On putting AP = x, PM = y and with the AM = s, let the force which pushes the body downwards at M be equal to P; then v = Pdx and with this integral thus taken ∫ so that it vanishes on putting x = 0, if the body indeed is put to begin its motion from rest at A, and dv = Pdx. Hence du = Pdx = dv and ddw = 0 , since du remains invariant in going from m to n. Therefore this equation is obtained (362) : dy dydv 2vd . ds = ds , the integral of which is : dy l av = 2l ds or vds 2 = ady 2 , and hence : ∫ ∫ dx 2 Pdx = ady 2 − dy 2 Pdx. On account of which this equation is produced for the brachistochrone line sought : dy = dx ∫ Pdx , ( a − ∫ Pdx )EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 285 in which the indeterminates x and y have been separated from each other. Moreover the length of this curve is found from this equation : ds = dx a . ( a − ∫ Pdx ) Q.E.I. Corollary 1. 368. Therefore at A, where the speed of the body disappears, either ∫ Pdx = 0 , and dy = 0, or the tangent to the curve at A is incident along the vertical at AP. But when it becomes Pdx = a , there the tangent to the curve is horizontal [as dy/ds = 1 at A]. ∫ Corollary 2. 369. Because ddw = 0 and du = Pdx, the [fundamental] equation (363) becomes : 2v = Pdy . r ds Pdy Now ds is the normal force[; in the figure here, the normal force Pdy is given by GI = P cos θ = ds ; p. 180] acting upon the curve at M , along the normal drawn towards the axis AP. Consequently, the normal force is equal to the centrifugal force and acting in the same direction. On account of which the brachistochrone curve has this property, that the total force exerted on the curve is twice as great as the normal force alone. Now in the following we will demonstrate that all brachistochrone lines have this property in place, either in a vacuum or in a medium with resistance. [Thus, the centrifugal force is the force of the body pressing on the curve due to the motion, which Pdy here has the same magnitude as the normal component of the weight ds along IG; thus Pdy the curve supplies a total normal force equal to 2 × ds along GI to produce the required centripetal force.] Corollary 3. 370. On account of the arbitrary nature of a an infinite number of brachistochrone curves are given all having the starting point A. And this letter a can be used, so that a curve from A passes through the point C, which is the line between A and C, upon which the time is a minimum.EULER’S MECHANICA VOL. 2. Chapter 2f. page 286 Translated and annotated by Ian Bruce. Corollary 4. 371. Because the curve AMC (Fig. 47) has a horizontal tangent somewhere, let that be BC and the other vertical axis CQ is taken at C. Let CQ = X, QM = Y and CM = S; then dX = −dx , dY = −dy , dS = − dS and ∫ Pdx = a − ∫ PdX with the integral ∫ PdX thus taken, so that it vanishes on putting X = 0. If the curve is referred to this axis CQ, this equation is found : dY = dX ( a − ∫ PdX ) ∫ PdX or dS = dX a . ∫ PdX Corollary 5. 333. Therefore all these curves have two similar and equal arcs on each side of the axis CQ. [p. 181] In a similar manner the curve is equally disposed on each side of the axis AB. On account of which curves of this kind have infinite diameters parallel to each other and places at a distance BC, unless perhaps the force acting is therefore taken, so that above A it is negative, in which case the curve CMA can tend upwards and the concave part is changed to downwards. Example 1. 373. Let the force acting be or P = g; the integral becomes ∫ Pdx = gx; thus with gb put in place of a, this equation is obtained for the brachistochrone for this hypothesis of the force acting : dy = dx x ( b− x ) or ds = dx b . ( b− x ) But if the equation is referred to the axis CQ, it becomes dY = dX ( b − X ) X or dS = dX b , X the integral of which is S = 2 bX . From which equation it is clear that the curve is a cycloid upon a horizontal base described by a circle of diameter b and turned downwards, as thus found by the most celebrated Johan Bernoulli and by other outstanding geometers now some time ago. Therefore if any two points A and M are given, the line, upon which a body descends from A to M the most quickly, can be found, if a cycloid is described having the same cusp at A and having the horizontal base passing through the point M ; that is easily brought about from a single described cycloid, since all cycloids are similar curves. Moreover the time, at which the body reaches M from A, is a minimum, equal to ∫ g( bx − x ) dx b 2 and the length of the curve AM is equal to :EULER’S MECHANICA VOL. 2. Chapter 2f. page 287 Translated and annotated by Ian Bruce. ∫ ( b− x ) = 2b − 2 b( b − x ). dx b Moreover, when PM = y = ∫ ( bxdx− xx ) , [p. 182] the time to traverse AM is equal to 2 y +2 ( bx − xx ) gb equal to the arc in the circle of diameter b, the versed sine of which is equal to x, times by 2 . gb [G.G. Leibniz, Cummunicatio