LEIBNIZ’S PAPER on the catenarycurve, was written at the instigation ofJacques Bernoulli, for the ActaEruditorum of Leipzig, June 1691.Following the example of BlaisePascal, who had initiated, in 1658, a contest for the construction of thecycloid, Gottfried Leibniz alsoprovoked the geometers of his time, bychallenging them to submit, at thefixed date of mid-1691, their geometricmethod for the construction of thecatenary curve. Leibniz later providedthe answer, followed by Jean Bernoulliand Christian Huyghens.

The two following papers are ahistorical account of the origin of thestudy of this transcendental curve, and,at the same time, the first physical-geometric construction showing thespecies-relationship between thecatenary and the logarithmic curves, as two companioncurves; one arithmetic, the other geometric. (All of thedifferentials of the catenary curve, are arithmetic means ofcorresponding differentials of the logarithmic curve; and, allof the differentials of the logarithmic curve, are geometricmeans of the catenary.)

This discovery of Leibniz, which was based on thequadrature of the hyperbola, is a beautiful example of themethod of PROPORTIONALITY AND SELF-SIMILARITY, whichhas been the hallmark of Platonic physical-geometry from thefirst applications of the Thales Theorem, to the laterconstructions of Carnot, Monge, and Poncelet, at the EcolePolytechnique. In a letter to Huyghens, Leibniz added this

insight concerning his discovery: “I have reduced everything tologarithms, not only becauseeverything is generated in a verysimple and natural way (so much so that the catenary curve seems to have been created for the purposeof generating logarithms), but alsobecause they make possible, bymeans of ordinary geometry, thediscovery of an infinity of realpoints, all constructible from asingle constant proportionapplicable in all situations.” In theActa of 1691, Leibniz emphasizedthat, with his work on the catenary,he was able to determine “the bestof all possible constructions for thetranscendentals.”

The search for a mathematicsthat would be the “least

inadequate” for describing the physical phenomena ofelliptical pathways of the planets, had been initiated byJohannes Kepler, but had remained incomplete. Leibniz,drawing upon work by Huyghens, Fermat, and the Bernoullibrothers, undertook to resume that unfinished agenda, whichwas premised on the Platonic assumption that the generativeprinciple in the universe was not only well-orderedproportionately, but also required a calculus (differential andintegral) for transcendental curves, whose physical conditionsare subjected to non-constant changes in curvature. This wasin direct opposition to, and conflict with, the straight-line,action-at-a-distance (“push me-pull me”) treatment of theproblem of gravitation, and of the pathway of light,

54

Two Papers on the Catenary CurveAnd Logarithmic Curve

(Acta Eruditorum, 1691)

G.W. Leibniz

TRANS LATIO N

G.W. Leibniz

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Click here for Full Issue of Fidelio Volume 10, Number 1, Spring 2001

© 2001 Schiller Institute, Inc. All Rights Reserved. Reproduction in whole or in part without permission strictly prohibited.

elaborated by Descartes, Newton, and their followers.Although Leibniz often makes a statement like, “given a

certain property, find the curve,” the discovery of principlethat Leibniz developed in the papers published in the Acta,and especially in his calculus of the catenary curve, was notaimed at the discovery of curves, per se. It was aimed at thediscovery of the “INTENTION,” or “PURPOSE,” of the curve.There are two levels at which this principle of discoveryapplies: one is the level of the integral, and the other, the levelof the differential.

From the higher standpoint of the integral, the purpose, orfinal causality of the curve, is a transfinite relative to thedifferential, incorporating within itself an ever-increasingdensity of singularities. And, as a transfinite, its purposeresides, ultimately, in the increase of the power of mankindover nature, with the intention of demonstrating the principleof sufficient reason in the best of all possible worlds. From thestandpoint of the differential, on the other hand, the intentionof the curve is to follow a non-linear direction whichexpresses the least-action pathway at every infinitesimally

small increment of action, as exemplified by the least-timecurvature of the pathway of light developed by PierreFermat, Christian Huyghens, and Jean Bernoulli, in theirdiscoveries of the non-linear curvature of light in thechanging density of a medium of refraction.

