the interior and lower side, or curve, of the arch of a bridge, &c. In contradistinction from the extrados, or exterior curve, or line on the upper side of the arch. See ARCH in this Suppl.

INVOLUTION and EVOLUTION, are terms introduced into geometry by the celebrated Mr Huyghens, to express a particular manner of describing curvilinear spaces which occurred to him when occupied in the improvement of his noble invention of pendulum clocks. Although he was even astonished at the accuracy of their motion, and they soon superceded all balance clocks, he knew that the wide vibrations were somewhat slower than the narrow ones, and that a circle was not sufficiently inclosed at the sides to render all the vibrations isochronous. The proper curve for this purpose became an interesting object. By a most accurate investigation of the motions of heavy bodies in curved paths, he discovered that the cycloid was the line required. Lord Brouncker had discovered the same thing, as also Dr Wallis. But we do not imagine that Huyghens knew of this; at any rate, he has the full claim to the discovery of the way of making a pendulum oscillate in a cycloidal arch. It easily occurred to him, that if the thread by which the pendulum hangs be suspended between two curved cheeks, it would alternately lap on each of them in its vibrations, and would thus be raised out of the circle which it describes when suspended from a point. But the difficulty was to find the proper form of those cheeks. Mr Huyghens was a most excellent geometer, and was possessed of methods unknown to others, by which he got over almost every difficulty. In the present case there was fortunately no difficulty, the means of solution offering themselves almost without thought. He almost immediately discovered that the curve in question was the same cycloid. That is, he found, that while a thread unwinds from an arch of a cycloid, beginning at the vertex, its extremity describes the complementary arch of an equal cycloid.

Thus he added to this curve, already so remarkable for its geometrical properties, another no less curious, and infinitely exceeding all the others in importance.

The steps by which this property was discovered are such direct emanations from general principles, that they immediately excited the mind of Mr Huyghens, which delighted in geometry, to prosecute this method of describing or transforming curve lines by evolution. It is surprising that it had not ere this time occurred to the ancient geometers of the last century, and particularly to Dr Barrow, who seems to have raked his fancy for almost every kind of motion by which curve lines can be generated. Evolution of a thread from a curve is a much more obvious and conceivable genesis than that of the cycloid invented by Mercennus, or that of the conchoid by Nicomedes, or those of the conic sections by Vieta. But except some vague expressions by Ptolemy and Gaffendus, about describing spirals... Involvion by a thread unslipped from a cylinder, we do not recollect any thing of the kind among the writings of the mathematicians; and it is to Huyghens alone that we are indebted for this very beautiful and important branch of geometry. It well deserves both of these epithets. The theorems which constitute the doctrines of evolution are remarkable for their perspicuity and neatness. Nothing has so much contributed to give us clear notions of a very delicate subject of mathematical difficulty, namely curvature, and the measure and variations of curvature. It had become the subject of very keen debate; and the notions entertained of it were by no means distinct. But nothing can give such a precise conception of the difference of curvature, in the different parts of a cycloid or other curve, as the beholding its description by a radius continually varying in length. This doctrine is peculiarly valuable to the speculator in the higher mechanics. The intensity of a deflecting force is estimated by the curvature which it induces on any rectilineal motion; and the variations of this intensity, which is the characteristic of the force, or what we call its nature, is inferred from the variations of this curvature. The evolution and involution of curve lines have therefore great claim to our attention. But a Work like ours can only propose to exhibit an outline of the subject; and we must refer our readers to those eminent authors who have treated it in detail.

Varignon, in the Memoirs of the French Academy for 1726, has been at immense pains to present it in every form; James Bernoulli has also treated the subject in a very general and systematic manner. Some account is given of it in every treatise of fluxions. We recommend the original work of Mr Huyghens in particular; and do not hesitate to say, that it is the finest specimen (of its extent) of physico-mathematical discussion that ever has appeared. Huyghens was the most elegant of all modern geometers; and both in the geometrical and physical part of this work, De Horologio Oscillatorio, he has preserved the utmost rigour of demonstration, without taking one step in which Euclid or Apollonius would not have followed him.

Such authors form the taste of the young mathematician, and help to preserve him from the almost mechanical procedure of the expert symbolical analyst, who arrives at his conclusion without knowing how he gets thither, or having any notions at all of the magnitudes of which he is treating.

There are two principal problems in this doctrine.

I. To ascertain the nature of the figure generated by the evolution of a given curve.

II. To determine the nature of the curve by whose evolution a given curve may be generated.—We shall consider each of these in order, and then take the opportunity which this subject gives of explaining a little the abstruse nature of curvature, and its measures and variations, and take notice of the opinions of mathematicians about the precise nature of the angle of contact.

The curve line ABCDEF (fig. 1.) may be considered as the edge of a crooked ruler or mould; a thread may be supposed attached to it at F, and then lapped along it from F to A. If the thread be now led away from A, keeping it always tight, it is plain that the extremity A must describe a curve line Abcdef, and involution, that the detached parts of the thread will always be tangents to the curve ABCDEF. In like manner will the curve line F'd'e'b'A' be described by keeping the thread fast at A, and unlapping it from the other end of the mould.

This process was called by Mr Huyghens the Evolution of the curve ADF. ADF is called the Evolute. Adf was named by him the Curve by Evolution. It has been since more briefly termed the Evolutrix, or unlapper. It has also been called the Involute; because, by performing the process in the opposite direction f'd'A, the thread is lapped up on the mould, and the whole space ADFf'd'A is folded up like a fan. The detached parts Cc, Dd, or C'e, D'd', &c. of the thread, are called Radii of the Evolute; perhaps with some impropriety, because they rather resemble the momentary radii of the evolutrix. We may name them the Evolved Radii. The beginning A of evolution may be considered as the vertex of the curves, and the ends F and f may be called the Terms.

There is another way in which this description of curve lines may be conceived. Instead of a thread Ff gradually lapped up on the mould, we may conceive Ff to be a straight edged ruler applied to the mould, and gradually rolled along it without sliding, so as to touch it in succession in all its points. It is evident, that by this process the point f will describe the curve f'd'A, while the point F describes the other curve F'd'a'.

This way of conceiving it gives a great extension to the doctrine, and homologates it with that genus of curve lines by which cycloids of all kinds are described, and which we may distinguish by the name of Provolution. For it is plain, that the relative motions of the points A and b are the same, whether the ruler BB' roll on the mould ABF, or the mould roll on the ruler; but there will be a great difference in the form of the line traced by the describing point, if we suppose the plane on which it is traced to be attached to the rolling figure. Thus, when a circle rolls on a straight line, a point in its circumference traces a cycloid on the plane attached to the straight line, while the point of the straight line which quitted the circle describes on the plane attached to the circle another line; namely, the involute of the circle. This mode of description allows us to employ a curved ruler in place of the straight one BB'; and thus gives a vast extension to the theory. But at present we shall confine ourselves to the employment of the straight line BB', only keeping in mind, that there is an intimate connection between the lines of evolution and of provolution.

By the description now given of this process of evolution and involution, it is plain,

1. That the evolution is always made from the convex side of the evolute.

2. That the evolved radii Bb, Cc, Dd, &c. are respectively equal to the arches BA, CA, DA, &c. of the evolute which they have quitted; and that bB', cC', dDd', &c. are always equal to the whole arch ADF.

3. That any point B of the lapped up thread describes during its evolution a curve line B'f'i' parallel to bcdef; because these curves are always equidistant from each other.

4. That if the thread extend beyond the mould as a tangent to it, the extremity a will describe a parallel or equidistant From this it appears that \( B \) is the complete evolatrix of \( FEDCB \), while \( b \) is the evolatrix of that arch, and the added tangent \( Bb \). In like manner, the lapped up thread \( ADF \), with the added part \( F' \), describes the evolatrix \( s' \).

5. If from any point \( C \) of the evolute there be drawn lines \( Cb, Ce, Cd, Cc, \) &c., to the evolutes, those which are more remote from the vertex are greater than those which are nearer. Draw \( Ba, cC, dD, eE \), touching the evolute. \( Cb \) is less than \( CB + Bb \); that is \((2)\), than \( Cc \). Again, \( DC + Cc \) is equal to \( Dd \), which is less than \( DC + Cd \). Therefore \( Cc \) is less than \( Cd \). Now let \( Cc \) cut \( Dd \) in \( r \). Then \( cr + rDE \) is greater than \( cE \). But \( cE \) is equal to \( dr + rDE \). Therefore \( cr \) is greater than \( dr \); and \( cr + rC \) is greater than \( dr + rC \), which is greater than \( cC \). Therefore \( cC \) is greater than \( cC \).

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

7. The evoluted radius cuts every arch of the evolatrix perpendicularly, or a right line drawn through the intersection at right angles touches the evolatrix in that point. Through any point \( d \) draw the line \( mdt \) at right angles to \( dD \). The part of it \( md \) next to the vertex is wholly without the curve, because it is without the circle described round the centre \( D \); and this circle is without the evolatrix on that side of \( d \) which is next the vertex \((6)\). Any point \( t \) on the other side of \( d \) is also without the curve. For let \( teE \) be another evoluted radius, cutting \( Dd \) in \( n \); then \( nd \) is less than \( nt \), because \( ndt \) is a right angle by construction; and therefore \( nd \) is acute. But because \( En + nd \) are greater than \( ED + Dd \), that is, than \( Ec \), and \( nd \) is greater than \( nc \). Therefore, since it is less than \( nt \), it follows that \( nc \) is much less than \( nt \), and \( t \) 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 same reasoning, it is demonstrated, that all the curves \( A'bdf, A'b'df', A'b'df'' \), are cut perpendicularly by the tangents to the evolute. Also all these curves intersect the evolute at right angles in their vertices.

It follows from this proposition, that from every point, such as \( i \), or \( j \), or \( k \), &c., in the space \( AOF \) comprehended by the evolute and its extreme tangents \( AO, FO \), two perpendiculars may be drawn to the evolatrix \( A'df \); and that from any point in the space within the angle \( A'df \) only one perpendicular can be drawn; and that no perpendicular can be drawn from any point on the other side of \( ADF \). Apollonius had observed these circumstances in the conic sections, 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 speculative geometry.

It also follows from this proposition, that if a curve \( A'bcd \) is cut by the tangents of \( ABCDEF \) at right angles in every point, it will be described by the evolution of that curve: For if the evolatrix, whose vertex is \( A \), be really described, it will coincide with \( A'bcd \) in \( A \), and have the same tangent; it therefore does not deviate from it, otherwise their tangents would separate, and would not both be at right angles with the lines touching the evolute. They must therefore coincide throughout.

