) Oldenburg had then in his possession two letters from Newton for Leibniz, dated Oct. 24 and 26; these he showed to Leibniz. (3) As Newton himself mentions, these were blotted and hastily written; and thus Leibniz asks, on this account, that Oldenburg should let him see the tract of Newton to which they refer; which tract Leibniz knew was in the possession of Oldenburg, that is, a copy of it. For the details of the argument, occupying ten quarto pages, see the above-mentioned book by Sloman, pp. 97-106.
The Latin, “Excerpsi ex Epist. Neutoni 20 Aug. 1672 ad Neuton,” as given by Gerhardt, seems somewhat unintelligible; especially the word Neuton. What Collins had (or what Oldenburg, as suggested by Sloman, had) was a copy of a manuscript that Newton had sent to Barrow. Gerhardt says, “so far as the script can be deciphered” ; perhaps the word Neuton is an error of transcription, or maybe an error on the part of Leibniz, due to the juxtaposition of the Neutoni which comes just before. In any case, Note 25 applies.
I do not think Gerhardt’s translation of the word excerpsi is correct.
Gerhardt does not state whether the extract is badly written (this would show that it had been done in a very great hurry, for Sloman says that Leibniz, in his matter for publication, wrote a beautiful hand), or whether spoilt by age; in the latter case, as old-time inks contained salts of iron, the manuscript might be restored by photography, by means of a special plate, that I understand is sometimes used for detecting forgeries in deeds and notes.
The letter was sent to Barrow to be sent on to Collins, probably with the object of being communicated through the latter to others; Collins seems to have been the regular channel of communication at this period, in a similar way to Mersenne.
So we find in a manuscript, dated July 11, 1677, first of all an allusion to Sluse’s method of tangents, “in which the equation is purged of irrational or fractional quantities” ; then the remark, “I have no doubt that the gentlemen I have just mentioned know the remedy that is necessary to apply”; then follows the rule for a quotient, and the remark that this will be sufficient for fractions; lastly the rule for powers, with the remark that this will be sufficient for irrationals. Later, he says, “This method has more advantage over all others that have been published than that of Slusius over all the rest, because it is one thing to give a simple abridgment of the calculation, and quite another thing to get rid of reductions and depressions.”
Thus, after the sight of Newton’s paper, his whole business has been to improve the method of Sluse.
I read it quite otherwise; he has had information of some kind, whether from Oldenburg direct or from Tschirnhaus, while in Paris, and visits London with the express intent of seeing the original papers.
See the third critical note, p. 181.
Could this possibly have had its rise in an effort on the part of Leibniz to understand fluxions, or rather the idea of fluxions as he had found it in Newton’s paper?
In 1582, Gregory XIII had directed 10 days to be suppressed from the calendar, then in accordance with the Julian system of intercalation, in order to allow the error which had crept into the time of the vernal equinox, by which Easter-day was settled, to be put right. The Gregorian calendar was introduced into all Catholic countries the same year, in Scotland in 1600, in the protestant states of Germany in 1700, but not in England until 1752. At the same time the commencement of the legal year in England was altered from May 25 to January 1; thus we frequently find two years given for dates between January 1 and May 25; while there are two days of the month given for all months of the year. For instance, February 1673 in the new Gregorian calendar would be only February 1672 in the Julian, distinguished by the letters O. S. (Old Style) ; and this date was written February 1672/3. Similarly the date November 18/28, 1676, was the 28th of November in the New Style, and the 18th in the Old Style, the number of the year being the same, since the day did not lie between the 1st of January and the 25th of May.
“Methodus Tangentium a Slusio publicata nondum rei fastigium tenet.” These are Leibniz’s words; Gerhardt omits to translate the word publicata, which probably refers to the publication in the Phil. Trans. of 1672, by Slusius, of the rules of his method, illustrated by examples. Sluse had probably improved upon this before 1676, but there is no evidence on this point. It would seem as if the subsequent work by Leibniz, culminating in the manuscript of July 11, 1677, was largely an attempt to perfect the rule of Sluse as a rule, and that Leibniz, if ever, did not appreciate the idea fundamental in the calculus, namely that of rates, until very much later.
[Translated from Dr. C. I. Gerhardt’s article, “Leibniz and Pascal,” in the Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1891 (Zweiter Halbband), pp. 1053-1068. My own notes are put in square brackets, to distinguish them from those given by Gerhardt.]