Indeed, light knows the least-action pathway to take,because it follows, according to a non-entropic law ofphysical space-time, a PROPORTIONAL ORDERING PRINCIPLE

which is coherent with a least-pathway and least-timemotion. It is this PROPORTIONAL ORDERING PRINCIPLE

which expresses the relationship between the differential andthe integral, between the evolute and the involute, andbetween the catenary curve and the logarithmic curve.

The following two articles have been translated from theFrench text, “G.W Leibniz: La naissance du calculdifferentiel, 26 articles des Acta Eruditorum. Introduction,traduction, et notes par Marc Parmentier” (Paris: Vrin, 1989),in consultation with the Latin original as it appears in, “G.WLeibniz: Mathematische Schriften,” ed. by C.I. Gerhardt(Hildesheim: Georg Olms Verlagsbuchhandlung, 1962).

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1. The String Whose Curve Is Described by Bending Under Its OwnWeight, and the Remarkable Resources That Can Be DiscoveredFrom It by However Many Proportional Means and Logarithmsfrom Acta Eruditorum, Leipzig, June 1691

The problem of the catenary curve,1 or funicularcurve, is interesting for two reasons: First, it fur-ther extends the science of discovery, in other

words the science of Analysis, which up to now has beenincapable of tackling such questions; second, it extendsthe progress of construction techniques. In point of fact, Ihave come to realize that the resourcefulness of this curveis only equal to the simplicity of its construction, whichmakes it the primary one among all the transcendentalcurves.

This curve can be constructed, and traced very simply,by a physical type of construction, that is, by suspending astring, or better, a small chain (of variable length). And, assoon as you can determine its curve, you can discover allof the proportional means, and all of the logarithms thatyou wish to find, as well as the quadrature of the hyper-bola. Galileo was the first who tried, without success, todiscover its nature; he mistakenly conjectured that it wasa parabola. Joachim Jung, the renowned philosopher andmathematician of this century, who, well before Descartes,had many enlightened ideas for the reform of the sci-ences, experimented with it, made some calculations, and

came up with the proof that it was not a parabola; butwithout coming to the solution for the real curve.

Since then, many people have tried to solve the prob-lem, but without success, until a very learned scientistrecently gave me the opportunity to deal with it. In fact,the well-known Bernoulli, after having successfully testeddifferent cases of curves with the Analysis of the Infinitewhich I had contributed with my differential calculus,asked me publicly, in the Acta of last May (p. 218ff), if Iwould examine the problem of the catenary curve, andsee if, with our calculus, I could come up with a determi-nation of the curve. After having graciously accepted todo the experiment, I have not only succeeded, unless I ammistaken, in becoming the first to solve this famous prob-lem, but, I have also found some remarkable applicationsfor this curve; which is why, following the example ofBlaise Pascal, I invited mathematicians to discover, forthemselves, the solution to this problem, by challengingtheir methods, to see if others could eventually find otherways to the solution, different from the one Bernoulli andI have used.

Only two people made it known that they had suc-

ceeded within the given time period, that is, ChristianHuyghens—unnecessary to stress the merit of his greatcontributions to the Republic of Letters—and the other,Bernoulli himself, in collaboration with his youngerbrother, whose intellect finds no equal but his own erudi-tion; Bernoulli’s contribution demonstrates that no futurediscovery from him, no matter how brilliant, should sur-prise us. I therefore judge that he has in fact proven, as Iannounced it, that our method of calculus does extend tothis curve, and that it further opens the way to solvingproblems which have up until now been considered for-midable. However, it is up to me to reveal my own

results; others can show later the results of their ownsolutions.

Here is a geometric construction for the curve, withoutthe use of a string, and without using any chain, andwithout any assumption of a quadrature; a constructionwhich should be considered the most perfect method forgenerating all the transcendental curves, and the mostappropriate for the purpose of Analysis. Given two seg-ments that have between them a determined invariableratio, represented here by D and K, as soon as you knowthe ratio of these two segments, the rest of the solution isderived by simple application of ordinary geometry.