8. The arches \( bcd \) and \( b'c'd' \), intercepted by the same radii \( Bb \) and \( Dd \), may be called concentric; and the angles contained between the tangents drawn thro' their extremities are equal. Thus the angle \( bcd \) is equal to \( b'c'd' \); but although equidistant, parallel, and containing the same angle between their tangents and between their radii, they are not similar. Thus, the arch \( bcd \) has a curvature at \( s \) that is the same with that of any circle whose radius is equal to \( Aa \); but the curvature at \( A \) is incomparable with it, and unmeasurable. The same may be said of the curvatures at \( s \) and at \( B \).

9. If a circle \( udx \) be described round the centre \( D \) with the radius \( Dd \), it both touches and cuts the evolatrix in the point \( d \), and no circle can be described touching the curve in that point, and passing between it and the circle \( udx \): For since it touches the curve in \( d \), its centre must be somewherer in the line \( dD \) perpendicular to \( mdt \). It cannot be in any point \( n \) more remote from \( d \) than \( D \) is; for it would pass without the arch \( dux \), and be more remote than \( du \) from the arch \( dux \) of the evolatrix. On the other side, it would indeed pass without the arch \( dux \), which lies within the arch \( dux \) of the evolatrix: but it would also pass without the curve. For it has been already demonstrated \((7)\) that \( nd \) is greater than \( nc \); and the curve would lie between it and the circle \( dux \).

Thus it appears, that a circle described with the evolved radius approaches nearer to the curve, or touches it more closely, than any other circle; all other circles either intersect it in measurable angles, or are within or without the curve on both sides of the point of contact. This circle \( udx \) has therefore the same curvature with the curve in the point of contact and coalescence. It is the equicurvature 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 evolatrix, and the point of the evolute is the centre of curvature at the point of contact with the evolatrix. The evolute is the geometrical locus of all the centres of curvature of the evolatrix.

This is the most important circumstance of the whole doctrine of the involution and evolution of curve lines. It is assumed as a self-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 described with shorter radius; for the same reason, all beyond the contact must be less incurvated, by reason of the greater radius. The curvature at the contact must be neither greater nor less than that of the circle. But we thought it better to follow the example of Huyghens, and to establish this leading proposition on the strictest geometrical reasoning, acknowledging the singular obligation which mathematicians are under to him for giving them so palpable a method of fixing their notions on this subject. When the evolute of a curve is given, we have not only a clear view of the genesis of the curve, with a neat and accurate mechanical method of describing it, but also a distinct comprehension of the whole curvature, and a connected view of its gradual variations.

We speak of curvature that is greater and lesser; and every person has a general knowledge or conception of the difference, and will say, that an ellipse is more curved at the extremities of the transverse axis than anywhere else. But before we can institute a comparison between them with a precision that leads to anything, we must agree about a measure of curvature, and say what it is we mean by a double or triple curvature. Now there are two ways in which we may consider curvature, or a want of rectitude: We may call that a double curvature which, in a given space, carries us twice as far from the straight line; or we may call that a double curvature by which we deviate twice as much from the same direction. Both of these measures have been adopted; and if we would rigidly adhere to them, there would be no room for complaint: but mathematicians have not been steady in this respect, and by mixing and confounding these measures, have frequently puzzled their readers. All agree, however, in their first and simplest measures of curvature, and say, that the curvature of an arch of a circle is as the arch directly, and as the radius inversely. This is plainly measuring curvature by the deflection from the first direction. In an arch of an inch long, there is twice as much deflection from the first direction when the radius of the circle is half the length. If the radius is about 57½ inches, an arch of one inch in length produces a final direction one degree different from the first. If the radius is 114½ inches, the deviation is but half of a degree. The linear deflection from the straight path is also one half. In the case of circles, therefore, both measures agree; but in by far the greatest number of cases they may differ exceedingly, and the change of direction may be greatest when the linear deviation is least. Flexure, or change of direction, is, in general, the most sensible and the most important character of curvature, and is understood to be its criterion in all cases. But our processes for discovering its quantity are generally by first discovering the linear deviation; and, in many cases, particularly in our philosophical inquiries, this linear deviation is our principal object. Hence it has happened, that the mathematician has frequently stopped short at this result, and has adapted his theorems chiefly to this determination. These differences of object have caused great confusion in the methods of considering curvature, and led to many disputes about its nature, and about the angle of contact; to which disputes there will be no end, till mathematicians have agreed in their manner of expressing the measures of curvature. At present we abide by the measure already given, and we mean to express by curvature or flexure the change of direction.

This being premised, we observe, that the curvature of all these curves of evolution where they separate from their evolutes, is incomparable with the curvature in any other place. In this point the radius has no magnitude; and therefore the curvature is said to be infinitely great. On the other hand, if the evolved curve has an asymptote, the curvature of the evolutrix of the adjacent branch is said to be infinitely small. These expressions becoming familiar, have occasioned some very intricate questions and erroneous notions. There can be little doubt of their impropriety: For when we say, that the curvature at A is infinitely greater than at B, we do not recollect that the flexure of the whole arch A b is equal to that of the whole arch A B, and the flexure at A must either make a part of the whole flexure, or it must be something disparate.

The evolutrix A b c d f (fig. 2.) of the common equilateral hyperbola exhibits every possible magnitude of curvature in a very small space. At the vertex A of the hyperbola it is perpendicular to the curve; and therefore has the transverse axis A A' for its tangent. The curvature of the evolutrix at A is called infinitely great. As the thread unlaps from the branch ABC, its extremity describes A b c. It is plain, that the evolutrix must cut the asymptote H at right angles in some point G, where the curvature will be what is called infinitely small; because the centre of curvature has removed to an infinite distance along the branch AF of the hyperbola. This evolutrix may be continued to the vertex of the hyperbola on the other side of the asymptote, by causing the thread to lap upon it, in the same way that Mr Huyghens completed his cycloidal oscillation. Or we may form another evolutrix A A' F', by lengthening the thread from G to F', the centre of the hyperbola, and supposing that, as soon as the curve A A' is completed, by unlapping the thread from the branch ABC, another thread laps upon the hyperbola A' F'. This last is considered as a more geometrical evolution than the other: For the mathematicians, extending the doctrine of evolution beyond Mr Huyghens's restriction to curves which had their convexity turned one way, have agreed to consider as one continued evolution whatever will complete the curve expressed by one equation. Now the same equation expresses both the curves AF and A'F', which occupy the same axis AA'. The cycloid employed by Huyghens is, in like manner, but one continuous curve, described by the continued revolution of the circle along the straight line, although it appears as two branches of a repeated curve. We shall meet with many instances of this seemingly compounded evolution when treating of the second question.

Since the arch A b d G contains every magnitude of curvature, it appears that every kind of curvature may be produced by evolution. We can have no conception of a flexure that is greater than what we see at A, or less than what we see at G; yet there are cases which seem to show the contrary, and are familiarly said, by the greatest mathematicians, to exhibit curvatures infinitely smaller still. Thus, let ABC (fig. 3.) be a conical parabola, whose parameter is AP. Let AEF be a cubical parabola, whose parameter is AQ. If we make AQ to AD as the cube of AP to the cube of AQ, the two parabolas will intersect each other in the ordinate DB. For, making AP = p, and AQ = q, and calling the ordinate of the conic parabola y, that of the cubic parabola z, and the indeterminate abscissa AD x, we have

\[ \frac{p^3}{q^3} : \frac{q^3}{x^3} = \frac{q^3}{z^3} : \frac{x^3}{z^3}, \text{ and } \frac{p}{q} = \frac{q}{z} : \frac{z}{x}; \]

but \( \frac{p}{q} : \frac{q}{z} = \frac{p}{z} : \frac{z}{x} \); therefore, by composition,

\[ \frac{p^3}{q^3} : \frac{q^3}{x^3} = \frac{q^3}{z^3} : \frac{x^3}{z^3}, \text{ and } \frac{p}{q} = \frac{q}{z} : \frac{z}{x}; \]

therefore \( x = y \), and the parabolas intersect in B.

Now, because in all parabolas the ordinates drawn at the extremity of the parameters are equal to the parameters, we can describe a circle, and demonstrate that it has all involutions to an arch lying without the parabola. These infinite final curvatures, therefore, are not warranted by our arguments, nor does it yet appear that there are curves which cannot be described by evolution. We are always puzzled when we speak of infinites and infinitesimals as something precise and determinate; whereas the very denomination precludes all determination. We take the distinguishing circumstance of those different orders for a thing clearly understood; for we build much on the distinction. We conceive the curvature of the cubical parabola as verging on that of the common parabola, and the one series of curvatures as beginning where the other ends. But Newton has shewn, that between these two series an endless number of similar series may be interposed. The very names given to the curvature at the extremities of the hyperbolic evolventrix have no conceptions annexed to them. At the vertex of the hyperbola there is no line, and at the intersection with the asymptote there is no curvature. These unguarded expressions, therefore, should not make us doubt whether all curves may be described by evolution. If a line be incurved, it is not straight. If so, two perpendiculars to it must diverge on one side, and must converge and meet on the other in some 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 described. Finally (which should decide the question), we shall see by and bye, that the cubic, and all higher orders of paraboloids, may be so described by evolution from curves having asymptotic branches of determinable forms.

Such are the general affections of lines generated by evolution. They are not, properly speaking, peculiar properties; for the evolventrix may be any curve lines whatever. They only serve to mark the mutual relations of the evolute with their evolventrices, and enable us to construct the one, and to discover its properties by means of our knowledge of the other. We proceed to show how the properties of the evolventrix 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 position: Therefore the length of the evolved arch must, in like manner, be known; and this, not only in toto, but every portion of it. Now this is not universally, or even generally the case. The length of a circular, parabolic, hyperbolic, arch has not yet been determined by any finite equation, or geometrical construction. Therefore their evolventrices cannot be determined otherwise than by approximation, or by comparison with other magnitudes equally undetermined. Yet it sometimes 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 instance in the very evolution which gave occasion to the whole of this doctrine; namely, that of the cycloid; which we shall therefore take as our first example.

Let ABC (fig. 5.) be a cycloid, of which AD is the axis, and A r i D the generating circle, and AG a tangent to the cycloid at A, and equal to DC. Let Involution. BKE touch the cycloid in B, and cut AG in K. It is required to find the situation of that point of the line BE which had unfolded from A?