“When I speak of the geometry of indivisibles,” says Leibniz, “I intend something far more comprehensive than the geometry of Cavalieri, which does not appear to me to be anything but an insignificant (mediocris) part of the geometry of Archimedes.” [The general statement appears to me to be nearer the truth than that of Gerhardt, who lays unjustifiable stress on the above remark of Leibniz. I have endeavored to show later that there is strong probability that the work of Cavalieri, which Leibniz in the Historia acknowledges to have read, was the Exercitationes Sex, and not the Geometria that was published ten years earlier; perhaps he read them both.]
[It seems to me that those who claim merely the symbolism of the Calculus as an “invention” of Leibniz are really detractors from his genius. I have endeavored to show in the chapters previous to this, that this symbolism, more especially as regards the sign of differentiation, was a gradual adaptation and development of ideas already preconceived for finite differences, until Leibniz had obtained a standardized symbolism for the infinitesimal calculus. This, in my opinion, evidences an immensely greater intellect than that necessary for an “invention” ; even if we do take the standpoint that he was helped by the work of his immediate predecessors. Perhaps Gerhardt’s word Erfindung might be better rendered by “construction” instead of “invention” or “discovery.”]
[There was absolutely nothing in Pascal to suggest the sign or the rules for differentiation, and Leibniz might just as easily have obtained his ideas on integration from Galileo or others as from Pascal.]
[According to the generally accepted account, Leibniz was in London at the end of the third week of October, 1676, on his way home, via Amsterdam. At that time he could not differentiate a product.]
[A point therefore to be carefully noticed is that the figure given for the characteristic triangle is totally different from that given in the “Bernoulli postscript”; it is also different from the figure used by him in the manuscript dated Oct., 1674, which is undoubtedly derived from the figure used by Pascal in the opening lemma to the Traitté des Sinus du quart de Cercle (compare the figures given on pp. 62 and 15) ; it is different from either of the figures used in the manuscripts of Oct. 29, Nov. 11, 1675 (see above, Chapter IV, pp. 78, 83, 102), the last of these being like Barrow’s Differential Triangle, as used by him throughout his theorems on quadratures. Does this point to a new supposition: namely, that Leibniz originally invented a certain characteristic triangle of his own, essentially different in small detail from that of Pascal, Barrow, or any one else; that then he gradually passed from this to that of Pascal, later to Barrow’s form; that he found this the most convenient of all; finally, through lack of memory, he ascribes the earliest form to Pascal, instead of to himself, making an erroneous apperception of the time at which he had discovered this early form? The point is referred to in a later note (40).]
It went by the name of “Compagnie” ; out of it grew, in 1666, the “Académie des Sciences.”
[Gerhardt no doubt here refers to French mathematicians; but the first-mentioned names, of those that lived in Paris, with the exception of Roberval hardly bear comparison with those of the three who did not live there.]
The writings of Roberval and Pascal bearing reference to this have been mentioned in the essay “Leibniz in London.” [Omitted in Chapter VI.]
Nova Stereometria Doliorum Vinariorum, inprimis Austriaci, figurae ominium aptissimae, et Usus in eo Virgae Cubicae compendiosissimus et plane singularis. Accessit Epitome Stereometriae Archimedeae Supplementum. Lin-cii an. M DC XV. See my Geschichte der Mathematik in Deutschland, pp. 109ff.
Roberval in a letter to the astronomer Hevelke (Hevelius) in Dantzig, writes: “Concerning analysis, in which I delight, I have far more [theorems] ; and no fewer concerning the doctrine of the infinite, which they now call the ‘doctrine of indivisibles’....” Published in: Huygens et Roberval. Documents nouveaux. Par C. Henry; (Leyden, 1879).
[By ordinate and abscissa, Gerhardt means what Pascal calls the axis and base of the segment. Pascal only considered the whole solid of revolution, and the semi-solid, their volumes, their centers of mass, and the centroids of their surfaces; but those for solids generated by a quarter of a revolution could have been deduced quite easily.]
[Pascal, in effect, obtained the general formula
x = 
(mx)/(x),
where stands for either a summation of finite quantities, or for the equivalent of integration. If this is to be ascribed to Pascal as an original contribution, then we must assume that he had never seen Cavalieri’s Exercitationes Sex, Exer. quinta, Theorems 6, 7, 8, and certain others of the fifty propositions that form this section of the book; the section being entirely devoted to centers of gravity, while the method is a direct anticipation of Pascal’s.]
[What is generally known as the Arithmetical Triangle is not mentioned in the Lettres de Dettonville; see Note 19, p. 204.]
[It may be of interest to note that the pseudonym of Amos Dettonville is an anagram on Lovis de Montalte; Lovis, or Louis de Montalte being the pseudonym under which Pascal’s Lettres provincials appeared.]
