Involution equidiftant curve aßyde, lying without Abcdef. From this it appears that By is the complete evolutrix of FEDCB, while bd cef is the evolutrix of that arch, and the added tangent B b. In like manner, the lapped up thread ADF, with the added part F, defcribes the evolutrix♪ y' B' A'.

Abedin A, and have the fame tangent; it therefore Involution. does not deviate from it, otherwise their tangents would feparate, and would not both be at right angles with the lines touching the evolute. They must therefore coincide throughout.

8. The arches bed and y, intercepted by the fame radii Bb and Dd, may be called concentric; and the angles contained between the tangents drawn thro' their extremities are equal. Thus the angle is equal to /po: but although equidiftant, parallel, and con. taining the fame angle between their tangents and between their radii, they are not fimilar. Thus, the arch a 6 has a curvature at a that is the fame with that of any circle whofe radius is equal to A ; but the cur

6. Hence it follows, that a circle defcribed round any point of the evolute, with a radius reaching to any point of the evolutrix, will cut the evolutrix in that point, and be wholly within it on the fide remote from the vertex, and without it on the fide next the vertex.

7. The evolved radius cuts every arch of the evolutrix perpendicularly, or a right line drawn through the interfection at right angles touches the evolutrix in that point. Through any point d draw the line mdt at right angles to d D. The part of it md next to the vertex is wholly without the curve, because it is with out the circle defcribed round the centre D; and this circle is without the evolutrix on that fide of d which is next the vertex (6). Any point on the other fide of d is alfo without the curve. For lette E be another evolved radius, cutting D din n: then nd is less than nt, because n dt is a right angle by conftruction; and - therefore nt dis acute. But because En + n D are greater than ED, En+nd are greater than ED+ Dd, that is, than Ee, and nd is greater than ne. Therefore, fince it is lefs than nt, it follows that ne is much lefs than nt, and lies without the curve. Therefore the whole line mdt is without the curve, except in the point d. It therefore touches the curve in d, and the radius Dd cuts it at right angles in that point. By the fame reasoning, it is demonftrated, that all the curves Abdf, a ßs f, A'b'd f', a'''', are cut perpendicularly by the tangents to the evolute. Alfo all these curves interfect the evolute at right angles in their vertexes.

It follows from this propofition, that from every point, fuch as s, or i, or o, &c. in the space AOF comprehended by the evolute and its extreme tangents AO, FO, two perpendiculars may be drawn to the evolu. trix Adf; and that from any point in the space with in the angle A of only one perpendicular can be drawn; and that no perpendicular can be drawn from any point on the other fide of ADF. Apollonius had obferved thefe circumftances in the conic fections, but had not thought of marking the boundary formed by the evolute ADF. Had he noticed this, he would certainly have discovered the whole theory of evolution, and its importance in fpeculative geometry.

It alfo follows from this propofition, that if a curve Abedef is cut by the tangents of ABCDEF at right angles in every point, it will be defcribed by the evolution of that curve: For if the evolutrix, whofe vertex is A, be really described, it will coincide with SUPPL. VOL. II. Part I.

The fame may be faid of the curvatures at 6 and at B. 9. If a circle udz be defcribed round the centre D with the radius D d, it both touches and cuts the evolutrix in the point d, and no circle can be described touching the curve in that point, and paffing between it and the circle udz: For fince it touches the curve in d, its centre must be somewhere in the line d D perpendicular to mdt. It cannot be in any point n more remote from d than D is; for it would pafs without the arch du, and be more remote than du from the arch de of the evolutrix. On the other fide, it would indeed pass without the arch dz, which lies within the arch de of the evolutrix: but it would also pass without the curve. For it has been already demonstrated (7) that nd is greater than ne; and the curve would lie between it and the circle d z.

Thus it appears, that a circle defcribed with the evolved radius approaches nearer to the curve, or touches it more closely, than any other circle; all other circles either interfect it in measurable angles, or are within or without the curve on both fides of the point of contact. This circle udz has therefore the fame curvature with the curve in the point of contact and coalef cence. It is the EQUICURVE CIRCLE, the circle of equal curvature, the osCULATING CIRCLE (a name given it by Leibnitz). The evolved radius of the evolute is the RADIUS OF CURVATURE of the evolutrix, and the point of the evolute is the CENTRE OF CURVATURE at the point of contact with the evolutrix. The evolute is the geometrical locus of all the centres of curvature of the evolutrix.