Draw BH parallel to the base DC of the cycloid, cutting the generating circle in H, and join HA. Describe a circle KEM equal to the generating circle AHD, touching AG in K, and cutting BK in some point E. It is known, by the properties of the cycloid, that BK is equal and parallel to HA, and that BH is equal to the arch A b H. Because the circles AHD and KEM are equal, and the angles HAK and AKE are equal, the chords AH and KE cut off equal arches, and are themselves equal. Because BHAK is a parallelogram, AK is equal to HB; that is, to the arch A b H, that is, to the arch K m E. But if the circle KEM had been placed on A, and had rolled from A to K, the arch engaged would have been equal to AK, and the point which was in contact with A would now be in E, in the circumference of a cycloid AEF, equal to CBA, having the line AG, equal and parallel to DC, for its base, and GF, equal and parallel to DA, for its axis. And if the diameter KM be drawn, and EM be joined, EM touches the cycloid AEF.

Cor. The arch BA of the cycloid is equal to twice the parallel chord HA of the generating circle: For this arch is equal to the evolved line BKE: and it has been shewn, that EK is equal to KB, and BE is therefore equal to twice BK, or to twice HA. This property had indeed been demonstrated before by Sir Christopher Wren, quite independent of the doctrine of evolution; but it is given here as a legitimate result of this doctrine, and an example of the use which may be made of it. Whenever a curve can be evolved into another which is susceptible of accurate determination, the arch of the evolved curve is determined in length; for it always makes a part of the thread whose extremity describes the evolatrix, and its length is found, by taking from the whole length of the thread that part which only touches the curve at its vertex.

This genesis of the cycloid AEF, by evolution of the cycloid ABC, also gives the most palpable and satisfactory determination of the area of the cycloid. For since BE is always parallel to AH, AH will sweep over the whole surface of the semicircle AHD, while BE sweeps over the whole space CBAEF; and since BE is always double of the simultaneous AH, the space CBAEF is quadruple of the semicircle AHD. But the space described in any moment by BK is also one-fourth part of that described by BE. Therefore the area GAEF is three times the semicircle AHD; and the space DHABC is double of it; and the space CBAG is equal to it.

Sir Isaac Newton has extended this remarkable property of evolving into another curve of the same kind to the whole class of epicycloids, that is, cycloids formed by a point in the circumference of a circle, while the circle rolls on the circumference of another circle, either on the convex or concave side; and he has demonstrated, that they also may all be rectified, and a space assigned which is equal to their area (See Principia, B. I. prop. 48, &c.). He demonstrates, that the whole arch is to four times the diameter of the generating circle as the radius of the base is to the sum or difference of those of the base and the generating circle. We recommend these propositions to the attention of the young reader who wishes to form a good taste in mathematical researches; he will there see the geometrical principles of evolution elegantly exemplified.

We may just observe, before quitting this class of curves, that many writers, even of some eminence, in their compilations of elements, give a very faulty proof of the position of the tangent of a curve described by rolling. They say, for example, that the tangent of the cycloid at E is perpendicular to KE; because the line KE is, at the moment of description, turning round K as a momentary centre. This, to be sure, greatly shortens investigation; and the inference is a truth, not only when the rolling figure is a circle rolling on a straight line, but even when any one figure rolls on another. Every point of the rolling figure really begins to move perpendicularly to the line joining it with the point of contact. But this genesis of the arch E r, by the evolution of the arch B b, shews that K is by no means the centre of motion, nor HK the radius of curvature. Nor is it, in the case of epicycloids, trochoids, and many curves of this kind, a very easy matter to find the momentary centre. The circle KEM is both advancing and turning round its centre; and these two motions are equal, because the circle does not slide but roll, the detached arch being always equal to the portion of the base which it quits. Therefore, drawing the tangents E r, M g, and completing the parallelogram E f M g, E f will represent the progressive motion of the centre, and E g the motion of rotation. EM, the motion compounded of these, must be perpendicular to the chord EK.

The investigation that we have given of the evolatrix of the cycloid has been somewhat peculiar, being that which offered itself to Mr Huyghens at the time when he and many other eminent mathematicians were much occupied with the singular properties of this curve. It does not serve, however, so well for exemplifying the general process. For this purpose, it is proper to avail ourselves of all that we know of the cycloid, and particularly the equality of its arch BA to the double of the parallel chord HA. This being known, nothing can be more simple than the determination of the evolatrix, either by availing ourselves of every property of the cycloid, or by adhering to the general process of referring every point to an abscissa by means of perpendicular ordinates. In the first method, knowing that BE is double of BK, and therefore KE equal to HA, and KA = BH, = H b A, = K m E, we find E to be the describing point of the circle, which has rolled from A to K. In the other method, we must draw EN perpendicular to AG; then, because the point E moves, during evolution, at right angles to BE, EK is the normal to the curve described, and NK the subnormal, and is equal to the corresponding ordinate H' I of the generating circle of the cycloid ABC. This being a characteristic property of a cycloid, E is a point in the circumference of a cycloid equal to the cycloid ABC.

Or, lastly, in accommodation to cases where we are supposed to know few of the properties of the evolute, or, at least, not to attend to them, we may make use of the fluxionary equation of the evolute to obtain the fluxionary equation of the evolatrix. For this purpose, take a point e very near to E, and draw the evolving radius b e, cutting E f (drawn parallel to the base DC) in o; draw e n parallel to the axis of the evolute, cut- Involving Eo in v; also draw bb parallel to the base, and Bd perpendicular to it. If both curves be now referred to the same axis CGF, it is plain that Bb, Bd, and db are ultimately as the fluxions of the arch, abscissa, and ordinate of the evolute, and that Ee, es, and vE, are ultimately as the fluxions of the arch, abscissa, and ordinate of the evolutrix. Also the two fluxionary triangles are similar, the sides of the one being perpendicular, respectively, to those of the other. If both are referred to one axis, or to parallel axes, the fluxion of the abscissa of the evolute is to that of its ordinate, as the fluxion of the ordinate of the evolutrix is to that of its abscissa. Thus, from the fluxionary equation of the one, that of the other may be obtained. In the present case, they may be referred to AD and FG, making CG equal to the cycloidal arch CBA. Call this a; AI, x; IB, y; and EB, or EB, z. In like manner, let Ft be u, tE = v, and tE = w; then, because DH = DA = AH, and DA and AH are the halves of CF and BE, we have

\[ \frac{DH}{DA} = \frac{a^2 - z^2}{4}. \]

So DI = \( \frac{a^2 - z^2}{4 \times \frac{1}{2} a} = \frac{a^2 - z^2}{2a}. \) But DI =

\[ \frac{a^2 - z^2}{2a}. \] Therefore Ft, or u, \( = \frac{a^2 - z^2}{2a}. \) Also \( \frac{u}{z} = \)

\[ \frac{a^2}{\sqrt{2au}}. \] Therefore we have \( \frac{u}{z} = \frac{a^2}{\sqrt{2au}}. \)

\[ \sqrt{\frac{a^2}{2au}} = \sqrt{GF} \cdot \sqrt{Ft}, \] which is the analogy competent to a cycloid whose axis is GF = DA.

It is not necessary to insist 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 case it is called a radiated curve, it is most convenient to consider its evolutrix in the same way, conceiving the ordinates of both as inscribed on the circumference of a circle described round the same centre. Spirals evolve into other spirals, and exhibit several properties 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 similar equiangular spiral, and is itself the evolutrix of a third. This is evident on the slightest inspection. Let Crq (fig. 6.) be an equiangular spiral, 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 unlooped from the arch Crq, it is plain that the first motion 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, since this is the case in every position of the point, it is manifest that its path must be a spiral PQR, cutting the radii in the same angle as the spiral Crq. James Bernoulli first discovered this remarkable property. He also remarked, that if a line PH be drawn from every point of the spiral, making an angle with the tangent equal to that made by the radius (like an involution angle of reflection corresponding with the incident ray SP), those reflected rays would all be tangents to another similar and equal spiral I v H; so that PH = PS. S and H are conjugate foci of an infinitely slender pencil; and therefore, the spiral I v H is the caustic by reflection of RQP for rays flowing from S. If another equal and similar spiral x v r roll on I v H, its centre x will describe the same spiral in another position w v z. All these things flow from the principles of evolution alone; and Mr Bernoulli traces, with great ingenuity, the connection and dependence of caustics, both by reflection and refraction, of cycloidal, and all curves of provolution, and their origin in evolution or involution. A variety of such repetitions of this curve (and many other singular properties), made him call it the spira mirabilis. He desired that it should be engraved on his tombstone, with the inscription Eadem mutata resurgo, as expressive of the resurrection of the dead. See his two excellent dissertations in Act. Erudit. 1692, March and May.

Another remarkable property of this spiral is, that if, instead of the thread evolving from the spiral, the foiral evolve from the straight line PC, the centre S will describe the straight line PS. Of this we have an example in the apparatus exhibited in courses of experimental philosophy, in which a double cone descends, by rolling along two rulers inclined in an angle to each other (see Grove's Nat. Phil. I. § 210). It is pretty remarkable, that a rolling motion, seemingly round C, as a momentary centre, should produce a motion in the straight line SP; and it shews the inconclusiveness of the reasoning, by which many compilers of elements of geometry profess to demonstrate, that the motion of the describing point S is perpendicular to the momentary radius. For here, although this seeming 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 speculative geometric curiosity to mark the intimate relation between the genesis of curves by evolution and provolution; it may be applied to important purposes both in science and art. Mr M'Laurin has given a very inviting example of this in his account of the Newtonian philosophy; where he exhibits the moon's path in absolute space, and from this proposes to investigate the deflecting forces, and vice versa. 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 Mells Watt and Boulton's ingenious contrivance for producing the rectilineal motion of a piston rod by the combination of circular motions. M. de la Hire, of the Academy of Sciences at Paris, has been at great pains to shew how all motions of evolution may be converted into motions of provolution, in a memoir in 1706. But he would have done a real service, if, instead of this ingenious whim, he had shewn how all motions of provolution may be traced up to the evolution which is equivalent to them. For there is no organic genesis of a curvilinear motion so simple as the evolution of a thread from a curve. It is the primitive genesis of a circle; and it is in evolution alone that any curvilinear motion is comparable with circular motion. A given curve line is an individual, and therefore its primitive organical genesis must also be individual. This is strictly true of evolution. A para- INV