suae pariter duarumque alienarum ad edendum sibi primum a Dn. Io. Bernoullio, deinde a Dn. Marchione Hospitalio communcatarum solutionum problematis curvis celerrimi descensus a Dn. Io. Bernoullio geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea, Acta erud. 1697, p. 201; Mathematische Schriften, herausgegeben von C. I. Gerhardt, 2. Abteilung, Band 1, Halle 1858, p. 301. Iac. Bernoulli, Solutio problematum fraternorum ………….una cum propositione aliorum, Acta erud. 1697, p. 211; Opera, Genevae 1744. p. 768. G. De L’Hospital, Solutio problematis de linea celerrimi descensus, Acta erud. 1697, p. 217. I. Newton, Epistola missa ad praenobilem virum D. Carolum Montague, in qua solvuntur duo problema mathematicis a Johanne Bernoulli math. cel. proposita, Phil. trans. (London) 1697; Acta erud. 1697, p. 223; Opuscula, Tom. I, Lausannae et Genevae 1744, p. 280. R. Sault, Analytical investigation of the curve of quickest descent, Phil. trans. (London) 1698, p. 425. I. Craig, The curve of quickest descent, Phil. trans. (London) 1701, p. 746. P. St.] Example 2. n 374. If the force acting P is as some power of the abscissa CQ, clearly P = X n , then f ∫ PdX = ( nX+1 ) f . Consequently, the brachistochrone curve AMC is expressed by this n+1 n equation : thus so that :EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 288 Whereby if n = 1 or n > 1, the curve CM or the line BC becomes infinitely great. Moreover the cusp of the curve A, or the place at which the motion starts, has to be taken: Algebraic curves are produced if n = 11+− 22m m with m denoting some positive number. Therefore in these cases n is a negative number less than one, yet thus, so that n + 1 is a positive number. Let m = 1, then n = − 13 . Whereby it makes the integral of which is [p. 183] Which equation free from irrationalities is of the sixth order. In a similar way other algebraic curves can be found, which under certain hypotheses are brachistochrones. Scholium 1. 375. From the solution of the given problems the solution of inverse problems follow, in which the force acting is sought directed downwards, such that the given curve is a brachistochrone. Moreover this curve must have the lowest point C as a horizontal tangent and for some point A, where the motion starts, the tangent is vertical. As if this equation dY = RdX is given for the motion, then Hence it is found that 3 dS If therefore the radius of osculation at M is put as r, since r = − dXddY there is obtained : adY . P = 2rdS Whereby the problem is solved by this single ratio: so that the radius of osculation of the curve at M to a given line, is thus as the sine of the angle that the tangent to the curve makes with the vertical, to the force acting, which is sought. Now the height corresponding to the speed that the body has at M is given by :EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 289 from which it follows that the speed of the body is proportional to the sine of that angle that the tangent to the curve makes with the vertical. As if the curve CMA is a circle described with radius c, then r = c and from which the force becomes : Therefore the force pulling the body down must be proportional to the abscissa AP, to which also the speed is proportional. Scholium 2. 376. The order requires that the brachistochrone line is found for the hypothesis of a force acting pulling downwards, so that we can now determine the brachistochrone lines for the hypothesis of centripetal forces. But the fundamental proposition (361) thus has been prepared, so that elements of the curve Mm and mμ (Fig. 46) and the orthogonal ordinates MP and mp are referred to the axis AP, which in the case of centripetal forces is not easily squared. Indeed it is seen that the elements MG and mH can be considered as converging to the centre of force; but this error arises from this, since the elements MG and mH cannot be parallel, as required by the fundamental proposition, and which it is wrong to disregard. It is evident that the situation might be restored by determining the radius of curvature, which, if MG and mH are parallel to each other, is equal to mG ; [recall in the inserted diagram above, that MG : d . Mm ds = 1 = ds . dθ , or r = 1 . dx = dx / d ( dy / ds ) , ignoring signs; ] which expression dx sin θ dθ dx sin θ dθ cannot be put in place, if MG and mH converge to a centre of force. Whereby, before we approach brachistochrones under the hypothesis of centripetal forces, [p. 185] we will derive a property from the fundamental proposition to accommodate the hypothesis of whatever the forces are acting. From which it is observed that the most celebrated Hermann in Phorononia and others, who gave the brachistochrones for centripetal forces, are in error, since they have used a principle not consistent with the truth, as will soon be indicated. [Iac. Hermann, Phoronomia, seu de viribus et motibus corporum solidorum et fluidorum, Amstelodami 1716, p. 81. Ioh. Machin, Inventio curvae, quam corpus descendens brevissimo tempore describeret, urgente vi centripeta ad datum punctum tendente, quae crescat vel decrescat iuxta quamvis potentiam distantiae a centro; dato nempe imo curvae puncto et altitudine in principio casus, Phil. trans. (London) 1718, p. 860. P. St. ]EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 290 PROPOSITION 42. Theorem. 