This is the most important circumftance of the whole doctrine of the involution and evolution of curve lines. It is affumed as a felf-evident truth by the precipitant writers of elements. It is indeed very like truth: For the extremity of the thread is a momentary radius during the process of evolution; and any minute arch of the evolute nearer the vertex must be conceived as more incurvated than the arch at the point of contact, because defcri. bed with fhorter radii: for the fame reason, all beyond the contact must be lefs incurvated, by reason of the greater radii. The curvature at the contact must be neither greater nor lefs than that of the circle. we thought it better to follow the example of Huyghens, and to establish this leading propofition on the Atricteft geometrical reafoning, acknowledging the fingular obligation which mathematicians are under to him for giving them fo palpable a method of fixing their notions on this fubject. When the evolute of a curve is given, we have not only a clear view of the genefis of

C

But

the

Involution. meters, the interfections q and p will be in a line A qp, which makes half a right angle with the axis AP. Therefore, when AQ is greater than AP, the point q is without the conical parabola, and the whole arch of the cubical parabola cut off by the ordinate DB is alfo without it; but when AQ is lefs than AP, q is within the conical parabola, as is alfo the arch q B. Therefore the remaining arch BEA is without it, and is therefore lefs incurvated at A. An endless number of conical parabolas of smaller curvature may be drawn by enlarging AP; yet there will ftill be an arch AEB of the cubical parabola which is without it, and therefore lefs incurvated. Therefore the curvature of a cubical parabola is lefs than that of any conical parabola: It is faid to be infinitely lefs, because an infinity of cubical parabolas of fmaller curvature than AEB may be drawn by enlarging AQ.

It may be demonftrated in the fame manner, that a paraboloid, whofe ordinates are in the fubbiquadrate ratio of the abfciffe, has an infinitely fmaller curvature at the vertex than the cubical parabola. And the curvature of the paraboloid of the next degree is infinitely lefs than this; and fo on continually. Nay, Sir Ifaac Newton, who firft took notice of this remarkable circumftance, demonftrates the fame thing of an endless fucceffion of paraboloids interposed between any two de. grees of this series. Neque novit (fays he) natura limi

tem.

If this be the cafe, all curves cannot be defcribed by evolution; for we have no conception of a radius of curvature that is greater than a line without limit. The theory of curvilineal motions delivered in the article DYNAMICS must be imperfect, or there must be curve lines which bodies cannot defcribe by any powers of nature. The theory there delivered profeffes to teach how a body can be made to defcribe the cubical para bola, and many other curves which have these infinitefimal curvatures; and yet its demonftrations employ the radius of curvature, and cannot proceed without it. We profefs ourselves obliged to an attentive reader (who has not favoured us with his name) for making this obfervation. It merits attention.

There must be fome paralogifm or misconception in all this language of the mathematicians. It does not neceffarily follow from the arch AEB lying without the arch AIB, that it is lefs incurvated at A; it may be more incurvated between A and B. Accordingly we fee, that the tangent BT of the conical parabola is lefs inclined to the common tangent AV than the tangent B of the cubical parabola is; and therefore the Hexure of the whole arch AEB is greater than that of the whole arch AIB; and we shall fee afterwards, that there is a part of AEB that is more incurvated than any part of AIB. There is nothing correfponding to this unmeaning and inconceivable fucceffion of feriefes of magnitudes of one kind, each of which contains an endlefs variety of individuals, and the greateft of one feries infinitely lefs than the smalleft of the next, &c. ; there is nothing like this demonftrated by all our arguments. In none of thefe do we ever treat of the curvature at A, but of a curvature which is not at A. At A we have none of the lines which are indifpenfably neceffary for the demonftration. Befides, in the very fame man. ner that we can defcribe a cubical parabola, and prove that it has an arch lying without the conical parabola,

com.