Involution has but one evolute. But there are infinite motions of provolution which will describe a parabola, or any curve line whatever; therefore there are not primitive organic modes of description. That this, however, is the case, may be very easily shewn. Thus let ABCD (fig. 7.) be a parabola, or any curve; and let abcd be any other curve whatever. A figure Emkhi may be found such, that while it rolls along the curve abcd, a point in it shall describe the parabola. The process is as follows: Let Bb, Cc, Dd, &c. be a number of perpendiculars to the parabola, cutting the curve abcd in so many points. The perpendiculars may be so disposed that the points a, b, c, &c. shall be equidistant. Now we can construct a triangle Ech fo, that the three sides Er, ch, and BE, shall be respectively equal to the three lines Ee, ef, ff. In like manner may the whole figure be constructed, having the little bases of the triangles respectively equal to the successive portions of the base Abcd, and the radii equal to the perpendiculars Bb, Cc, Dd, &c. Let this figure roll on this base e. While the little side ef moves from its present position, and applies itself to ef, the point E describes an arch E of a circle round the centre e, and, falling within the parabola, is somewhere between E and F. Then continuing the provolution, while the next side hi turns round f till i applies to g, the point E describes another arch Fo round f, first rising up and reaching the parabola in F, when the line bE coincides with fF, and then falling within the parabola till the point b begins to rise again from f by the turning of the rolling figure round the point g. Reversing the motion, the sides hi, b, e, ef, &c. apply themselves in succession to the portions gf, fe, cd, &c. of the base, and the point E describes an undulating line, consisting of arches of circles round the successive centres g, f, e, &c. These circular arches all touch the parabola in the points G, F, E, &c. and separate from it a little internally. By diminishing the portions of the base, and increasing the number of the triangular elements of the rolling figure without end, it is evident that the figure becomes ultimately curvilinear instead of polygonal, and the point E continues in the parabola, and accurately describes it. It is now a curvilinear figure, having its elementary arches equal to the portions of the base to which they apply in succession, and the radii converging to E equal to the perpendiculars intercepted between the curve ABCD and the base. It may therefore be accurately constructed.

It is clear, that practical mechanics may derive great advantage from a careful study of this subject. We now see motions executed by machinery which imitate almost every animal motion. But these have been the result of many random trials of wipers, snail-pieces, &c. of various kinds, repeatedly corrected, till the desired motion is at last accomplished. But it is, as we see, a scientific problem, to construct a figure which shall certainly produce the proposed motion; nor is the process by any means difficult. But how simple, in comparison, is the production of this motion by evolution. We have only to find the curve line which is touched by all the perpendiculars Bb, Cc, Dd, &c. This naturally leads us to the second problem in this doctrine, namely, to determine the evolute by our knowledge of the involute; a problem of greater difficulty and of greater importance, as it implies, and indeed teaches, the curvature of lines, its measure, and the law of its involution, variation in all particular cases. The evolute of a curve is the geometrical expression, and exhibition to the eye, of both these affections of curve lines.

Since the evolved thread is always at right angles to the evolutrix and its tangent, and is itself always a tangent of the evolute, it follows, that all lines drawn perpendicular to the arch of any curve, touch the curve which will generate the given curve by evolution. Were this evolved curve previously known to us, we could tell the precise point where every perpendicular would touch it; but this being unknown, we must determine the points of contact by some other method, and by this determination we ascertain so many points of the evolute. The method pursued is this: When two perpendiculars to the proposed curve are not parallel (which we know from the known position of the tangents of our curve), they must intersect each other somewhere on that side of the tangents where they contain an angle less than 180°. But when they thus intersect, one of them has already touched the evolute, and the other has not yet reached it. Thus let bs, cs (fig. 1.) be the two perpendiculars: being tangents to the evolute, the point s of their intersection must be on its convex side, and the unknown points of contact B and E must be on different sides of s. These are elementary truths.

Let sE approach toward bB, and now cut it in x. The contact has shifted from E to D, and x is still between the contacts. When the shifting perpendicular comes to the position sC, the intersection is at i, between the contacts B and C. And thus we see, that as the perpendiculars to the involute gradually approach, their contacts with the evolute also approach, and their intersection is always between them. Hence it legitimately follows, that the ultimate position of the intersection (which alone is susceptible of determination by the properties of the involute) is the position of the point of contact, and therefore determines a point of the evolute. The problem is therefore reduced to the investigation of this ultimate intersection of two perpendiculars to the proposed curve, when they coalesce after gradually approaching. This will be best illustrated by an example: Therefore let ABC (fig. 8.) be a parabola, of which A is the vertex, AH the axis, and AV one-half of the parameter; let BE and CK be two perpendiculars to the curve, cutting the axis in E and K, and intersecting each other in r; draw the ordinates BD, CV, and the tangent BT, and draw BF parallel to the axis, cutting CK in F, and CN in O.

Because the perpendiculars intersect in r, we have rE : EB = EK : BF. If therefore we can discover the ratio of EK to BF, we determine the intersection r. But the ratio of EK to BF is compounded of the ratio of EK to BO, and the ratio of BO to BF. The first of these is the ratio of equality; for DE and VK are, each of them, equal to AV, or half the parameter. Take away the common part VE, and the remainders EK and DV are equal, and DV is equal to BO; therefore EK : BF = BO : BF; therefore rE : EB = BO : BF, and (by division) BE : rE = FO : OB. Now let the point C continually approach to B, and at last unite with it. The intersection r will unite with a point of contact N on the evolute. The ultimate ratio of FO to OB, or of rE : EB, is evidently that of ED Involution to DT, or ED to 2DA; therefore BE : EN = ED : 2DA, or as half the parameter to twice the abscissa.

Thus we have determined a point of the evolute; and we may, in like manner, determine as many as we please.

But we wish to give a general character of this evolute, by referring it to an axis by perpendicular ordinates. It is plain that V is one point of it, because the point E is always distant from its ordinate DB by a line equal to AV; and therefore, when B is in A, E will be in V, and r will coincide with it. Now draw VP and NQ perpendicular to AH, and NM perpendicular to VP; let EB cut PV in t; then, because AV and DE are equal, AD is equal to VE, and VE is equal to one half of DT. Moreover, because BD and NQ are parallel, DE : EQ = BE : EN = DE : DT; therefore DT = EQ, and VE = 1/2 EQ, and therefore = 1/2 VQ; therefore VT = 1/2 of MT, and 1/2 MV. This is a characteristic property of the evolute. The subtangent is 1/2 of the abscissa; in like manner, as in the common parabola, it is double of the abscissa. We know therefore that the evolute is a paraboloid, whose equation is \( ax^3 = y^3 \); that is, the cube of any ordinate MN is equal to the parallelopiped whose base is the square of the abscissa VM, and altitude a certain line VP, called the parameter. To find VP, let CR be the perpendicular to the parabola in the point where it is cut by the ordinate at V; draw the ordinate RS of the paraboloid, and RG perpendicular to AH. Then it is evident, from what has been already demonstrated, that VK is 1/4 of KG, and 1/4 of VG; therefore KG = 4 VK, and (in the parabola) VC = 2 VK. Also, because KV : VC = KG : GR, we have GR = 2 KG = 8 VK; therefore VP × RG = 8 VP × VK. But VP = 27 VK; therefore, because in the paraboloid VP × VS = SR, or VP × RG = VG; we have 8 VP × VK = 27 VK × VK, and 8 VP = 27 VK; or VK : VP = 8 : 27; or VP = 1/27 AV, or 1/27 of the parameter of the parabola ABC. The evolute of the conical parabola is the curve called the semicubical parabola, and its parameter is 1/27 of the conical parabola.

This investigation is nearly the same with that given by Huyghens, which we prefer at present to the method generally employed, because it keeps the principle of inference more closely in view.

Mr Huyghens has deduced a beautiful corollary from it. Since the parabola ABC is described by the evolution of the paraboloid VNR, the line RC is equal to the whole evolved arch RNV, together with the redundant tangent line AV. If therefore we take from CR a part C' equal to the redundant AV, the remainder R is equal to the arch RNV of the paraboloid. We may do this for every position of the evolved radius, and thus obtain a series of points V, S, T, U, of the evolute of the parabola. We have even an easier method for obtaining the length of any part of the arch of the paraboloid, without the previous description of the parabola ABC. Suppose P the arch of the paraboloid, and yz the tangent; make P = 1/27 of the parameter, and describe the arch P' of a circle; then draw from every tangent yz a parallel line xw, cutting the circle in u. The length of the arch P' is equal to yz + uw. The celebrated author congratulates himself, with great justice, on this neat exhibition of a right line equal to the arch of a curve, without the employment of any line higher than the circle. It is the second curve that has been so rectified, the cycloid alone having been rectified by plain geometry a very few years before by Sir Christopher Wren. It is very true, he candidly admits it, that this very curve had been rectified before by Mr William Neill, a young gentleman of Oxford, and favourite pupil of Dr Wallis; also by Mr Van Heuraet, a Dutch gentleman of rank, and an eminent mathematician. But both of these gentlemen had done it by means of the quadrature of a curve, constructed from the paraboloid after the manner of Dr Barrow, *Lett. Geom. XI.* Nor was this a solitary discovery in the hands of Mr Huyghens, as the rectification of the cycloid had been in those of Sir Christopher Wren; for the method of investigation furnished Mr Huyghens with a general rule, by which he could evolve every species of paraboloid and hyperboloid, two classes of curves which come in the way in almost every discussion in the higher geometry. He observes, that the ratio of Bf to Ec, being always compounded of the ratios of Bf to Ba, and of Bb to Ec; and the ultimate ratio of Bf to Bo being that of TE to TD, which is given by the nature of the paraboloid, we can always find the ratio of BE to BN, if we know that of Dd to Ec. In all curves, the ratio of Dd to Ec (taken indefinitely near), is that of the subtangent to the sum of the subtangent and ordinate of a curve constructed on the same abscissa, having its ordinates equal to the subnormals Dd, Ec, VK, &c. In the conic sections the ratio is constant, because the line so constructed is a straight line; and, in the parabola, it is parallel to the axis. See farther properties of it in Barrow's *Lett. Geom. XI.*

From this investigation, Mr Huyghens has deduced the following beautiful theorem:

Let a be the parameter of the paraboloid, x its abscissa, and y its ordinate; and let the equation be \( ax^3 = y^3 \); let the radius of the evolute meet the tangent through the vertex A in Z. We shall always have BN = \( \frac{a}{m} \) BE + \( \frac{m+n}{m} \) BZ. Thus,

\[ \begin{align*} \text{If } a \times y &= 1 \\ a^2 \times y &= 1 \\ a^3 \times y &= 1 \\ a^4 \times y &= 1 \\ a^5 \times y &= 1 \\ a^6 \times y &= 1 \\ \end{align*} \]

then BN = \( \frac{1}{BE} + \frac{1}{BZ} \)

This is an extremely simple and perspicuous method of determining the radius of the evolute, or radius of curvature; and it, at the same time, gives us the rectification of many curves. It is plain that every geometrical curve may be thus examined, because the subnormals DE, VK are determined; and therefore their differences are determined. These differences are the same with the differences of Dd and Ec; and therefore the ratio of Dd to Ec is determined; that is, the subsidiary curve now mentioned can always be constructed.