Pascal published what he had written to de Carcavi along with the five essays in the following year, under the title of: Lettres de A. Dettonville, contenant quelques unes de ses Inventions de Geometrie. Scavoir, La Resolution de tous les problems, touchant la Roulette qu‘il avoit proposez publiquement au mois de Juin, 1658. L’Egalité entre les Lignes courbes de toutes sortes de Roulettes et des Lignes Elliptiques. L‘Egalité entre les Lignes Spirales et Paraboliques, demonstrée à la maniere des Anciens. La Dimension d’un Solide formé par le moyen d‘une Spirale autour d’un Cone. La Dimension et le Centre de Gravité des Triangles Cylindriques. La Dimension et le Centre de Gravité de l’Escalier. Un Traitté des Trilignes et leurs Onglets. Un Traitté des Sinus et des Arcs de Cercle. Un Traitté des Solides Circulaires. A Paris, M DC LIX. This writing contains the essays of Pascal of the year 1658 together with communications to Huygens, de Sluse, and an unnamed correspondent. From the correspondence of Huygens in the years 1658 and 1659, which is printed in that truly great work: Oeuvres Complètes de Christiaan Huygens publiées de la Société Hollandaise des Sciences, we see that a great movement arose among contemporary mathematicians through Pascal’s problems, as well as through the printed works that we have mentioned. Leibniz expresses himself thus: “By this time, the controversy [referring to Gregory St. Vincent] had cooled down; when lo! fresh movements in the realm of geometry are stirred up through the whole of France, by Blaise Pascal, a man of the highest genius, and one who at that time had come nearer to the reputation of Galileo and Descartes than any one else.”—This writing of Pascal was recommended for study to Leibniz by Huygens.
[As given by Pascal in his letter to de Carcavi, containing the particulars of his method for centers of gravity and the definitions of “trilignes” and “onglets,” the problems proposed in June were:
- To find the dimension and the center of gravity of the space CYZ.
- To find the dimension and the center of gravity of its semi-solid of rotation about the base ZY, i. e., the solid formed by the triligne CYZ when rotated about the base ZY through half a turn only.
- To find the dimension and the center of gravity of the solid of revolution about the axis CZ.
To which are added the three proposed in the Histoire de la Roulette at the commencement of October:
- To find the dimension and the center of gravity of the curved line CY.
- To find the dimension and the center of gravity of the surface of the semi-solid about the base.
- To find the dimension and the center of gravity of the surface of the semi-solid about the axis.]
By “Triligne” Pascal intends a plane figure bounded by two straight lines perpendicular to one another and a curved line. One of these perpendicular lines is called the axis and the other the base of the figure. If upon such a figure as a base there is erected a right solid, and this solid is cut by a plane which passes through the axis, or the base, then the portion of the solid that is cut off is called an “onglet.” A “double onglet” is obtained if, through the solid formed by production on the other side of the base, there is drawn a plane with the same inclination. [The last sentence does not make it clear that the second cutting plane also passes through the axis, or the base, as the case may be; nor that the plane is anticlinic and not parallel to the first plane; nor that Pascal took in general the inclination of the planes to the plane of the triligne to be 45°. I have therefore tried to represent the onglet and the double onglet in a diagram, see above.
ABC is the triligne, OABC is the onglet of (the axis or base) AB, and OBPCA is the double onglet of AB; the angles OAC, PAC are half right angles.]
By Sinus Pascal intends the ordinates multiplied by the indefinitely small portions of the arc. [This is a very misleading statement; for Pascal especially distinguishes between sines and ordinates, and thus makes a considerable advance over his contemporaries. He defines them at the same time for finite section and for infinitesimal section; the distinction is made perfectly obvious in a diagram if we use finite section, say, division into four equal parts, of the quadrant of a circle as a special case of a triligne. Now the sum of the sines or the ordinates are defined as the sum of the rectangles (for, as with all cases of indivisibles, that is what it comes to), formed by the sines or the ordinates respectively multiplied by the corresponding equal sectional parts. Thus, to speak of the sum of the sines as being the ordinates multiplied by the small portions of the arcs is quite wrong. Though only in rare cases is the space drawn, Pascal’s idea of the sum of the sines is that of the space formed by straightening the arc and erecting at each point of division the corresponding sine. Now, as Pascal says in Prop. 1 of the Traitté des Trilignes, the sum of the ordinates, which have to be applied to the base, makes the figure itself; while in Prop. 1 of the Traitté des Sinus du quart de Cercle, he shows that the sum of the sines (as a special case of the general theorem quoted in iii by Gerhardt supra, p. 534) of a quadrant is equal to the square on the radius. Thus, in modern notation,
The concluding paragraph of the Traitté des Solides Circulaires runs thus: “All these results arise from the fact that the straight lines OI are ordinates, that is to say that they are equally distant and proceed from equal divisions of the diameter; this brings it about that the simple sum of the ordinates is the same thing as the space intercepted between the extremes. But this is not true for the sines, since the distances between adjacent ones are not equal to one another, and thus the sum of the sines is not equal to the space intercepted between the extremes; there must be no mistaken idea on this point.” We find the same care taken by Barrow; but Tacquet breaks down in determining the surface of a cone through not understanding the necessity of this point, and in consequence condemns the method of indivisibles.]