we can defcribe a circle, and demonftrate that it has al- Involution fo an arch lying without the parabola. These infinitefimal curvatures, therefore, are not warranted by our arguments, nor does it yet appear that there are curves which cannot be defcribed by evolution. We are always puzzled when we fpeak of infinites and infinitefimals as of fomething precife and determinate; whereas the very denomination precludes all determination. We take the distinguishing circumftance of thofe different orders for a thing clearly understood; for we build much on the diftinction. We conceive the curvature of the cubical parabola as verging on that of the mon parabola, and the one feries of curvatures as be ginning where the other ends. But Newton has fhewn, that between these two serieses an endless number of fimilar feriefes may be interpofed. The very names gi ven to the curvature at the extremities of the hyperbolic evolutrix have no conceptions annexed to them. At the vertex of the hyperbola there is no line, and at the interfection with the affymptote there is no curvature. These unguarded expreffions, therefore, fhould not make us doubt whether all curves may be described by evolution. If a line be incurvated, it is not ftraight. If fo, two perpendiculars to it muft diverge on one fide, and muft converge and meet on the other in fome point. This point will lie between two other points, in which the two perpendiculars touch that curve by the evolution, of which the given arch of the curve may be defcribed. Finally (which fhould decide the question), we fhall fee by and bye, that the cubic, and all higher orders of paraboloids, may be fo defcribed by evolution from curves having affymptotic branches of determinable forms.

Such are the general affections of lines generated by evolution. They are not, properly fpeaking, peculiar properties; for the evolutrixes may be any curve lines whatever. They only ferve to mark the mutual relations of the evolutes with their evolutrixes, and enable us to conftruct the one, and to discover its properties by means of our knowledge of the other. We proceed to fhew how the properties of the evolutrix may be determined by our knowledge of the evolute.

This problem will not long occupy attention, being much limited by the conditions. One of the first is, that the length of the thread evolved must be known in every pofition: Therefore the length of the evolved arch muft, in like manner, be known; and this, not only in toto, but every portion of it. Now this is not univerfally, or even generally the cafe. The length of a circular, parabolic, hyperbolic, arch has not yet been determined by any finite equation, or geometrical conftruction. Therefore their evolutrixes cannot be determined otherwife than by approximation, or by comparifon with other magnitudes equally undetermined. Yet it fometimes happens, that a curve is discovered to evolve into another of known properties, although we have not previously discovered the length of the evolved arch. Such a discovery evidently brings along with it the rectification of the evolute. Of this we have an inftance in the very evolution which gave occafion to the whole of this doctrine; namely, that of the cycloid; which we shall therefore take as our firft example.

Let ABC (fig. 5.) be a cycloid, of which AD is the axis, and AHD the generating circle, and AG a tangent to the cycloid at A, and equal to DC. Let

C 2

BKE

Involution, ting Eo in v; alfo draw bbi parallel to the bafe, and Bd perpendicular to it. If both curves be now referred to the fame axis CGF, it is plain that Bb, Bd, and db are ultimately as the fluxions of the arch, abfcifs, and ordinate of the evolute, and that Ee, ev, and མ v E, are ultimately as the fluxions of the arch, abfciffa, and ordinate of the evolutrix. Alfo the two fluxionary triangles are fimilar, the fides of the one being perpendicular, refpectively, to those of the other. If both are referred to one axis, or to parallel axes, the fluxion of the abfciffa of the evolute is to that of its ordinate, as the fluxion of the ordinate of the evolutrix is to that of its abfciffa. Thus, from the fluxionary equation of the one, that of the other may be obtained. In the prefent cafe, they may be referred to AD and FG, making CG equal to the cycloidal arch CBA. Call this a; AI, x; IB, y; and AB, or EB, z. In like manner, let Ft beu, E = v, and FE =w; then, becaufe DH' = DA' — AH', and DA and AH are the Al

halves of CF and BE, we have DH' =

4 But DI =

Alfo w =

a u

น ( √2au =√Fa: √ u = √GF: ✔Ft, which is the analogy competent to a cycloid whofe axis is GF DA.

It is not neceffary to infift longer on this in this place; because all these things will come more naturally before us when we are employed in deducing the evolute from

its evolutrix.