56

F H L

3C

C

p 1C 1(C)

1(j)

(j)

3(j)

P

E

G

A

Q

1J3J J M

3N N 1N O 1(N) (N) 3(N)

1B

B(C)

3B3(C)

R

mu

tb

v

1j

3jj

c

K D

Catenary

curve

Logar ithmic curve

FIGURE 1. The catenary curve and logarithmic curve.

57

FIGURE 1. The Catenary Curve andLogarithmic CurveGiven an indefinite straight line ON parallel to the hori-zon, given also OA, a perpendicular segment equal toO3N, and on top of 3N, a vertical segment 3N3j, whichhas with OA the ratio of D to K, find the proportionalmean 1N1j (between OA and 3N3j); then, between 1N1jand 3N3j; then, in turn, find the proportional meanbetween 1N1j and OA; as we go on looking for secondproportional means in this way, and from them thirdproportionals, follow the curve 3j-1j-A-1(j)-3(j) in sucha way that when you take the equal intervals 3N1N,1NO, O1(N), 1(N)3(N), etc., the ordinates 3N3j, 1N1j,OA, 1(N)1(j), 3(N)3(j), are in a continuous geometricprogression, touching the curve I usually identify as loga-rithmic. So, by taking ON and O(N) as equal, elevate overN and (N) the segments NC and (N)(C) equal to thesemi-sum of Nj and (N)(j), such that C and (C) will betwo points of the catenary curve FCA(C)L, on which youcan determine geometrically as many points as you wish.

Conversely, if the catenary curve is physically construct-ed, by suspending a string, or a chain, you can constructfrom it as many proportional means as you wish, andfind the logarithms of numbers, or the numbers of loga-rithms. If you are looking for the logarithm of numberOv, that is to say, the logarithm of the ratio between OAand Ov, the one of OA (which I choose as the unit, andwhich I will also call parameter) being considered equal tozero, you must take the third proportional Oc from Ovand OA; then, choose the abscissa as the semi-sum of OBfrom Ov and Oc, the corresponding ordinate BC or ONon the catenary will be the sought-for logarithm corre-sponding to the proposed number. And reciprocally, if thelogarithm ON is given, you must take the double of thevertical segment NC dropped from the catenary, and cutit into two segments whose proportional mean should beequal to OA, which is the given unity (it is child’s play);the two segments will be the sought-for numbers, one larg-er, the other smaller, than 1, corresponding to the proposedlogarithm.

Another method: After you have found NC, as I havesaid, take OR (the R point being taken from the horizon-tal AR, such that OR is equal to OB or NC), the sum andthe difference of segments OR and AR will be the twonumbers, the one larger, the other smaller, than 1, corre-sponding to the given logarithm. Indeed, the differencebetween OR and AR is equal to Nj, and their sum to(N)(j); just as OR and AR are, in turn, the half-sum andthe semi-difference between (N)(j) and Nj .2

Here is the solution to the main problems usually posedfor a given curve. To draw the tangent at a given point C.

On the horizontal straight line AR, going throughsummit A, take R such that OR is equal to OB which isknown, the straight line CT which is anti-parallel to OR(cutting the axis OA at point T) will be the tangent we arelooking for. In short, I call here anti-parallel, the straightlines OR and TC, which make with the parallels AR andBC, the angles ARO and BCT, not equal angles, but com-plementary angles. The right triangles OAR and CBT arethus similar triangles.3

Find the Segment Equal to an Arc of theCatenary CurveIf you draw a circle with center O, and radius OB, cuttingthe horizontal straight line going through A and R, ARwill be equal to the given arc AC. We also see from whatprecedes, that cv will be equal to the portion of the curveCA(C). If that portion were twice the value of the para-meter, that is to say, if AC or AR were equal to OA, itsinclination on the horizon at point C, in other words theangle BCT, would be 45 degrees, and the angle CT(C)would consequently be a right angle.

Find the Quadrature of the Area between theCatenary Curve and One or More StraightLines

After having found point R, as we did above, rectangleOAR will be equal to the area of the quadriline AONCA.The quadrature of any other sector can be derived in thesame way. We can also find that the arcs are proportionalto the areas of the quadrilines.