[The effect is as Gerhardt states, but these sums are differently defined by Pascal in his letter to de Carcavi. The triangular sum of the numbers or magnitudes A, B, C, D, starting with A, (which should be stated), is the sum of all of them, plus the sum of all of them except the first, A, plus the sum of all except the first two, A and B, and so on; this is represented by Pascal as in the margin, and he goes on to show that this is equal to the first taken once, the second twice, and so on. Thus defined, the reason why they are named triangular numbers is obvious. The pyramidal sum is similarly defined as the triangular sum of all, plus the triangular sum of all except the first, plus the triangular sum of all except the first two, and so on. As if there were built up a pyramid having the first triangular sum as its bottom layer, the second triangular sum as the next layer, and so on; thus defined, the origin of the name pyramidal is obvious. Pascal then shows that this is the sum of the quantities taken respectively once, three times, six times, and so on, according to the sequence of the triangular numbers. Then using the property that twice a triangular number diminished by its ordinal number is equal to the square of that ordinal (i. e., n(n + 1)—n = n2), he also shows that, if two such pyramidal sums of quantities are taken, and from one of them the bottom layer is removed (i. e., the first triangular sum), then the sum of the two is equal to the sum of the quantities respectively multiplied in succession by the squares of the natural numbers. There is no connection between this and what is usually known as the Arithmetical Triangle of Pascal.]
[Pascal simply states the results, as deduced, not from the theorem quoted by Gerhardt, but (together with the theorem quoted) from the preliminary lemma that the radius is to the sine as the hypotenuse of the infinitesimal triangle is to its base: in modern notation, r: y = ds: dx, or r dx = y ds, where y is a sine and not an ordinate in Pascal’s sense. All the following theorems are particular cases of the formula ∫yn ds = r.∫yn-1dx.]
Descartes had spoken disparagingly about Pascal’s “Essay on the Conics.” Perhaps Pascal’s decided opposition to Descartes may be traced back to this. Pascal’s niece, Marguerite, writes: “M. Pascal used to speak very little about science; however, when the occasion for doing so occurred, he would state his opinion on those matters about which people were speaking to him. For example, with reference to the philosophy of Descartes, he merely said what he thought. He was of the same opinion as Descartes concerning automatism, but far from being so on the “subtle matter,” which he ridiculed. But he could not put up with his (Descartes’s) method of explaining the formation of the universe, and he often said: “I cannot pardon Descartes. In the whole of his philosophy, he would have been highly pleased to have dispensed with God; but he could not help making use of him to give a fillip to set the universe in motion. That being done, he had no further use for God.” (Fougére, Lettres, Opuscules et Mémoires de Madame Perier et de Jacqueliue, soeurs de Pascal, et de Marguerite Perier, sa nièce. Paris, 1845, p. 458). [It is more probable that Pascal used geometry, as Barrow did, because he both preferred it and thought it more rigorous than analysis. With regard to the remark on method, Gerhardt does not intend to convey the impression that Pascal abandoned for the more strictly geometrical method of moments the mechanical idea of the balance, with which he commences. By the way, to the best of my belief, the word “moment” is never used by Pascal.]
[I have gone carefully through the “Lettres of Dettonville” and I find no mention of Archimedes except in one place, namely, Prop. 1 of the Traitté des Solides Circulaires; and the whole of this is devoted to volumes of solids and their centers. Nor can I find any place where Pascal determines the surface of a sphere, at least not by reducing it to an equivalent plane figure, I have however shown that Barrow does do this (see above, Chapter III, p. 58). Surely Leibniz must be confusing the work of Pascal with that of Barrow on quadratures, the latter being so similar to the former in places that Barrow might easily be suspected of “borrowing” from Pascal; much more easily indeed than Leibniz could be so suspected with regard to either, in spite of his own assertion with regard to Pascal. See Notes 23, 24.]
[These are far more like Barrow’s results than those of Pascal; while the style is entirely Barrovian and quite different from that of Pascal.]
[there is no rectification of curves in Pascal; the whole of this sentence would however serve as a summary of the work of Barrow on rectification.]