When the ordinates of a curve converge to a centre, in which cafe it is called a radiated curve, it is moft convenient to confider its evolutrix in the fame way, conceiving the ordinates of both as infifting on the circumference of a circle defcribed round the fame centre. Spirals evolve into other spirals, and exhibit several proproperties which afford agreeable occupation to the curious geometer. The equiangular, logarithmic, or loxodromic spiral, is a very remarkable example. Like the cycloid, it evolves into another equal and fimilar equiangular spiral, and is itself the evolutrix of a third. This is evident on the flightest inspection. Let Crap (fig. 6.) be an equiangular fpiral, of which S is the centre; if a radius SC be drawn to any point C, and another radius SP be drawn at right angles to it, the intercepted tangent CP is known to be equal to the whole length of the interior revolutions of the spiral, though infinite in number. If the thread CP be now unlapped from the arch Crg, it is plain that the firft mo. tion of the point P is in a direction PT, which is perpendicular to PC, and therefore cuts the radius PS in an angle SPT, equal to the angle SCP; and, fince this is the cafe in every pofition of the point, it is manifeft that its path must be a spiral PQR, cutting the radii in the fame angle as the spiral Crap. James Bernoulli firft difcovered this remarkable proper ty. He also remarked, that if line PH be drawn from every point of the fpiral, making an angle with

the tangent equal to that made by the radius (like an angle of reflection correfponding with the incident ray SP), thofe reflected rays would all be tangents to another fimilar and equal fpiral Iv H; fo that PH PS. S and H are conjugate foci of an infinitely flender pencil; and therefore the fpiral I v H is the cauftic by reflec tion of RQP for rays flowing from S. If another equal and fimilar fpiral xvy roll on Iv H, its centre z will defcribe the fame foiral in another pofition wuz. All these things flow from the principles of evolution alone: and Mr Bernoulli traces, with great ingenuity, the connection and dependence of cauftics, both by reflection and refraction, of cycloidal, and all curves of provolution, and their origin in evolution or involution. A variety of fuch repetitions of this curve (and many other fingular properties), made him call it the SPIRA MIRABILIS. He defired that it fhould be engraved on his tombftone, with the infcription EADEM MUTATA RESURGO, as expreffive of the refurrection of the dead. See his two excellent differtations in Aa. Erudit. 1692, March and May.

Another remarkable property of this fpiral is, that if, instead of the thread evolving from the fpiral, the fpiral evolve from the ftraight line PC, the centre S will defcribe the ftraight line PS. Of this we have an example in the apparatus exhibited in courses of experimental philofophy, in which a double cone defcends, by rolling along two rulers inclined in an angle to each other (fee Gravefande's Nat. Phil. I. fic). It is pretty remarkable, that a rolling motion, feemingly round C, as a momentary centre, fhould produce a motion in the ftraight line SP; and it fhews the inconclufiveness of the reafoning, by which many compilers of elements of geometry profefs to demonftrate, that the motion of the defcribing point S is perpendicular to the momentary radius. For here, although this feeming momentary radius may be shorter than any line that can be named, the real radius of curvature is longer than any line that can be named.

But it is not merely an object of fpeculative' gcometric curiofity to mark the intimate relation between the genefis of curves by evolution and provolution; it may be applied to important purposes both in fcience and in art. Mr M Laurin has given a very inviting example of this in his account of the Newtonian philofophy; where he exhibits the moon's path in abfolute fpace, and from this propofes to investigate the deflecting forces, and vice verfa. We have examples of it in the arts, in the formation of the pallets of pendulums, the teeth of wheels, and a remarkable one in Meffis Watt and Boulton's ingenious contrivance for producing the rectilineal motion of a pifton rod by the combination of circular motions. M. de la Hire, of the Academy of Sciences at Paris, has been at great pains to fhew how all motions of evolution may be converted into motions of provolution, in a memoir in 1706. But he would have done a real fervice, if, instead of this ingenious whim, he had fhewn how all motions of provolution may be traced up to the evolution which is equivalent to them. For there is no organic genefis of a curvilineal motion fo fimple as the evolution of a thread from a curve. It is the primitive genefis of a circle; and it is in evolution alone that any curvilineal motion is comparable with circular motion. A given curve line is an individual, and therefore its primitive organical genefis muft also be individual. This is ftrictly true of evolution.

A para

bola