Find the Center of Gravity of the CatenaryCurve, or of a Portion of That CatenaryAfter having established the fourth proportional Ou ofthe arc AC, in other words AR, of the ordinate BC and ofthe parameter OA, let us add to it the abscissa OB; thenthe half-sum OG will generate the center of gravity G ofthe catenary CA(C). Furthermore, by taking the intersec-tion E of the tangent TC with the horizontal straight linegoing through A, and by completing the rectangle GAEP,P will be the center of gravity of arc AC. The center ofgravity of any other arc C1C will be at the distance AMfrom the axis, pM being the perpendicular segment tothe horizontal line going through the summit, takenfrom the intersection point p of the tangents Cp and1Cp; but we can also get it from the centers of gravity ofthe arcs AC and A1C. We can further deduce BG, corre-sponding to the lowest possible position of the center of

58

Iwas thrilled to discover in my reading of them, theconcordance between three solutions to the probleminitiated by Galileo and revived by M. Bernoulli; it is

a guarantee of exactitude which will convince those whodo not go into the details of such questions. Therefore,even if there is no opportunity to compare them, here,point by point, their agreement on the fundamentals is

obvious. The three of us have established the law of tan-gency, as well as the rectification of the catenary. Idemonstrated, a long time ago, in the Acta of June 1686(p. 489), (by means of a new type of contact which I havecalled “osculation”) how to measure the curvature of acurve by using the radius of its osculating circle; that is,among all of the tangent circles, the one which is the

gravity of a string, of a chain, or of any other flexible butnon-extensible line, of the given length cv, suspendedfrom points C and (C). For any other figure other thanthe curve CA(C) which I am now interested in, the centerof gravity will be further up.

Find the Center of Gravity of the AreaBetween the Catenary Curve and One orMany Straight Lines

Take the half Ob of OG, and then complete the rectanglebAEQ; Q will be the center of gravity of the quadrilineAONCA. We can easily deduce from this the center ofgravity of any other figure taken between the catenarycurve and one or many straight lines. The remarkableresult is that not only the quadriline figures like AONCAare proportional to the arcs AC, as I have already noted it,but the distances between their centers of gravity and thehorizontal straight line going through O, that is OG andOb, are proportional, the first always being double of thesecond; as for their distance to axis OB, that is PG andQb, their proportionality is purely and simply equality.

Find the Volume and Surface of SolidsGenerated by Rotation Around Any FixedStraight Line Delimited by the CatenaryCurve and One or Many Straight LinesAs one can see, this result is gotten from the two preced-ing problems. If the catenary curve CA(C) rotates aroundaxis AB, the area generated will be equal to the circlewhose radius is the root of the double rectangle EAR. Wecan also discover the value of other surfaces and volumesby the same method.

Because I wished to be brief, I omit here a number of

theorems and problems which are already implicit inwhat I have just elaborated, and which can easily bederived from it. Given, for example, two points C and 1Cof a catenary curve, and given p the intersection of thetangents at these points, draw from points 1C, p, C, thesegments 1C1J, pM, CJ, perpendicular to the horizontalstraight line AEE going through the summit, then weshall have

(1JJ 3 AC) - (1CC 3 1JM) = 1BB 3 OA.

This could also be an opportunity for introducing infi-nite series. For example, parameter OA being consideredas unity, establish the notation a for arc AC, the segmentAR, and y as the ordinate BC; we shall get:

, etc.,

a series which can be established from a simple rule. Bymaking use of what we have just said, we can furtherdeduce the rest from the characteristic elements of thecurve. For example, by considering as known the summitA, another point C, and the length AR of arc AC, whichlimits it, it is possible to get the parameter AO of thecurve, that is in substance point O: in fact, since B is alsoknown, let us trace BR and then draw segment Rm, suchthat angle BRm is equal to angle BRA. Under such condi-tions, the straight line Rm (which you have extended) willcut the axis BA (extended) to the desired point O.3

I think what I have said includes the essential, and willpermit anyone to deduce everything that needs to be stat-ed about this curve. I am excluding myself from the taskof going through the demonstrations, in order to avoidunnecessary prolixity, and, moreover, because they wouldbe self-evident to anyone who has understood the calcu-lus that I have just explained, and which forms the basisof our new Analysis.