377. Whatever the forces acting should be, that line is a brachistochrone, upon which the moving body presses with a force that is twice as great, as either the centrifugal force alone or the normal force alone. Demonstration. Whichever and however many the forces acting should be, all these can be resolved into pairs, of which one pulls along the line MG (Fig. 46), and the other along MP. Let that one pulling along MG be equal to P and the one that pulls along MP be equal to Q , and calling AP = x, PM = y and AM = s and likewise the height corresponding to the speed at M is equal to v. From these two forces the tangential force is equal to Pdx −Qdy and the normal force is equal to ds Pdy + Qdx . On this account we have ds dv = Pdx − Qdy . When this expression is compared with that produced above (364), where we put dv = Pdx + Qdy + Rds ; [p. 186] Q is negative [as it acts in the opposite direction, i.e. inwards towards the axis AP;], Pdy + Qdx . But 2rv is the ds Pdy + Qdx centrifugal force, by which the curve at M is pressed, and is the normal force. ds and R = 0. Hence it follows from this to be the case that 2rv = Whereby since the centrifugal force is equal to the normal force, the total compressing force, which acts on the curve, is twice as great as either the centrifugal force alone, or the normal force alone. Q.E.D. [See (369)] Scholium 1. 378. In the following chapter we demonstrate that this same proposition is in place for a medium with some resistance; which we could demonstrate here by the same labour; but since resistance has been designated to the following chapter, it has been considered rather to transfer this theorem there.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 291 Corollary 1. 379. Therefore from this proposition it is easy to determine the brachistochrone for the hypothesis of any forces acting. And now we can present this from the other part above, where we determined the curves in which the total force given had the ratio to the centrifugal force. Corollary 2. ∫ ∫ 380. Since dv = Pdx − Qdy , then v = Pdx − Qdy with these integrals taken so that they vanish on making x and y = 0, [p. 187] if indeed the motion should start from rest at A. Corollary 3. 381. If therefore here the value for v is substituted, this equation is found for the brachistochrone curve : 2 ∫ Pdx − 2 ∫ Qdy Pdy + Qdx

. r ds 3 ds (363) with dx taken constant, since r is put to lie on the part Now we have r = dxddy opposite the axis AP; hence this equation is obtained : Corollary 4. 382. Since this equation is a differential of the second order and thus requires to be integrated twice, whatever the constant that has to be added for the one integration, the other must be effected, so that on making x = 0 also makes y = 0. Therefore an infinity of brachistochrone curves are produced for the hypothesis of the forces acting. And with an arbitrary constant it is able to be effected that the curve can pass through a give point. Corollary 5. 383. The time, in which a body from A arrives at M , is equal to [p. 188] which expression indeed before been found for the equation of the curve; now this time must be the minimum amongst all the other times for all the curves connecting the points A and M. Scholium 2. 384. Again as in whatever hypothesis of the forces acting these curves are described freely, in which the centrifugal force is equal and opposite to the normal force, thus these curves are brachistochrones, in which the normal force is also equal to the centrifugal force, but acting towards the same region. And as that property is common to all the curves freely described in a resistive medium also, thus this property is extended too to all the brachistochrone lines in a resistive medium.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 292 PROPOSITION 43. Problem. 