[Gerhardt states that the Centrobaryc Method, as considered by Leibniz in the manuscripts dated October 25, 26, 29, and November 1, 1675, shows clear connection with the work of Pascal. He asserts that, from a consideration of Archimedes, Pascal was enabled to extend the method of the ancients ; he does not seem to be aware of what Cavalieri had done and published as the fifth section of his Exercitationes Sex ; or else, knowing all about this, he suppresses that knowledge for fear of discrediting the statements of Leibniz concerning the methods of Cavalieri.
The striking points about the work of Cavalieri in question are as follows. He opens by defining gravity as a property of a body, a descensive force. He then defines a heavy body as one possessing this property, and in a note on the definition, he adds that these must be taken to include surfaces, lines, and
points. Then he gives the definition of “moment” in its mechanical sense. “The moment of a weight is its endeavor to descend, no matter at what distance it is hung.” This is followed by the note: “Since this moment is different at different distances, as will be seen in what follows, it is to be understood from this that the same weight may have different moments.” He then defines uniform and uniformly variable (difformis) weights, such as a parallelogram in which the density varies as some power of the distance from one side; also he defines the centers of gravity and equilibrium. In Prop. 6 he shows that the moments of bodies are compounded of the ratio of their weights and the ratio of their distances. In Prop. 8 et seq., he combines the doctrine of indivisibles with that of moments to find the centers of gravity of surfaces, chiefly by means of “analogous figures” ; thus, a uniform triangle is analogous to a parallelogram whose “difformity is of the first species,” i. e., the density varies as the distance from one edge. He shows that, if the difformity is of the nth species, i. e., if the density varies as the nth power of the distance from the edge, then the medial line is divided by the center of gravity into parts in the ratio of 1 to n + 1, although it is stated rather differently, and only worked out for the first few values; then, using the idea of moments he proceeds from one degree to another in the case of the triangle, where the axis of moments (limes) is a parallel to the base through the vertex, and in the following proposition, the base itself; next the semicircle and the hemisphere are dealt with, whether uniform or varying as the distance from the center. In Prop. 36, he lays down the idea that the axis of moments may be outside the figure under consideration; and then proceeds to consider cylinders, cones, parabolic conoids, and the sphere, and truncated portions of them; and finally he finds the moment of a portion of a hyperbola about the asymptote which is not the base of the portion considered. It is interesting to note that Cavalieri, when speaking of the difformity of weight, uses the phrase “incrementum difforme gravitatis,” i. e., the word incrementum is employed to connote a gradual increase that follows a definite law. Also it is worthy of remark that he employs the notation, o. l., o. p., o. q., o. c., etc. for “all the lines,” “all the planes,” “all the squares,” “all the cubes,” etc.
From the above it will be seen that Cavalieri has given a fairly comprehensive account of the use of moments for the determination of the center of gravity; thus he not only gives far more than Pascal, but anticipates him. Leibniz’s matter is far more like that of Cavalieri than that of Pascal; though he seems to be reading Pascal at the time he wrote the third part of the “Analytical Quadrature,” by the method of moments, for the last figure in this manuscript (see above, Chapter IV, p. 89), with the explanatory diagram that I have added on the right of it, is strongly reminiscent of the idea of the onglet of Pascal; although it may have arisen from Cavalieri’s work. The great point about this batch of manuscripts of October and November, 1675, is that nearly every figure has the tangent drawn to the curve; now the tangents are never drawn or used either by Cavalieri or by Pascal. A secondary consideration, but still one of importance, is that the subject-matter of these manuscripts is like nothing in Cavalieri or Pascal, as far as the “center of gravity method” is concerned. As we find Pascal’s Infinitesimal Triangle idea in the figure of Leibniz’s manuscript of October, 1674, I take it that this was the time at which he finished reading his Pascal. Hence, I imagine that in October, 1675, he had got a good knowledge of Descartes’s algebraical geometry, and began to study Cavalieri’s Exercitationes Sex; he did not get very far in this before he appreciated the power given by the method of moments; then, probably wearied by Cavalieri’s prolix demonstrations, he laid the book aside, and applied Cartesian analysis to the method of moments, running the idea for all it was worth. If this is the case, these manuscripts represent real original research, and are not study notes like some of the others.]