There is a singular result from this rule, which would hardly have been noticed, if the common method for determining BN had alone been employed. The equation of the paraboloid is so simple, that the increase of the ordinates and diminution of curvature seem to keep pace together; yet we have seen that, in the vertex of the cubical parabola, the curvature is less than any circular curvature that can be named. In the legs, the curvature certainly diminishes as they extend farther; Involution there must therefore be some intermediate point where the curvature is the greatest possible. This is distinctly pointed out by Mr Huyghens's theorem. The evolute of this paraboloid (having \( ax^2 = y^3 \)) is a curve ONRNQ (fig. 9.) consisting of two branches RO, and RQ, which have a common tangent in R; the branch RQ has the axis AE for its asymptote. The thread unfolding from OR, its extremity, describes the arch BC, and then, unfolding from RQ, it describes the small arch CB'A. When B' is extremely near A, the thread has a position B'NE, in which BN is very nearly \( \frac{1}{2} \)BE. At C, if CE be bisected in G, GR is \( \frac{1}{2} \)CZ. Here CR the radius of curvature is the shortest possible. The evolutes of all paraboloids consist of two such branches, if \( m + n \) exceeds 2.

Such is the theory of evolution and involution as delivered by Mr Huyghens about the year 1672. It was cultivated by the geometers with success. Newton prized it highly, and gave a beautiful specimen of its application to the description, rectification, and quadrature of epicycloids, trochoids, and epicycles of all kinds. But it was eclipsed by the fluxionary geometry of Newton, which included this whole theory in one proposition, virtually the same with Mr Huyghens's, but more comprehensive in its expression, and much more simple in its application. Adopting the unquestionable principle of Mr Huyghens, that the evolved thread is the radius of a circle which has the same flexures with the curve, the point of the evolute will be obtained by finding the length of the radius of the equicurve circle. The formula for this purpose is given in the article FLUXIONS OF THE ENCYCLOPEDIA BRITANNICA; but is incorrectly stated \( \frac{a+4}{2\sqrt{a}} \), instead of \( \frac{a+4x}{2\sqrt{a}} \). The theorem also from which it is deduced \( r = \frac{x^2}{y} \) is incorrectly printed, and is given without any demonstration, thereby becoming of very little service to the reader. For which reason, it is necessary to supply the defect in this place.

Therefore let \( A b c d E f \) (fig. 10.) be a circle, of which C is the centre, and ACE a diameter; let the points b, c, d, e, f, of the circumference be referred to this diameter by the equidistant perpendicular ordinates b, c, d, e, f; draw the chords b, c, d, e, f, producing d, e till it meet the ordinate b in a, produce c, g to the circle in f, and join b, f, d, f; draw b, b, c, c, perpendicular to the ordinates; then b, b, c, c, d, d, b, c, e, d, e, are ultimately proportional to the first fluxions of the abscissa AE, the ordinate c, g, and the arch A c; also a, b, the difference between d, m, and c, b is ultimately as the second fluxion of the ordinate. The triangle a, b, c is similar to b, d, f; for the angle a, b, c is equal to the alternate angle b, c, f, which is equal to b, d, f, standing on the same segment. The angle a, b, c is equal to b, d, f, standing on the segment b, c, d; therefore the remaining angles b, a, c and d, b, f are equal; therefore a, b, c = b, d, f = c, d, f = d, f. Now let the ordinates b, i, and d, k, continually approach the ordinate c, g, and at last unite with it; we shall then have b, c ultimately equal to \( \frac{1}{2} \)b, d, and c, g ultimately equal to \( \frac{1}{2} \)d, f. Therefore, ultimately, a, b, c = b, c, g, and c, g = \( \frac{b^2}{a} \).

Let \( u, v, w \) represent the variable abscissa, ordinate, involution, and arch. We have, for the fluxionary expression of the ordinate of the equicurve circle, \( v = \frac{w}{v} \) (\( v \) must have the negative sign, because, as the arch increases, \( v \) diminishes). In the next place, it is evident that, ultimately, \( b, b, c = c, g, c, c, \) and \( c, c = \frac{c, g \times b, c}{b, b} \). If \( r \) be the radius of the equicurve circle, we have \( u : w = v : r \), and \( r = \frac{v}{u} \). But we had \( v = \frac{w}{u} \). Substitute this in the present equation, and we obtain \( r = \frac{w}{u} \). Lastly, observe that \( \omega^2 = u^2 + v^2 \), and \( \omega = \sqrt{u^2 + v^2} = \frac{u^2 + v^2}{\sqrt{u^2 + v^2}} \). Therefore \( \omega^2 = u^2 + v^2 \), and we have \( r = \frac{u^2 + v^2}{u^2 + v^2} \), as the most general fluxionary expression of the radius of a circle, in terms of the sine, cosine, and arch.

When a curve and a circle have the same curvature, it is not enough that the first fluxions of their abscissae, ordinates, and arches, are the same. This would only indicate the position of their common tangent. They must have the same deflection from that tangent. This is always equal to half of the second fluxion of the ordinate. Therefore the circle and curve must have the same second fluxion of their ordinates. Therefore let D, b, c, d, F be any curve coinciding with, or osculated by, the circle A, b, c, d. Let its axis be DG, parallel to the diameter AE; and let c, b be its ordinate. Let D, n be \( x, c, b = y, \) and \( b, c = z \). We have \( x, y, z \), respectively equal to \( u, v, w \). Therefore the radius of the osculating circle is \( r = \frac{x^2}{y} \) or \( r = \frac{x^2 + y^2}{x^2 - y^2} \), for all curves whatever. (We recommend the careful perusal of the celebrated 2d corollary of the 10th proposition of the 2d book of Newton's Principia, where the first principles of this doctrine are laid down with great acuteness.)

Instead of supposing the ordinates equidistant, and consequently \( x \) invariable, we might have supposed the ordinates to increase by equal steps. In this case \( y \) would have had no second fluxion. The radius would then be \( \frac{x^2}{y} \). Or, lastly, we might suppose (and this is very usual) the arch \( z \) to increase uniformly. In this case \( r = \frac{x^2}{y} \): For because \( x^2 + y^2 = z^2 \), by taking the fluxion of it, \( 2x \cdot x + 2y \cdot y = 0 \), and \( y = \frac{x^2}{y} \); and therefore \( r = \frac{x^2}{y^2 - x^2} = \frac{x^2}{y^2 + x^2} \).

Having thus obtained the radius of curvature, and consequently a point of the evolute, we determine its form form by reference to an abscissa, without much farther trouble: It only requires the drawing \( C \rho \) perpendicular to the axis of the proposed curve, and giving the values of \( C \rho \) and \( D \rho \). If we suppose \( x \) constant, then,

\[ cC = \frac{z^2}{x} \]

we have \( D \rho = Dn + s \rho = \frac{x}{y} \)

\[ Dn + \frac{y}{x} \times cC = x + \frac{z^2}{x} - \frac{y}{x} \]

and \( pC = \frac{c}{x} \times cC - cn = \frac{z^2}{x} - y \).

But if we suppose \( y \) constant; then, \( cC \) being \( \frac{z^2}{y} \), we have

\[ D \rho = x + \frac{z^2}{x}, \text{ and } pC = \frac{z^2}{y} - y. \]

And if \( z \) be constant, then, \( cC \) being \( \frac{z^2}{x} \), we shall have \( D \rho \)

\[ = x + \frac{z^2}{x}, \text{ and } pC = \frac{z^2}{x} - y. \]

These formulae are so many general expressions for determining both the curvature of the proposed curve and the form of its evolute. They also give us the rectification of the evolute; because \( cC \) is equal to the evolved arch, or to that arch, together with a constant part, which was a tangent to the evolute at its vertex, in those cases where the involute has a finite curvature at its vertex; as in the common parabola.

Let us take the example of the common parabola, that we may compare the two methods. The equation of this is \( ax = y^2 \), or \( ax^2 = y \). This gives \( y = \frac{a}{x} \times x^{-\frac{1}{2}} \), and (making \( x \) constant)

\[ y = -\frac{1}{2} \times \frac{a}{x} \times x^{-\frac{3}{2}} = -\frac{a}{4x^{\frac{3}{2}}}. \]

Wherefore

\[ z = \sqrt{x^2 + y^2} = \frac{x}{2} \sqrt{1 + \frac{a^2}{x^2}}, \text{ and the radius of } \]

\[ \text{curvature } \left( = \frac{z}{x} \right) = \frac{a + 4x}{2\sqrt{a}}. \]

At the vertex, where \( x = c \), the formula becomes \( = \frac{a}{2} \).

Again, \( D \rho = x + \frac{z^2}{x} \) becomes \( \frac{a}{2} + 3x \);

and therefore \( V \rho = 3x \), the abscissa of our evolute.

Likewise \( c \rho \), its ordinate, \( \left( = \frac{z^2}{x} - y \right) \)

\[ = \frac{4a}{\sqrt{a}}; \text{ and } C \rho^2 = \frac{16x^2}{a}; \text{ and } C \rho^3 \times a = 16x^2. \]

But \( V \rho = 3x \), and \( V \rho = 27x \). Therefore \( C \rho^2 \times \)

\[ \text{with } a = x^2, = \frac{1}{4} \text{th } V \rho^2, \text{ and } \frac{1}{8} \text{th } C \rho^2 = V \rho. \]

Therefore the evolute VC is a semicubical parabola, whose parameter is \( \frac{a}{2} \); as was shown by Mr Huyghens. The arch VC is \( \frac{a + 4x}{2\sqrt{a}} = \frac{a}{2} \).

We shall give one other example, which compr-

Suppl. Vol. II. Part I. Involvion, and draw ST perpendicular to PT, cutting PT in t; and P perpendicular to SP. Let the arch GP be = z, the radius SP = y, and the perpendicular ST = p. Then, it is plain, that PP, op, PT, are ultimately proportional to z, y, p. The triangles PC, and TPT or TP, are also ultimately similar; as also the triangles PST and PPT. Therefore, ultimately,

\[ \frac{Tt}{PP} = \frac{PT}{PC} \]

also

\[ \frac{PP}{PP} = \frac{PS}{PT} \]

therefore \( Tt : PP = PS : PC \), or \( p : y = y : r \), and

\[ r = \frac{y^2}{p} \]; an expression of the radius of curvature, extremely simple, and of easy application.