2. Solutions to the Problem of the Catenary, or Funicular Curve,Proposed by M. Jacques Bernoulli in the Acta of June 1691Acta Eruditorum, Leipzig, September, 1691

59

closest to the curve, and whichforms, with the curve itself, thesmallest possible angle of contact;the famous Huyghens (whilenoticing that the centers of thosecircles are always located on thecurves that he was the first toinvent; that is, evolutes whosedevelopment generate involutes)took the idea of applying my the-ory to this curve, and looked forthe radius of curvature of thecatenary, that is, its osculating cir-cle, and in doing so, he discoveredits evolute; this curve is alsoshown in the solution of theBernoullis.5

Furthermore, the Huyghenssolution also gives the distancebetween the center of gravity andthe axis of the catenary; the solu-tion of the Bernoullis, along with mine, not only givesthe distance to the axis, but also to the basis, and to anyother straight line; thus permitting to locate that centerpoint as well as the quadrature of the area encompassedby the catenary. To this, I have even added to my solu-tion the center of gravity of this last figure, that is, of itsarea. M. Huyghens gives the construction of the curve bysupposing the following quadrature: xxyy = a4 - aayy,while M. Jean Bernoulli, and myself, have related thecatenary to the quadrature of the hyperbola; this lastone makes an absolutely judicious use of the quadratureof a parabolic curve, while for my part, I have reducedeverything to logarithms; I have determined in this waythe type of expression, as well as the best of all possible con-structions, for transcendentals. Indeed, all you need toknow is a unique constant proportion, which willenable you to discover an infinity of points, using onlyordinary geometry, and without any more need ofquadrature or rectification. One might enjoy noticing,in my construction, this singular and elegant concor-dance between the catenary and logarithms. Further-more, M. Huyghens (giving us the hope of a considerablesimplification with the use of a Table of Sines), madethe observation to the effect that the problem could alsobe reduced to a sum of secants, uniformly growing byminimal increments. I had made the same remark inthe past, and since I can still recall that it was also fromsuch increments that one could determine the rhombicor loxodromic curve for the purpose of navigation, sucha curve, which I remember having established a numberof years back by means of logarithms, I have dug out

my old draft papers which I havefinally published in the Acta Eru-ditorum of last April (p. 181).6

So, it turns out that thefamous Basle professor, M.Jacques Bernoulli, who had pre-cisely put the problem of thecatenary back on the agenda, hasalso forwarded a study on loxo-dromic curves, plus many re-markable discoveries, includingthe solution to this problemfound by his brother last June (p. 282), and where he showed aconstruction of the loxodromiccurve in which he used the rulesof my calculus with respect to thequadrature of the curve of abscis-sa z, and of ordinate y, followingmy differential equation:

When he finds out how I have reduced the problem tothe quadrature of the hyperbola, that is, to logarithms, hewill admit, I think, that this brings the final touch to thisinvestigation, and that all that remains to be done is tofacilitate practical applications, and bring this discoverymore to the reach of everyone.

I have to point out, here, that certain errors, which Ihave made in the construction of the rhombic curve that Ipublished last April, must be corrected. In point of fact:p. 181, line 12, 1L2L must be replaced by 1L3L; line 25,1d3L by 2d3L; and, p. 182, line 20, the ratio

must be replaced by the ratio

, etc.,

These are things that the context would have obvious-ly reestablished.

I find that M. Jacques Bernoulli has developed some-thing very elegant in January (p. 16 of the Acta), on theequality of certain portions of dissimilar curves. As forthe length of the finite curve, while describing aninfinity of loops, in the Acta of June (p. 283), it is notindeterminate since it is equal to a finite curve, and wecan follow it by a uniform movement in a finite time. I

Christian Huyghens

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60

refer on this point to what he hashimself declared in January (p.21), that one cannot obtain the(general) rectification of anyclosed geometric curve. I knowthat another great man also triedto proved the impossibility ofdetermining the indefinite areaquadrature of any closed geomet-ric curve; however it became evi-dent to M. Huyghens, as well as tomyself, that the question was farfrom resolved. And, unless I ammistaken, there exist counter-examples to which, nonetheless,the same reasoning can beapplied. I hope the author willnot be offended by this remark,which is inspired only by the loveof truth and not by any spirit ofcontradiction, because it does notdiminish in any way the great merits of his otherresults.