[The misreadings of Gerhardt, as given in his Geschichte der höheren Analysis (see above, Chapter IV, p. 65) are uncorrected even in 1891, the date of this essay, thirty-six years after the publication of the Geschichte! We should have “AB 
DC 
x” and “AB (= x)’—see the figure on the right (above) which is mine, while that on the left is the one that Gerhardt gives as that of Leibniz; again Gerhardt’s “id est ac in omn.x, sive a(cb2/2),” which makes Leibniz write nonsense, should be “id est a c in omn.x, sive a cb2/2,” the “a” being the preposition “away from” and not the length of a line; thus corrected we not only have a sensible reading but the whole paragraph is correct; I have made the correction when translating. Also with regard to Gerhardt’s statement that Leibniz starts from an alternative rendering of Prop. 2 of Pascal’s Traitté des Trilignes, it is worthy of remark that Pascal’s figure is altogether different from that of Leibniz; and this is only natural, because there is no similarity between the theorems, nor is there any relation between the methods of proof. Pascal’s proof is equivalent to the modern method of a change in the independent variable by a conversion to a double integral followed by a change in the order of integration, and is geometrical; that of Leibniz is equivalent to integration by parts, and is merely an example of the theorem of moments.
Thus (Pascal),
∫yx dx = ∫(∫x dx) dy = ∫½x2dy,
and
(Leibniz), fyx dx= [½x2y]—∫½x2dy;
where Pascal’s integrals are taken over the same area as one another, and those of Leibniz over complementary areas. It seems therefore ridiculous to say that “Leibniz commences with Prop. 2.... which he expresses as follows.”]
[This, means the result obtained geometrically by means of the triangle AZC, in the passage to which Note 23 refers.]
[The connection between number, ratio, and infinitesimal is peculiar.]
[Note the word “useful” (utile) : the “long s” is introduced merely as a convenient abbreviation in accordance with Leibniz’s usual idea of obtaining simplification by means of symbols.]
[I have discussed this fully in my translation of Gerhardt’s essay, “Leibniz in London” (see above, Chapter VI, p. 180). I have shown there that at least it is highly probable that the d in x/d stands for a certain length, namely the subtangent.]
[Note that, in spite of Gerhardt’s opening remarks about the algorithm of the calculus being due to reading Pascal, the symbols of integration and differentiation have not been mentioned in anything quoted by Gerhardt in this essay, except in the paragraph just above.]
[See critical notes on this point, Chapter VI, pp. 172-184. I believe some of those who read what is there given will, while giving Leibniz full credit for the introduction and development of the symbols ∫ and d, that made the calculus of Leibniz the powerful instrument it was, still find it hard if not impossible to agree with Gerhardt in his assertion that the ideas of Leibniz were not very strongly influenced by the best points of every single author that he studied, and more especially by the Lectiones Geometricae of Barrow and the Exercitationes Sex of Cavalieri.]
[So far I have failed to find any information as to the error into which Reginaldus fell; he does not appear to be mentioned by either Cantor or Zeuthen.]
[The Geometry of Cavalieri is indeed practically all quadratures; but Torricelli himself says (quoted by Tommaso Bonaventura in his preface to an edition of the Lezione Accademiche, 1715), in his preface to a Tract on Proportion, that he has used indivisibles for tangents as well as for quadratures; Roberval, through his own efforts at concealing his methods, we know comparatively little about; but the germ of Fermat’s method is the same as that of Barrow’s, namely the Differential Triangle; lastly it is probable that Huygens’s knowledge was considerably more than he let anybody know (and so too with Gregory)—cf. Leibniz’s words, “suppressing their analysis,” a few lines later. It is to be observed that Leibniz deliberately speaks of the mathematicians of France and Italy only; “at the present time,” 1679, he must have been aware that Barrow had complete geometrical knowledge, at any rate, of all the matters in question.]
[The Horologium was published in March or April, 1673, and the presentation of a copy to Leibniz was undoubtedly made after his return from his first visit to London (Cantor says that the dedication was dated March 25, 1673; see Cantor, III, p. 138). Hence, the date at which Leibniz obtained the Characteristic Triangle can be assigned to some time at least not later than the beginning of October, 1673; and therefore the inclusion of this in the manuscript dated Aug., 1673 (see above, Chapter IV, p. 59), marks the exact date of its discovery.]
[In the “Bernoulli postscript” (see p. 14) Leibniz states that he “sought a Dettonville from Buotius, a Gregory St. Vincent from the Royal Library, and started to study geometry in earnest.” In the Historia (see p. 37) Leibniz says that “in order to obtain an insight into the geometry of quadratures, he consulted the Synopsis Geometriae of Honoratus Fabri, Gregory St. Vincent, and a little book by Dettonville (Pascal).” In his letter to the Marquis de l‘Hospital he says, “At the start I only knew the indivisibles of Cavalieri, and the ‘ductions’ of Father Gregory St. Vincent, along with the ‘Synopsis of Geometry’ of Father Fabri” (see below, p. 220). I suggest that the correct explanation of these inconsistencies is that he did get the Dettonville from Huygens as stated here, the St. Vincent from the Royal Library, and the work that he obtained from Buotius was the Exercitationes Sex of Cavalieri.]