The logarithmic or equiangular spiral PQR (fig. 6.) affords an easy example of the use of this formula. The angle SPT, which the ordinate makes with the curve, is everywhere the same. Therefore let a be our tabular radius, and b the sine of the angle SPT. We have

\[ ST = \frac{by}{a} \]; and therefore \( PC \left( = \frac{y^2}{p} \right) = \frac{ay}{b} \]

This is to SP or y in the constant ratio of a to b, or of SP to ST; that is, ST : SP = SP : PC, the triangles SPT and PCS are similar, the angles at P and C equal, and C is a point of an equiangular spiral \( p \) round the centre S.

It is not meant that the construction pointed out by this theory of involution, expressed in its most general and simple form, is always the best for finding the centre of the equicurve circle. Our knowledge of, or attention to, many other properties of the curve under consideration, besides those which simply mark its relation to an abscissa and ordinate, must frequently give us better constructions. But evolution is the natural genesis of a line of varying curvature. Moreover, in the most important employment of mathematical knowledge, namely, mechanical philosophy, it is well known, that the most certain and comprehensive method of solving all intricate problems is by reference of all forces and motions to three co-ordinates perpendicular to each other. Thus, without any intentional search, we have already in our hands the very fluxionary quantities employed in this doctrine; and the expression which it gives of the radius of curvature requires only a change of terms to make it a mechanical theorem.

Thus have we considered the two chief questions of evolution and involution. We have done it with as close attention to geometry as possible, that the reader's mind may become familiar with the \( \phi \) as \( \sigma \) while acquiring the elementary knowledge, which is to be employed more expeditiously afterwards by the help of the symbolical analysis. Without such ideas in the mind, the occupation is oftentimes as much divested of thought as that of an expert accountant engaged in complex calculations; the attention is wholly turned to the rules of his art.

It now remains to consider a little the nature of this curvature of which so much has been said, and about which so many obscure opinions have been entertained. We mentioned, in an early part of this article, the unwarranted use of the terms of infinite and infinitesimal magnitude as applicable to curvature, and shewed its impropriety by the inconsistencies into which it leads mathematicians. Nothing threw so much light on this involution subject as Mr Huyghens's Geometry of Evolution; and we should have expected that all disputes would have been ended by it. But this has not been the case; and even the most eminent geometers and metaphysicians, such as the Bernoullis and Leibnitz, have given explanations of orders of curvature that can have no existence, and explanations of that coalescence which obtains between a curve line and its equicurve circle, which are not warranted by just principles.

These errors (for such we presume to think them) arose from the method employed by the geometers of last century for obtaining a knowledge of the magnitude and variation of curvature. The scrupulous geometers of antiquity despised ever being able to compare a curve with a right line. The moderns, although taught by Des Cartes to define the nature of a curve by its equation, allowed that this only enabled them to exhibit a series of points through which it passed, and to draw the polygon which connects these points, but gave no information concerning the continuous incurved arches, of which the sides of the polygon are the chords. They could not generally draw a tangent to any point, or from any point; but they could draw a chord through any two points. Des Cartes was the first who could draw a tangent. He contrived it so, that the equation which expresses the intersections of the curve with a circle described round a given centre should have two equal roots. This indicates the coalescence of two intersections of the common chord of the circle and the curve. Therefore a perpendicular to the radius so determined must touch the curve in the point of their union. This was undoubtedly a great discovery, and worthy of his genius. It naturally led the way to a much greater discovery. A circle may cut a curve in more points than two: It may cut a conic section in four points; all expressed by one equation, having four roots or solutions. What if three of these roots should be equal? This not only indicates a closer union than a mere contact, but also gives indication of the flexure of the intervening arch. For, before the union, the intersections were in the arch both of the curve and of the circle; and therefore the distinction between the union of two and of three intersections must be of the same kind with that between a straight line and an arch of this circle. The flexure of a circle being the same in every part, it becomes a proper index; and therefore the circle, which is determined by the coalescence of three intersections, was taken as the measure of the curvature in that point of the curve, and was called the circle of curvature, the equicurve circle. There is a certain progress to this coalescence which must be noticed. Let ABD (fig. 4.) be a common parabola, EBF a line touching it in B, and BO a line perpendicular to EBF. Taking some point O in the other side of the axis for a centre, a circle may be described which cuts the curve in four points a, b, c, and d. By enlarging the radius, it is plain that the points a and b must separate, as also the points c and d. Thus, the points b and c approach each other, and at last coalesce in a point of contact B, with the parabola, and with its tangent. In the mean time, a and d have retired to A and D. If we now bring the centre O nearer to B, the new circle will fall wholly within the last circle ABD; and therefore both... A and D will again approach to each other, and to B, which still continues a point of contact. It is plain that A will approach faster to B than D will do. At length, the centre being in o, the point A coalesces with B, and we obtain a circle B, touching the curve in B, and cutting it in s. Consequently the arch B is wholly within, and B is wholly without the parabola; and the circle both touches and cuts the parabola in B. Here is certainly a closer union, at least on the side of a. But perhaps a farther diminution of the circle may bring it closer on the side of D. Join B.

Let a smaller circle be described, touching the parabola in B, and cutting it in s. Draw s parallel to B. It may be demonstrated that the new circle cuts the parabola in s. Now the arch between c and s being without the parabola, the arch B C must be within it; and therefore this circle is within the parabola on both sides of B, and is more incurvated than the parabola. We have seen, that a circle greater than B is without the parabola on both sides of B; and therefore is less incurvated than the parabola. Therefore the individual circle B is neither more nor less curved than the parabola in the point B. Therefore the circle indicated by the coalescence of three intersections is properly named the equicurvature circle; and, since we measure all curvatures by that of a circle, it is properly the circle of curvature, and its radius is the radius of curvature.

Had B been the vertex of the axis, every intersection on one side of B would have been similar to an intersection on the other, and there would always have been two pairs of roots that are equal; and therefore when three intersections coalesce, a fourth also coalesces, and the contact is said to be still closer.

What has now been thrown with respect to a conic section is true of every curve. When two intersections coalesce, there is a common tangent; when three coalesce, there is an equal curvature, and no other circle can pass between this circle and the curve. There cannot be a coalescence of four intersections, except when the diameter is perpendicular to the ordinates, and those are bisected by the diameter.

Mr Leibnitz, who valued himself for metaphysical refinement, and never fails to claim superiority in this particular, notices the important distinction between a simple contact and this closer union, in a very well written dissertation, published in the Acta Eruditorum, July 1686. He calls the contact of equal curvatures an osculation, and the circle of equal curvature the osculating circle, and delivers several very judicious remarks with the tone of a master and instructor. He also speaks of different degrees or orders of osculation, each of which is infinitely closer than the other, as a thing not remarked by geometers. But Sir Isaac Newton had done all this before. The first twelve propositions of the Principia had been read to the Royal Society several years before, and were in the Registers. The Principia had received the imprimatur of the Society in July 1686; but was almost printed before that time. In the Scholium to the 11th Lemma, is contained the whole doctrine of contact and osculation; and in the lemma and its corollaries, is crowded a body of doctrine, which has afforded themes for volumes. The author glances with an eagle's eye over the whole prospect, and points out the prominent parts with the most compressed brevity; but with sufficient precision for marking out the more important objects, and particularly the different orders of curvature. This lemma and its corollaries are continually employed in the twelve propositions already mentioned. In 1671 he had written the first draught of his method of fluxions, where this doctrine is systematically treated; and Mr Collins had a copy of it ever since 1676. It is well known that Leibnitz, when in London, had the free perusal of the Society's records, and information at all times by his correspondence with the secretary Oldenburg and Mr Collins. His conduct respecting the theorems concerning the elliptical motion of the planets, and the resistance of fluids, leave little room to doubt of his having availed himself in like manner of his opportunity of information on this subject. He gives a much better account of the Newtonian doctrine on this subject than in those other instances, it being more suited to his refining and paradoxical disposition.

In this and another dissertation, he considers more particularly the nature of evolution, and of that osculation which obtains between the evolutrix and the circle described by the evolved radius. He says, that it is equivalent to two simple contacts, each of which is equivalent to two intersections. An osculation produced in the evolution of a curve is therefore equivalent to four intersections. And he advises, with an air of authority, the mathematicians to attend to these remarks, as leading them into the rectifices of science. He is mistaken, however; and the listening to him would prevent us from forming a just notion of osculation, and from conceiving with distinctness the singular fact of a circle both touching and cutting a curve in the same point. James Bernoulli lost his friendship, because he presumed to say that the presence of four intersections in an osculation is not warranted by the equation expressing those intersections.

Mr Leibnitz was misled by the way in which he had considered the osculation in the evolution of curves. It merits attention. From any point within the space ADFOA (fig. 1), two perpendiculars may be drawn to the evolutrix A b df; and therefore two circles may be described round that point, each touching the curve. Each contact is the union of two intersections. Therefore, as the centre approaches the evolute, the contacts approach each other, and they unite when the centre reaches the evolute. Therefore the osculation of evolution is equivalent to four intersections.

But when two such circles are described round a point s, so that both may touch the evolutrix A b df, the point s is in the intersection of one evolved radius with the prolongation of another. The contact at the extremity b of the prolonged radius b B is an exterior contact, and the arch of the circle crosses the evolutrix, from without inwards, in some point more remote from A. The contact at the extremity e of the radius e E is an interior contact; and if e e be greater than the straight line EA, the arch of this circle crosses the curve, from within outwards, in some point nearer to A. Thus each contact is accompanied by an intersection on the side next the other contact, sometimes beyond it, and sometimes between the contacts. As the contacts approach, the intersections also approach, still retaining their characters as intersections, as the contacts still continue contacts. Also the circle next to A crosses from without inwards, and that next to f crosses from Involution within outwards. They retain this character to the last; and when the contacts coalesce, the two circles coalesce over their whole circumference, still however, crossing the curve in the same direction as before; that is, without the curve on the side of A, and within it on the side of f. The contacts unite as contacts, and the intersections as intersections. Thus it is that the osculating circle both touches and intersects the curve in the same point.