385. If the body is always drawn towards the centre of forces C (Fig. 48), to find the brachistochrone line AM upon which the body reaches M from A the quickest. Solution. The line AC is drawn from the point A, at which the motion begins, to the centre of forces C, [p. 189] likewise MC is drawn with the perpendicular CT to the tangent MT. Putting AC = a, CM = y, CT = p, the centripetal force at M = P and the speed at M corresponds to the height v. With these in place, then we have dv = − Pdy and v = − Pdy , ∫ with this integral thus taken, so that it vanishes on placing y = a. Moreover Pp the normal force is equal to y ; to this the centrifugal force must be equal and acting in the same direction ; for then the brachistochrone curve comes about, as we have shown in the preceding proposition. Therefore the curve must be convex towards the centre C and the radius of osculation falls on the side away from the centre C. Whereby, ydy since this expression dp shows the radius of osculation, in as much as it falls towards the centre, now the expression for the − ydy radius of osculation in our case is dp . Therefore the centrifugal force is equal to Pp − 2vdp 2dp ∫ Pdy = ydy , to which the normal force must be put equal ; from which this ydy y 2dp Pdy equation arises : p = Pdy , the integral of this is : ∫ which is the equation for the curve sought between y and p. But if from the centre C the ys arc MP is drawn, and this is called a , then it becomes :EULER’S MECHANICA VOL. 2. Chapter 2f. page 293 Translated and annotated by Ian Bruce. Hence the equation becomes : and with the value of p substituted from the above equation, there is obtained : which is the equation between y and the arc of the circle s described with the radius a, which is expressed by the angle ACM, from which the construction of the curve sought follows. Q.E.I. [p. 190] Corollary 1. ∫ pp 386. Since the height corresponding to the speed is given by : v = − Pdy = b , the speed of the body at some place is as the perpendicular from C sent to the tangent, and in a similar manner, where in free motion the speed is inversely proportional to this perpendicular. Corollary 2. Pp 387. Let the radius of osculation at M = r; then 2rv = b from the condition of the 2 yv 2 py problem. Hence there is obtained r = Pp = bP . Moreover since at the start of the curve at A, we set p = 0, or AC is the tangent to the curve, and the radius of osculation at A is equal to zero also, except perhaps the centripetal force P likewise vanishes at A. Corollary 3. 388. The body has the maximum speed at the place where dp = 0; moreover there from the equation for the curve it makes dy = 0. Whereby the body moves the quickest in that place where the straight line CM is normal to the curve. Therefore the curve beyond this point recedes from the centre C. Scholium 1. 389. Therefore the speed of the body at the individual points of the brachistochrone is not proportional to the sine of the angle, which the tangent to the curve makes with the p direction of the centripetal force ; [p. 191] for the sine of this angle TMC is y , now the speed has been found to be proportional to p . Indeed this property is in place if the centre of forces is at an infinite distance and the directions of the forces acting are parallelEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 294 dy to each other, as is understood from Prop. 41, where the speed is as ds , i. e. as the sine of the angle, which the element of the curve makes with the direction of the force acting. Moreover the most celebrated Hermann in Comm. Acad. Petrop. A 1727 has attributed this common property to all the brachistochrones in vacuo as in a resistive medium. And on this account not only these lines, which he gave in a medium with resistance for brachistochrones, are not such, but also those in vacuo found for centripetal forces. − Pdy p2 Moreover in this case he found the equation ∫ b = 2 and now clearly in disagreement y with our equation. [Recall that Johannes Bernoulli’s solution had relied on Fermat’s least time principle from optics, whereby one can derive the law of refraction at a plane interface between refracting media, and which can be applied for the uniform force case in mechanics, but which it was unwise to extend to other cases.] Example 1. 390. Let the centripetal force be in proportion to the distance of the body from the centre y of force ; the formula for the force becomes P = f . Whereby this gives Which is the equation for the brachistochrone according to this hypothesis of the centripetal force between p and y. Now the other equation between the arc described with the radius, which is a measure of the angle ACM, and y is this : The lowest point of this curve or nearest to the centre is obtained by putting either [p. 192] dy = 0 or p = y; moreover then it becomes : therefore this is the minimum distance of the curve from the centre C. The radius of osculation of this curve at any point is equal to : Therefore at the point nearest the centre the radius of osculation is a maximum, clearly equal to : The tangent of the angle ACM is put equal to t , with the total sine equal to 1; then we have ds = 1+dttt ; again on putting a this equation is found :EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 295 From which it is understood that the curves are algebraic, as long as b +b2 f is a square number. Again, the length of the curve AM is generally equal to : and in this case it is given by : From which equation it follows that the brachistochrone curve AM is a hypocycloid, which is generated by the rotation of a circle, the diameter of which is equal to : described on the concave side of the periphery AE with centre C and with radius AC. Therefore since b can be taken as you wish, it is apparent that all the hypocycloids arising on the periphery AE are brachistochrones. Example 2. 