[I think the passage throws considerable light on the character of these manuscripts, besides explaining how it was that Leibniz seems to have taken a very long time to study the works of the authors mentioned. I look on these manuscripts, not as “study notes” merely, nor yet as true “research,” but as a mixture of each. I suggest that there is quite enough evidence to make it safe to assert that the characteristic of Leibniz’s method of study was to read a very small portion of an author at a time, then to break off and follow out the train of ideas suggested to him by the passage to the furthest limit, before proceeding further with his reading; thus he is led to his own original developments. For instance, note in the next sentence how he says he “tried to find a new sort of center.” This is very characteristic; he is not satisfied with merely acquiring knowledge, even at this early stage, but at once seeks to utilize each point, as he grasps it, to obtain something new, something original previously undiscovered. Cf. the study notes on the work of Pascal, given below under III.]
[That is, a “homothetic center.”]
[As I have been unable to find the word “moment” defined, or even mentioned, in any place except in the Exercitationes Sex of Cavalieri, I suggest that this is fairly good circumstantial evidence for the reading of this work by Leibniz before he discovered the theorem in question.]
[Observe that this is not the figure used in the manuscript of October, 1674 (see above, Chapter IV, p. 62), the latter being a diagram that one would naturally expect him to have obtained from the figure in the lemma that commences Pascal’s Traitté des Sinus du quart de Cercle (cf. Note 6, p. 196) ; but is a figure such as one would expect Leibniz to abstract from those given by Barrow, either from Lect. XII, prop. 1, 2, 3, or from Lect. XI, prop. 1 (see Chapter IV, p. 58, and Chapter I, p. 16, respectively. In the latter especially we have the right-angled triangle used by Leibniz on page 39, quoted by Gerhardt in the article translated in the present number). I therefore suggest that Leibniz worked at Barrow and Pascal conjointly, and applied Descartes’s analysis to their geometrical theorems. If this is not the case, Leibniz was at fault, for Pascal was discussing sines and not ordinates (see Note 18, p. 202) ; i. e., Pascal was integrating with regard to θ and not with regard to x. Observe also that the figure as given is not correct; the rectangle should be that having AC, CD as adjacent sides.]
[Note that the area is taken to be produced by the assemblage of lines applied in order, in the true Cavalierian style.]
[Query: urged thereto by a question on the part of Huygens, as to whether Leibniz could now find the properties of the auxiliary curve (see p. 18).]
[This fits in perfectly with my suggestion that Leibniz attacked Barrow’s Lectiones at several different times. Having, as I think, taken Barrow’s advice given in the preface, he sampled the first few propositions of each lecture, and obtained from those of Lect. XI and XII his Characteristic Triangle. This could I think have been definitely settled if Gerhardt had only given the figure used by Leibniz in the manuscript dated August, 1673. Assuming for the time being that my suggestion is correct and that Leibniz is merely confusing the author that he read at this time, I suggest that characteristically he broke off his reading of Barrow, pursued the idea he had obtained, and made out those theorems on quadratures that he speaks of; this so improved his geometry that later he was able to read Barrow thoroughly and appreciate all that was in it, and to find that his theorems had been anticipated. I also suggest that it was on this second or third reading that he came across the theorem that led to his Arithmetical Tetragonism. A fresh reference to Barrow to find if there were any other ideas that he could develop, considerablv later, having already found him a mine of information, would then probably be the occasion on which the marginal notes in his own notation were inserted by Leibniz.]
[Leibniz seems to have got these men in true perspective, Cavalieri, Fermat, Gregory, and Barrow, as far as the infinitesimal calculus is concerned. But I doubt whether he, even after he came to his fullest appreciation of Barrow’s geometrical theorems, or indeed any other person except Bernoulli, ever appreciated the real inwardness of these theorems, or that Barrow’s tangent problems could be used, in the manner I have shown in the appendix to my Barrow, to draw a tangent to any curve given by an equation in either Cartesian or polar coordinates.]
[This I take to mean the principle that differentiation and integration are inverse operations; for it is practically certain that in November, 1675, he could not differentiate a product; otherwise, as previously argued, he would have verified his solution of the unfortunate equation, x + y2/2d = a2/y, which he gives as
(y2 + x2) (a2—yx) = 
,
by differentiation, as he did with a previous solution that did not contain a product.]
[From this probably arose the first germ of the idea of the Quadratrix, in the sense used by Leibniz.]
[Substitute Barrow and Mercator in conjunction, and we have a feasible suggestion for explaining the first method of proof for the Arithmetical Quadrature of the Circle; the method that Leibniz does not seem ever to have divulged.]