My character leads me to personally celebrate whole-heartedly, and with real pleasure, the men who haveacquired, or will acquire great merit in participating inthe Republic of Letters, because I think this is the mostjustified price that must be given for their works, andwhich can constitute for them, as for others, an incen-tive for the future. I cannot hide the immense joybrought me by the work erected by the famous Bernoul-li, with his younger and very ingenuous brother, basedon the new calculus that I have initiated; more especial-ly, as I had not yet met anybody who had made use of it,with the exception of the very quick-witted Scotsman,John Craig.

But, thanks to their brilliant inventions, I hope to seeextended into the works of the mind, the use of thismethod which to my view, as well as to their ownadmission, is extremely rich in possibilities. There is nodoubt that with this method, Mathematical Analysisshall be brought to its perfection, and that the problemsof transcendentals, which up to now have been excluded,should come under its purview. So, M. Bernoulli hasmade this profound remark, which is, that at eachinflection point, the proportion between t and y, that isto say, between dx and dy, takes the greatest or thesmallest value that can be assigned. In all eventuality, Ihave no doubt that he will uncover some results whicheven I do not suspect myself; because there still remainmany points which I am not able to concern myselfwith, and on which I am not able to pronounce myself

conclusively with the necessaryprecision.

Just as the works of Pascal andHuyghens gave me the opportuni-ty to make discoveries throughthese kinds of reflections, andfrom which I gradually achievedsome results, which would havebeen difficult to attribute to suchworks directly; similarly, it seemsto me that all that I have accom-plished will give rise to more pro-foundly hidden discoveries thatothers will make. So, I sincerelythank the famous Bernoulli forhaving formulated the problemsrelated to the catenary, and tocontinue to do so, in cases wherethe catenary is of variable thick-ness, where the string is extensi-ble, or where the heavy string is

replaced by an elastic band, or, finally, the case of thecurve formed by a sail in the wind. I only wish I had thefree time to debate these questions with him, but respon-sibilities of a totally different nature forbid me entirely todo so, and so, it is with difficulty that I have been able torecently find the time to put together and finalize thesolution to the problem which he asked me to solve morethan a year ago.

Finally, since he attempted to imagine (p. 290) the cir-cumstances that led me to these ideas, and which works Ihad been using to help me, I insist on revealing to himmy sources in all honesty. Advanced geometry was a totalstranger to me until I met Christian Huyghens, in Paris, in1672, and to whom I publicly acknowledge in this article,as I did in personal letters, I owe the most, after Galileoand Descartes. After having read his Horologium Oscilla-torium, as well as the Letters of Dettonville (that is, Pascal),and the works of Gregoire de Saint Vincent, I acquiredsuddenly from them a great light, quite unexpected onmy part, and also for that of those who knew I was anovice in this domain. I was very open to these results,and I soon began to give a few outlines on them. This ishow a considerable number of theorems appeared to mespontaneously, and which were only corollaries of a newmethod.

I later found a few, among others, from Jacques Grego-ry and Isaac Barrow. But I noticed that their origins werenot sufficiently clear, and that a more profound thingneeded to be discovered, which was not thought possiblebefore; that is, that the most elevated part of geometrycould one day be submitted to Analysis. I have revealed

Jean Bernoulli

The

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certain elements of this, a fewyears ago, more for public interestthan for personal glory, and maybeit would have been a better serviceto keep my name out of it. How-ever, I prefer to see that my seedsgrow and bear fruit also in thegardens of others. Even thoughmy hands were tied, and I couldnot busy myself with this as Ishould have, there was a higherdomain for which new avenuesneeded to be opened; so, this iswhat was important in my eyes:That is, the case of developingmethods is always more crucial,than particular problems, althoughit is the latter which usually bringapplause.