391. Let the centripetal force be proportional to the square of the distances, so that the force is given by P = y2 ; hence it becomes : f2 [p. 193] Which is the equation for the brachistochrone according to this hypothesis of the centripetal force. Now the other equation between the arc s and y is this : Therefore the lowest point of this curve is determined by putting dy = 0 with the aid of this cubic equation ay 3 + bf 2 y = abf 2 . Moreover this equation between s and y is sufficient for the curve sought to be constructed.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 296 Scholium 2. 392. Therefore from these, which have been presented in this and in the preceding proposition, it is understood, how under any hypothesis of the forces acting, that line can be found upon which a body arrives at a given point in the shortest possible time from a given point. Therefore it is now necessary to determine that line, upon which a body arrives the quickest, not to a given point, but to a given line from a given point, which curve reasonably is one from an infinite number of brachistochrones; but which we will indicate in the following proposition. PROPOSITION 44. Theorem. 393. A body from a given point A (Fig. 49) arrives the quickest at some given line BM on the brachistochrone AM, which crosses the given line BM at right angles, and this under the hypothesis of some force acting. [p. 194] Demonstration. Let AM be that line on which the body from A arrives the quickest at the line BM ; it is evident in the first place that this line is a brachistochrone; for if the line is given, upon which the body the body arrives more quickly at M from A, by that the question is rather satisfied. Besides this line AM crosses at the point M of the curve BM at right angles; for unless it crosses at right angles, by drawing a shorter normal mn, as mn < mM, the body can arrive quicker at the curve BM along Amn rather than along AmM. Whereby without exception it is necessary that the place be found, so that the given curve AM stands normally to the curve. Consequently, the body upon that curve of an infinitude of brachistochrones drawn from A to the curve BM arrives at the curve BM the quickest which crosses curve BM at right angles. Q.E.D. Corollary 1. 394. Therefore if an infinity of curves are sought, upon which arrives at BM from A in a given time, then it is necessary that the given time is greater than the time for the brachistochrone AM; for otherwise the problem becomes impossible.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 297 Corollary 2. 395. If it should happen that more brachistochrone curves are normal to the curve BM, [p. 195] the more maximum and minimum times are produces also. For this method indicates minima as well as maxima times. Corollary 3. 396. Since the time for a brachistochrone curve AM is a minimum, it is understood from the method of maxima and minima, if two nearby brachistochrones are considered standing normal to the curve BM, the times for these are equal to each other. Corollary 4. 397. It is hence again evident, if the curve BM is of such a kind that all the brachistochrones drawn from the point A cut at right angles, the times for all the brachistochrones drawn as far as the curve BM must be equal to each other. Corollary 5. 398. On account of which the curve, which for all the brachistochrone curves drawn from the point A, which is the isochronous arc, or the curve that cuts the curves at the same time of travel, that too cuts all the brachistochrones at right angles, or it is the orthogonal trajectory of these curves. Corollary 6. 399. And it is evident in turn too, if the curve [p. 196] which from the infinitude of curves the isochronous arc cuts, is the orthogonal trajectory of this, then this infinitude of curves are all brachistochrones. Scholium. 400. It is easily understood that this proposition has a place too in a medium with resistance ; for in a like manner it is apparent that the time to pass through the normal element mn on the curve BM to be less that the time to pass through the element mM, which is not perpendicular ; moreover in the whole of this demonstration the force has been in place. Whereby if from the infinitude of curves drawn from the point A the law of the forces acting and the resistance can be found, in which these curves are all brachistochrones, likewise the orthogonal trajectory of these curves can also be shown by seeking only the curve the isochronous arc cuts from these curves. And this method of finding orthogonal trajectories now, the most celebrated Johan Bernoulli used in Act. Lips. A 1697. [see the note after (366)]EULER’S MECHANICA VOL. 2. Chapter 2f. page 298 Translated and annotated by Ian Bruce. PROPOSITION 45. Problem. 