At f the oscillation is indeed closer than anywhere else. The variation of curvature is less there than anywhere else, because the radius changes more slowly. It is this circumstance that determines the closeness of contact. If a circle osculates a curve, it has the same curvature. If this curvature does not change in the vicinity of the contact, the curve and circle must coincide; and the deviation of the circle (the curvature of which is everywhere the same) from the curve must proceed entirely from the variation of its curvature. This, therefore, is the important circumstance, and is indeed the characteristic of the figure as a care line; and its other properties, by which the position of its different parts are determined, may be ascertained by means of the variation of its curvature, as well as by its relation to co-ordinates. Of this we have a remarkable instance at this very time. The orbit of the newly discovered planet has been ascertained with tolerable precision by means of observations made on its motions for three years. In this time it had not described the 20th part of its orbit; yet the figure of this orbit, the position of its transverse axis, the place and time of its perihelion, were all determined within 100th part of the truth by the observed variation of its curvature. It therefore merits our attention in the close of this article. We know of no author who has treated the subject in so instructive a manner as Mr. M'Laurin has done, by exhibiting the theorem which constitutes Newton's 11th lemma in a form which points this out even to the eye (see M'Laurin's Fluxions, Chap. xi. § 363, &c.). We earnestly recommend this work to the young geometer, as containing a fund of instruction and agreeable exercise to the mathematical genius, and as greatly superior in perspicuity and in ideas which can be treasured up and recollected, when required, to the greatest part of the elaborate performances of the eminent analysts of later times. By expressing everything geometrically, the author furnishes us with a sort of picture, which the imagination readily reviews, and which exhibits in a train what mere symbols only give us a momentary glimpse of.

"As, of all right lines which can be drawn through a given point in the arch of a curve, that alone is the tangent which touches the arch so closely that no right line can pass between them; so, of all circles which touch a curve in a given point, that circle alone has the same curvature which touches it so closely that no circle can pass between them. It cannot coincide with the arch of the curve; and therefore the above condition is sufficient for making it equicurve. As the curve separates from the tangent by its flexure or curvature, it separates from the equicurve circle by its change of curvature; and as its curvature is greater or less according as it separates more or less from its tangent, so the variation of its curvature is greater or less according as it separates more or less from its equicurve circle. There can be but one equicurve circle at one point of a curve, involuted otherwise any other circle described between them through that point will pass between the curve and the equicurve circle.

"When two curves touch each other in such a manner that no circle can pass between them, they must have the same curvature; because the arch which touches one of them so closely that no circle can pass between them, must touch the other in like manner. But circles may touch the curve in this manner, and yet there may be indefinite degrees of more or less intimate contact between the curve and its equicurve circle." This is shown by the ingenious author in a series of propositions, of which a very short abridgment must suffice in this place.

Let any curve EMH (fig. 11.), and a circle ERB, touch a right line ET on the same side at E. Let any right line TK', parallel to the chord EB of the circle, meet the tangent in T', the curve in M, and a curve BKF (which passes through B) in K. Then, if MT × TK be everywhere equal to TE', the curvature of EMH in the point E is the same as that of the circle ERB; and the contact of EM and ER is so much the closer the smaller the angle which is contained at B between the curve BKF and the equicurve circle BQE.

Let TK meet the circle in R and Q. Then, because RT × TQ = TE', it must be RT × TQ = MT × TK; and RT : MT = TK : TQ. The line BKF may have any form. It may cross the circle BQR in B, as in the figure. It may touch it, or touch EB, &c. Let us first consider what situations of the point M correspond with the position of K, in that part of the curve BKF which lies without the circle BRE. Let TK move toward EB, always keeping parallel to it, till it coincide with it, or even pass it. Then, while the point K describes KB, it is evident that since TK is greater than TQ, TM must be less than TR, and the point M must always be found between T and R. The arch ME of the curve must be nearer to the tangent than the arch RE of the circle. If any circle be now described touching TE in E, and cutting off from EB a smaller chord than EB, it is clear that the whole of this segment must be within the segment BRE; therefore this smaller circle does not pass between ERB and the curve EMH. But since we see that the curve lies without the circle, in the vicinity of E, perhaps a greater circle than ERB may pass between it and the curve. A greater circle, touching at E, must cut off a chord greater than EB. Let Erb be such a circle, cutting EB in b, and TQ in q. Tq is necessarily greater than TQ. For since b is beyond B, and the arch BKF lies in the angle QBb, the circle Erb must cross the curve FKB in some point; suppose F. Then while K is found in the arch FB, the point q must be beyond K, or Tq must be greater than TK. Now Tr × Tq = TE' = TM × TQ. Therefore TM : Tr = Tq : TQ. Therefore Tq being greater than TQ, Tr must be less than TM, and the point r must lie without the curve, and the arch Er does not pass between EMH and the circle ERB. In like manner, on the other side of EB, it will appear, that when the curve BKF falls within the circle which touches EMH in E, and cuts off the chord EB, the arch of the curve corresponding to the arch BKF, lying within the circle, also lies within the circle. For T'K' being less than TQ, TM' is greater than TR, and the curve is within the circle. And, by similar reasoning, it is evident that a circle cutting off a greater chord falls without both the circle ERB and the curve, and that a circle less than ERB must necessarily leave some part of the curve BKF without it; and therefore TK must be greater than TQ, and the corresponding point r must be without the curve. All circles therefore touching TE in E fall without both ER and EM, or within them both, according as they cut off from EB a chord greater or less than EB, and no circle can pass between them when the rectangle MT × TK is always equal to ET, and the focus of the point K passes through B; that is, ERB is the equicurve circle at E.

This corroborates the several remarks that we have made on the circumference of a circle touching and cutting a curve in the same point. No other circle can be made to pass between it and the curve, and it therefore has the same curvature. This may therefore be taken as a sufficient indication of the equicurve circle; the character peculiarly assured to it by the nature of evolution. It must be noted, however, that the curve is supposed to have its concavity in the vicinity of the contact turned all the same way. For if the contact be in a point of contrary flexure, even a straight line will both touch and cut it in that point.

The reader cannot but remark, that MK is always the chord of a circle touching TE in E, and passing through M.

Let EM be another curve, touching TE in E, such that the conjugate curve KB, which always gives TM × TK = TE², also passes through B. Then, by what has now been demonstrated, the two curves EM and EM have the same equicurve circle ERB, and consequently the same curvature in E. Then, because the rectangles RT × TQ, MT × TK, and MT × TK, are equal, we have TM : TM = TK : TK. Therefore if the arch Bk passes between BK and QB, the curve EM must pass between the curve EM and the circle ER. EM must therefore have a closer contact with ER than EM has with it; and the smaller the angle QBK is which is contained between the curve and its equicurve circle, the closer is the contact of the curve EM and its equicurve circle ER. Thus the length of the chord EB determines the magnitude or degree of curvature at E, when compared with another; and the angle contained between the equicurve circle and the conjugate curve BKF determines the closeness of the contact of the curve with its equicurve circle (the angle TEB being supposed the same in both).

It appears, from the process of demonstration, that the curve EMH falls without or within the equicurve circle according as its conjugate curve BKF does. Also, when BKF cuts BQR, HME also cuts it. But if FQB is on the same side of QB on both sides of the intersection B, the curve HME is also on the same side of it on both sides of the contact E. It is also very clear, that the contact or approach to coalescence between the curve and its circle of curvature, is so much the closer as the conjugate curve BKF comes nearer to the adjoining arch of this circle. It must be the closest of all when KB touches QB, and it must be the least so when KB touches EB, or has EB for an asymptote. The space QBK is a sort of magnified picture of the space MER; and we have a sensible proportion of TQ to TK as the representation of the proportion of TM to TR, quantities which are frequently evanescent and insensible. When QBK is a finite angle, that is, when the tangents of BQ and BK do not coincide, the angle QBK can be measured. But no rectilineal angle can be contained as an unit in the curvilinear angle MER. They are incomensurable, or incomparable. Let the curve KB touch the circle QB without cutting it. This angle is equally incomparable with the former QBK; yet it has a counterpart in MER. This must be incomparable with the former in the same manner; for there is the same proportion between the individuals of both pairs. Thus it appears plainly, that there are curvilinear angles incomparable with each other. Yet are they magnitudes of one kind; because the smallest rectilineal angle must certainly contain them both; and one of them contains the other. But, further, there may be indefinite degrees of this coalescence or closeness of contact between a curve and a circle. The first degree is when the same right line touches both. This is a simple contact, and may obtain between any curve and any circle. The next is when EMH and ERB have the same curvature, and when the conjugate curve FKB intersects the circle QB in any assignable angle. This is an osculation. The third degree of contact, and second of osculation, is when the curve KB touches the circle QB, but not so as to osculate. The fourth degree of contact, and third of osculation, is when KB and QB have the same curvature or osculate in the first degree of osculation. This gradation of more and more intimate contact, or (more properly speaking) of approximation to coalescence, may be continued without end, "necesse non est natura limitem," the contact of EM and ER being always two degrees closer than that of BK and BQ. Moreover, in each of those classes of contact there may be indefinite degrees. Thus, when EM and ER have the same curvature, the angle QBK admits of indefinite varieties, each of which affects a different closeness of contact at E. Also, though the angle QBK should be the same, the contact at E will be so much the closer the greater the chord EB is.

For TR : TM = TK : TQ

Therefore RM : TR = KQ : KT

Or RM : KQ = TR : TK = TR × TQ : TK × TQ = TE² : TK × TQ.

Therefore, when TE is given, RM (which is then the measure of the angle of contact) is proportional to KQ directly, and to the rectangle TK × TQ inversely; and when KQ is given, RM is less in proportion as KT × TQ is greater. In the very neighbourhood of E and B, it is plain that KT × TQ is very nearly equal to EB², and therefore ultimately RM : KQ = ET² : EB².

It will greatly assist our conception of this delicate subject, if we view the origin of these degrees of contact as they are generated by the evolution of lines. A thread evolving from a polygon EDCBA (fig. 13.) describes with its extremity a line edbea, consisting of successive arches of circles united in simple contacts. If it evolve from any continuous curve CBA, after having evolved from the lines ED, BC, the arch eb will be united with the circular arch de by osculation of the first degree. If any other curve FC touch this evolute in a simple contact, and if the two curves FCBA and DCBA are both evolved, they will touch each other in Involution: a simple osculation in that point where they have the same radius. If FC touches DC in a simple osculation, the evolved curves will touch in an osculation of the second degree; and, in general, the osculation of the two generated curves is a degree closer than that of their evolutes; and in each state of one of the osculations, there is an indefinite variety of the other, according to the length of its radius of curvature. All this is very clear; and shows, that these degrees of contact do not indicate degrees of curvature, one of which infinitely exceeds another; for they are all finite.