[It is impossible for me to conjecture exactly which of his ideas is here referred to by Leibniz; for he calls a mere method by the name of “a calculus,” and what we should call a dodge for some particular kind of example by the name of “a method.” I think it may be possible that the “transmutation of figures” is referred to.]
[Notice that Leibniz says that he has not derived any help from Barrow for his methods (je n’ay tiré aucun secours pour mes methodes). This is less even than he might have said with perfect truth; for the methods of Barrow would have been a veritable hindrance to Leibniz’s analytical development. Even when using the Differential Triangle method, and literals for the lengths of his lines, the whole of the working is geometrical in the examples of the method given by Barrow, and not analytical.]
[See Notes 35, 36.]
[Perhaps this is meant to include Barrow.]
[Notice the words “by chance” (par hasard); these seem to point to a conclusion that Leibniz read the Pascal in a very desultory manner; this conclusion gets corroborated by the extract given by Gerhardt under the heading III. It is worthy of remark that the “by chance,” or “incidentally” (as I have rendered Leibniz’s word forte in the letter to Tschirnhaus), is made to refer to Pascal. “Forte Pascalius demonstrabat,” etc., i. e., “Incidentally Pascal was proving,” etc. I think it may be asserted that Pascal missed absolutely nothing that was pertinent to his purpose; whereas Barrow certainly missed the opportunity of being the discoverer of the series for the inverse tangent, and thereby the quadrature of the circle, by not applying Mercator’s method of division and integration to the result of one of his examples of the Differential Triangle method; as also after giving the method of “transmutation of figures” he missed those things to which it led.]
[In a manuscript dated October, 1674 (see above, Chapter IV, p. 61), Leibniz is using x and y for the variable ordinate and abscissa; while in a manuscript dated August, 1673, he considers “the classification of curves laid down by Descartes.” In this manuscript, according to Gerhardt, Leibniz has already constructed the “characteristic triangle,” but Gerhardt does not give the particular variant that Leibniz uses in this manuscript. I believe that this will prove to be of the Barrow type, when reference can be made to the original; for the title of the manuscript is strongly suggestive of Barrow, being: Methodus nova investigandi Tangentes. . . .ex datis applicatis, etc.; and Pascal’s work does not mention tangents.]
[That is, as the Characteristic Triangle, leading to integrations, is ascribed to the influence of the work of Pascal, so the Differential Calculus is ascribed to the influence of the work of Descartes. Is this the diplomatic characteristic in Leibniz peeping out? He is writing to a Frenchman, and attributes his work to the respective influences of two Frenchmen. Note that Leibniz goes on to state that the source of inspiration was summation of series by differences, suggesting the origin of the symbol dx.]
[In the manuscripts that we have had under consideration, Leibniz does not appear to have made any practical use of the Quadratrix.]
[It is precisely this point which formed the really great improvement in the reckoning section of the infinitesimal calculus. It is just this improvement that is due to Leibniz in analysis, and to Barrow in geometry; although Leibniz did not accomplish anything of the kind until 1676 or 1677. Newton’s method by means of series for fractions and roots does not bear comparison, let alone the futility of ascribing Leibniz’s method to a perusal of Newton’s work.]
[All that is any good in the following is to be found in Pascal; I think this corroborates the suggestion I have made as to Leibniz’s way when studying a book. It looks here as if he had read about twenty pages of Pascal, and about the same number of pages of Cavalieri’s section on centers of gravity; moved thereto probably or possibly by Pascal’s remark “. . . .the principle of indivisibles, which cannot be rejected by any one having pretensions to rank as a geometer.” Then he proceeds to work out his own combination of the two ideas, without bothering to see what else either of these authors had to say on the matter.]
[Leibniz tacitly assumes that all the points are occupied; this is necessary for the success of the notion of triangular sums.]
[Something very like this is indeed investigated fairly thoroughly in a manuscript dated October 25, 1675 (see above, Chapter IV, p. 65). Hence these extracts from Pascal were certainly made before that time, though probably not long before.]
[This is the rendering for “productum fieri aequale” ; he probably means that what is produced on the one side, i. e., the sum of the moments on one side of A, should be equal to the sum of the moments on the other side. But this endeavor to obtain something new seems rather futile.]
[It would have been interesting to have seen what this simple rule was. Probably nothing more than the propositions given by Pascal as Prop. 1, 2, 3 of his method of the balance; this would corroborate my suggestion that Leibniz did not study Pascal very steadily or thoroughly (cf. Notes 37, 43, 52, 57 on pp. 216, 218, 220, 223 respectively).]