401. Between all the curves joining the points A and C (Fig. 50) and equal in length, to determine that curve AMC, upon which a body arrives the quickest from A to C, according to the hypothesis that a uniform force g is acting directed downwards. [p. 197] Solution. With the vertical AP and the horizontal PM drawn, and calling AP = x, PM = y and AM = s, the time in which the arc AM is completed is equal to ds . Now by the ∫ gx method of isoperimetrics, concerning which I gave a singular dissertation with general formulas, from which any problems are able to be easily resolved, in the Comment. Acad. Petr. 1733, [Problematis isoperimetrici in latissimo sensu accepti solutio generalis, p. 123; Opera Omnia, series I, vol. 25; E027], two quantities are to be considered, the arc AM = s = ds and the time to pass ∫ through AM, equal to ∫ dsgx , of which the one with respect to the other has to be a minimum or a maximum. For the same is returned, either if between all the curves of equal length, that curve is sought which has the shortest descent time, or if between all the curves for which the descent is made in the same time that curve is sought which is the shortest. Moreover, from my formulas, ∫ ds gives this quantity diff . ds , and dy ∫ dsgx gives diff . ds gx , of which the one can be put equal to any multiple of the other. dy Hence this equation is obtained to be integrated : or Now by taking the square the equation becomes : hence it becomes :EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 299 From which the curve sought can be found. Q.E.I. Corollary 1. 402. In the equation found there are two arbitrary quantities in there a and m, from which it can be put into effect, that the curve passed through a given point C and that likewise it is of the given length[p. 198]. But then this curve is completed the quickest among all the other curves of the same length passing through A and C. Corollary 2. 403. If a and m are put infinitely large, a cycloid is produced, which not only among all the curves of the same length, but among all the curves entirely, is completed the quickest. Corollary 3. 404. If m is put equal to zero, then there is produced dy = 0 or a vertical straight line. But if a becomes equal to zero, then there arises some straight line drawn through the point A. For it is the straight line among all the lines which is completed in the same time, the minimum or the shortest. Corollary 4. 405. If m is put equal to 1, an algebraic curve is produced ; for the equation becomes : the integral of which is : This curve is indeed rectifiable; as it becomes : Also as well the time for the algebraic arc AM can be expressed ; for it becomes :EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 300 Corollary 5. [p. 199] 406. If on this curve x = a4 is taken, there the tangent is horizontal and the vertical straight line at that point of the curve is the diameter of the curve. Moreover at that place 4a and the length of the curve at that point is equal to 2a . And the time, in which y = 15 5 the arc is completed here, is equal to 4 a . Therefore, in the same time, the body 3 g . descends through the vertical height 4a 9 Scholium. 407. Now with these items concerned with the quickest speed of descent dispatched, we progress to the these curves to be considered, upon which more descents have a given relation between each other to be compared. Here it especially concerns the question of tautochronous curves, upon which either all descents, or the whole oscillations, are made in the same time to the lowest point. Then to this other difficult questions can be added that illustrate the strength of the method that we are using. PROPOSITION 46. Problem. 408. To find the general law of tautochronous curves, upon which all descents to the point A are completed in the same time, with the descent taken to start from any point on the curve AM (Fig. 51). [p. 200] Solution. The right line AP is taken for the axis, calling the part of the curve AM = s, and let the height corresponding to the speed at A be equal to b, and the height corresponding to the speed at M equal to v ; the time in which the arc AM is completed is equal to ds , which integral has thus been taken ∫ v so that it vanishes on putting s = 0. Then, if v is put equal to zero in that integral, the descent time is had from that place in which the speed is zero, as far as the point A or the whole descent time ; that has to be expressed by the same amount, whatever the quantity b should be. Therefore neither must this quantity b be present in the expression for the time nor in the expression for the curve AM, since the same descent time must be produced by these same