In conclusion, I will only addone thing, even if it is not on thissubject. I would like M. Bernoulli to consent to examineclosely the article on the measurement of forces, which Iopposed to M. Papin, especially near the end, where Ithink I have noticed the origin of the common error. He

was right, last July (p. 321), tounderscore the fact that no ele-ment of a force disappears with-out reappearing somewhere else;but force and quantity of motionare two different things; andaside from the fact that the morean obstacle is hard, the less thepotential is dissipated, it isabsolutely certain that the smallimpediments can be diminishedin any given proportion, and thatthe resistances from rubbing, thatis to say, owing to friction, are notproportional to the speed (as Iindicated in my Schediasma deresistentia). Even though thereexists resistance of the medium,nothing forbids us to imagineoscillations in empty space, free ofair, or in a medium as thin as you

want; finally, we must free the human mind from arbi-trary contingencies, in order to bring out the underlyingnature of the thing itself.

—translated by Pierre Beaudry

61

Jacques Bernoulli

TRANSLATOR’S NOTES1. The identification of the hanging chain by the name “catenary”

was established by Christian Huyghens, in a letter to Leibniz, dat-ed November 18, 1690.

2. The reader should note that the proportional means developed byLeibniz correspond to the arithmetic and the geometric means,and that the descriptive expression “semi-sum” signifies the arith-metic mean. Leibniz obtained the proportionality between thetwo curves by using his divider as a differential calculator, to gen-erate those two means. He calculated that, for any two segments,say NC and (N)(C), taken vertically under the catenary curve,which are equal to OB, and are equally situated on each side ofthe central axis, he could find their geometric mean AR by gener-ating a circle whose radius and arithmetic mean is OB. The short-er segment Nj, under the logarithmic curve, will be derived bysubtracting the geometric mean AR from the arithmetic mean OBof that circle. The longer segment (N)(j), under the logarithmiccurve, will be gotten by adding the geometric mean AR to thearithmetic mean OB. Thus, the logarithmic curve is the geomet-ric mean of the catenary curve, and the catenary curve is thearithmetic mean of the logarithmic curve.

3. This method of finding the tangent to a curve, without the curveitself, is one of the most profound discoveries of Leibniz. It wasHuyghens who initiated the method of discovering a curve by theproperty of its tangents; that is, discovering the evolute at theintersection of two perpendiculars generated from its involute.Here, Leibniz applies a similar property of tangents, which is torelate the tangent at right angle to its anti-parallel. Generally,Leibniz treats the problem of inversion of tangents, from the van-tage point of the intention of the differentials oriented towardtheir final cause.

4. Note that the shapes of the two curves are not only variables ofeach other, but their curvature will also be subject to variation bychanging the ratio of K to D. At the limit, and following Leibniz’sprinciple of continuity, if the ratio of K to D were to become 1:1,then both curves would be transformed into a curves of zero curva-ture; that is, a single, horizontal straight line. The ratio of K and Dchosen by Leibniz in this construction is 3:1.

5. A note on osculation. The reason why the notion of osculation is soimportant, is that it involves directly the application of the Par-menides Paradox. This is because the very idea of discovering anosculating circle to a given curve, leads you to the discovery of theevolute of that curve, as well as to an infinity of similar curves of thesame family. In other words, the discovery of the evolute, impliesthe discovery of a One of a Many.

6. Leibniz and the construction of the sine curve. According to theActa Eruditorum of 1694, Leibniz developed a construction for thesine curve as derived from the circle, using the Roberval method oftransferring the sine of the circle along the sine curve of a cycloid,and in so doing, he was able to determine the quadrature, that is, hewas able to construct the entirety of an area perfectly equivalent toa quarter of a circle.

On the one hand, such a true definite of quadrature is uniquelypossible, only when you treat the sines of such a quadrature as indivisi-bles, as an actual completed infinite sum; that is, an infinite which isdetermined in such a way that between two infinitesimals of thatsum, there is no possibility of inserting a third. However, on the otherhand, an indefinite quadrature could never have a completed infinitesum, and therefore, one cannot add infinitesimals to such an indefi-nite sum, nor can one reduce their indefinite totality to zero: nothingfinite can be added to, or subtracted from, that which is infinite.

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