The reader will do well to remark, that the magnitude, which is the subject of the above proportions, which is really of the same kind in them all, and considered as susceptible of various degrees and orders of infinitesimals, is not curvature, but lineal extension. It is RM, the subtense of the angle of contact MER. It is the linear separation from the tangent, or from the equicurve circle. It is, however, usually considered as the measure of curvature, or the proportions of this line are given as the proportions of the curvature. This is inaccurate; for curvature is unquestionably a change of direction only. As this line has generally been the interesting object in the refined study of curve lines, especially in the employment of it in the discussions of mechanical philosophy, it has attracted the whole attention, and the language is now appropriated to this consideration. What is called, by the most eminent mathematicians, variation of curvature, is, in fact, variation of the subtense of the angle of contact. But it is necessary always to distinguish them carefully.

Variation of curvature is the remaining object of our attention.

Curvature is uniform in the circle alone. When the curvature of the arch EMH (fig. 11.) decreases as we recede from E, the arch, being less deflected from its primitive direction ET then the arch ER, must separate less from the line ET, or must fall without the arch ER. The more rapidly its curvature decreases, the describing point must be left more without the circle. It must be the contrary, if its curvature had increased from E toward M. It may change its curve equably or unequally. If equably, there must be a certain uniform rate, which would have produced the same final change of direction, in a line of the same length, bending it into the uniformly incurved arch of a circle. It is not so obvious how to estimate a rate of variation of curvature; and authors of eminence have differed in this estimation. Sir Isaac Newton, who was much interested in this discussion, in his studies on universal gravitation, seems to have adopted a measure which best suited his own views; and has been followed by the greater number. He gives a very clear conception of what he means, by stating what he thinks a case of an invariable rate of variation. This is the equiangular spiral, all the arches of which, comprehended in equal angles from the centre, are perfectly similar, although continually varying in curvature. He calls this a curve equally variable, and makes its rate of variation (estimated in that sense in which it is uniform) the measure of the rate of variation in all other curves. Let us see in what respect its variation of curvature is constant. It may be described by the evolution of the same spiral in another position (see fig. 6.), and the ratio between the radius of the evolute and that of the evolventrix is always the same; or (which amounts to the involution of the same thing) the arch of the evolventrix bears to the evoluted arch of the evolute a constant ratio. The curvature of the spiral changes more rapidly in the same proportion as the ratio of the evolved arch to the arch of the evolventrix generated by it is greater, or as it cuts the radii in a more acute angle. These arches may be infinite; therefore the fraction fluxion of evolventrix expresses the rate of the variation of curvature in this form. Now let abcd (fig. 13.) be any other curve, and ABCD its evolute; let p be the centre of curvature at the point B of the evolute, and Bo the evolved arch; draw the radius pB, Bo, BM, m; join pm, and draw Bq perpendicular to pm. It is evident that m and Bo have the same ratio with BM and Bp; and that these two small arches may be conceived as being portions of the same equiangular spiral (perhaps in another position), of which q is the centre; and that q is in the curve of another of the same. For qB = qB = qB = qM, = pB : BM; therefore the ratio of these infinitesimal arches m and Bo will express the rate of variation in any curve. This is evidently equivalent to saying, that the variation of curvature is proportional to the fluxion of the radius of curvature directly, and the fluxion of the curve inversely. For m and Bo are ultimately as those fluxions,

\[ \frac{Bo}{m} = \frac{r}{z}, \]

where z is the arch of the spiral, and r the evolved radius of the other. Accordingly, this is the enunciation of the index of variation given by Newton (See Newton's Fluxions, Prob. VI. § 3.). Therefore, what Newton calls a uniform variation of curvature, is not an increase or diminution by equal arithmetical differences, but by equal proportions of the curvature in every point. The variation of curvature in similar points of similar arches is supposed to be the same.

It is evident that this ratio is the same with that of radius to the tangent of the angle pMB, or of 1 to its tabular tangent. The tangent therefore of this angle corresponding to any point of a curve is the measure of the variation of curvature in that point. Now it may be shown (and it will appear by and bye), that the fluxion of TK in fig. 11., or the ultimate value of KQ, is always \( \frac{d}{ds} \) of the fluxion of the radius of curvature. Therefore the tangent of the angle QBK is always \( \frac{d}{ds} \) of that of pMB; and therefore the angle QBK, which we have seen to be an index of the closeness of contact, is also the index of the variation of curvature (See MacLaurin, § 386.).

Sir Isaac Newton has given specimens of the use of this measure in a variety of geometrical curves, by means of a general expression of \( \frac{f}{x} \). Thus, in the curve ABC (fig. 8.), let AB be \( x \), AD = \( y \), DB = \( r \), BN = \( s \), and BE = \( p \); we have

\[ \frac{Nn}{Bb} = \frac{r}{s}. \]

Now DB = \( y \); \( p = Dd : Bb = x : s \). Therefore \( z = \frac{px}{y} \), and \( \frac{r}{s} = \frac{px}{y} \). Now, in every curve which we can express by an equation, we can obtain all these quantities \( p, y, r, s, \) and \( z \), and can therefore obtain the measure of of the variation of curvature. It also deserves particular notice, that this investigation of \( \frac{r}{z} \) is equivalent with finding the centre and radius of curvature of the evolute, by which the curve under consideration is generated; or with finding the centre \( q \) (fig. 13.) of an equiangular spiral, which will touch our curve in \( m \), its evolute in \( B \), and the evolute of the evolute in \( p \), if put into different positions when necessary. This leads to very curious speculations, for which, however, we have no room. It has been said, for instance, that the curvature at the intersection of a cycloid with its base is infinitely greater than that of any circle. If the evolution of the cycloid begin from this point, the curvature of its evolutrix will be infinitely greater still upon the same principles; and we shall have one infinitely greater than this by evolving it. Yet all these infinities, multiplied to infinity, are contained in the central point of every equiangular spiral! In like manner, there are evolutes which coincide with a straight line, and others of infinitely greater rectitude, and all they are curves. Can this have any meaning? And can it be reconciled with the legitimate reasoning from the same principles, that all these curvatures and angles of contact are producible by evolution; and that they may be, and certainly are every day described, by bodies moving in free space, and acted on by accelerating forces directed to different bodies?

The parabola (conical) is the most simple of all the lines of unequally varying curvature, and becomes a very good standard of comparison. In the parabola \( ABC \) (fig. 8.) let the parameter be \( 2a \). The equation is then \( 2ax = y^2 \); \( DE = a \); \( p \); or \( BE = \sqrt{a^2 + j^2} \); \( DQ = a + z \times \) (by what was formerly demonstrated).

Moreover, \( DB : BE = DQ : BN \); and \( BN = \frac{pa + zp}{a} \); \( r \). These equations give \( 2ax = 2y \); \( 2z \); \( 2p \); and \( \frac{ap + zp + 2p}{a} = r \). Now making \( x = 1 \), and reducing the equations, we obtain \( y = \frac{a}{j} \); \( p = \frac{a}{j} \); \( r = \frac{a}{j} \); and \( r = \frac{ap + zp + 2p}{a} \).

With these values of \( y \); \( p \); \( r \), we obtain a numerical value of \( \frac{r}{p} \) most readily. Thus, in order to obtain the index of variation of curvature in the point where the ordinate at the focus cuts the parabola, make \( a = 1 \).

Then \( 2x = y^2 \); \( x = \frac{1}{4} \); \( y = \sqrt{2x} = 1 \); \( j = \frac{a}{j} \);

\( x = \frac{1}{4} \); \( p = \sqrt{a^2 + j^2} = \sqrt{2} \); \( p = \frac{a}{j} = \sqrt{\frac{5}{3}} \);

and \( r = \frac{ap + zp + 2p}{a} = \sqrt{2 \times 3} \). Therefore \( \frac{r}{p} = 3 \), the index of variation in the point \( B \) when \( D \) is the focus of the parabola; that is to say, the fluxion of the radius of curvature is three times the fluxion of the curve.

The index of variation, where the ordinate is equal to the parameter, is had by making \( x = 2 \). This gives \( y = 2 \); \( j = \frac{1}{4} \); \( p = \sqrt{5} \); \( p = \sqrt{\frac{5}{3}} \); and \( r = 3\sqrt{5} \).

Wherefore \( \frac{r}{p} = 6 \), which is the index of variation.

Moreover, since \( p \) and \( r \) are in a constant ratio, it appears that the index of variation of curvature in the parabola is proportional to the ordinate \( y \). It is always \( \frac{r}{p} = 6 \); and thus, with very little trouble, we can describe the evolute of its evolute, i.e., of the semicubical parabola.

In like manner, it may be shewn, that in all the conic sections \( \frac{r}{p} \) is always proportional to the rectangle of the ordinate \( DB \) and the subnormal \( DE \), or to \( DB \times DE \). In the parabola, whose equation is \( 2ax = y^2 \), we have \( \frac{r}{p} = \frac{3}{a} \). In an ellipse, whose equation is \( 2ax - bx^2 = y^2 \), we have \( \frac{r}{p} = \frac{3 - 3b}{a} \times DB \times DE \), and in the hyperbola, whose equation is \( 2ax + bx^2 = y^2 \), \( \frac{r}{p} = \frac{2 + 3b}{a} \times DB \times DE \). This ratio, in all the three sections, is always as the tangent of the angle contained between the diameter and the normal at the point of contact. By this we may compare them with a parabola. In the cycloid at the point \( E \) (fig. 5) \( \frac{r}{p} \) is \( \tan \angle EKM \), &c., &c.

All these things may be traced in the observations made on figs. 11. and 12. When the angle \( BET \) is a right angle, the angle \( KBQ \) indicates it directly, its tangent being always \( \frac{r}{p} \). It is easy also to see, that when the curve \( EMH \) is a parabola, the line \( BKF \) is a straight line parallel to \( ET \). It is also plain, that by the same steps that we proved that no circle can pass between this parabola and its equicurve circle \( ERB \), so no other parabola can pass between them. Indeed the same reasoning will prove that no curve of the same kind can pass between any curve and its osculating circle. In many cases, it is more easy to reason from the curvature of a curve, by comparing it with an equicurve parabola than with an equicurve circle; particularly in treating of the curvilinear motions of bodies in free space, actuated by deflecting forces.

If \( EMH \) be an ellipse or hyperbola, \( BKF \) is another ellipse or hyperbola (\( M \).Laurin, § 373.)

We have thus endeavoured to introduce our readers into this curious branch of speculative geometry. An introduction is all that can be expected from a work of this kind. We have enlarged on particular points, in proportion as we thought that the notions entertained on the subject were inadequate, or even vague and indistinct; and we hope that some may be incited to acquire clearer conceptions by going to the fountain head. We conclude, by recommending to the young geometer the perusal of the Fluxions of Sir Isaac Newton, after after he has read M'Laurin's Chapter with care. He will probably be surprized and delighted with seeing the whole compassed by a master's hand into such narrow compass with such beautiful perspicuity.