curves, however b is changed. Now let the quantity v be composed from the letter z, related to the curve and with the curve AM only depending on the abscissa AP and the applied line PM , and not involvingEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 301 b, and from the letter h, which is composed from b and from constant quantities. Moreover let v be such a function of of h and z, that it vanishes on putting z = h and so that it becomes equal to b, if z = αh with α taken as some number. Again on putting ds = pdz for the equation of the curve sought, p must be such a quantity, in which neither b nor h is contained, since these lettes cannot arise in the equation of the curve. Therefore we have for the expression of the time to complete AM, ∫ v with this pdz integral thus taken, so that it vanishes on putting v = b or z = αh . Hence this integral, if z = h is put in that, [p. 201] gives the whole descent time, in which h cannot be present. Now this comes about, if ∫ v is a function of h and z of zero dimensions, or if pdz pdz is a v function of zero dimensions. Let v be a function of m dimensions of h and z; then it must m−2 be that p = Cz 2 with C denoting a constant quantity not depending on b. Hence as often as such a function can be ascertained for v, so this equation is obtained for the curve sought: if the constant is needed, so that s vanishes if either x or y vanishes at z. Q.E.I. Corollary 1. 409. Therefore where this method can be used, it is required that v is an expression from finite quantities and that this expression can be changed into a homogeneous function from constant h and z. Corollary 2. 410. On account of this it is necessary that v = b, if we put z = αh , where h is not found in the constant quantity added. Therefore it is sufficient, if we had considered making v = b by putting z = αh , nor is there a need [to add a constant], as the integration is completed. Corollary 3. 411. Also it is understood that the curve at A must have a tangent normal to the directing of the force acting ; [p. 202] also unless this is the case, the time to descend an infinitely small arc becomes small too. Scholion. 412. This solution prevails not only if as the figure indicates, the curve is set out by orthogonal coordinates; for nothing is different, for whatever quantities we wish to set out for the nature of the curve, as long as b does not enter into z. Moreover z can contain lines and any quantities depending on the curve. Therefore for this method in a vacuum, under any hypothesis of the forces acting, tautochronous lines can always be found, since the speed can always be expressed by finite quantities. But if, as in mediums with resistance it is customary to happen, the speed cannot be shown by finite quantities, then this method cannot be used, but another way is desired which is successful only if the speed is given by a differential equation.EULER’S MECHANICA VOL. 2. Chapter 2f. page 302 Translated and annotated by Ian Bruce. PROPOSITION 47. Problem. 413. If the body is acted on by some downwards force, to find the tautochrone line upon which all the descents are made in the same time. Solution. [p. 203] On putting AP = x, PM = y and AM = s (Fig. 51) let the speed at A correspond to the height b and at M it corresponds to the height v. Again let the force acting at M = P; then it becomes v = b − Pdx with the integral Pdx thus taken, in ∫ ∫ order that it vanishes on making x = 0. Now if on putting b = h and Pdx = z , v is a function of one ∫ dimension of h and z and vanishes on making z = h and it becomes v = b by making z = 0. Therefore m = 1 and thus this equation is found for the curve sought (408) : ∫ ds = Cdz and s = 2 az = 2 a Pdx. z If the equation between x and y is desired, then on account of ds = aPdx , a ∫ Pdx Q.E.I. Corollary 1. 414. Since the integral ∫ Pdx vanishes on putting x = 0, to tangent to the curve at A is horizontal, unless P vanishes at A. And the curve has a vertical tangent somewhere on the curve and there generally a cusp ; this comes about where the integral becomes ∫ Pdx = aP . For there dy = 0. 2 Corollary 2. 415. At the lowest point of this curve at A the radius of osculation is equal to the ydy , since at A, s and y become equal. Whereby at A the radius of subnormal dx = sds dx osculation is equal to 2aP, where P denotes the force acting at the point A.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 303 Corollary 3. [p. 204] 416. From the radius of osculation at A and the force acting at A, the time for the ascent or descent is found for the infinitely small arc equal to : (172). And the time of each descent is equal to this time. Hence, under the hypothesis of gravity equal to 1, a pendulum of length 2a completes an infinitely small descent in the same time. Example 1. 417. Let the force acting everywhere be constant, and certainly P = g; the integral becomes Pdx = gx and s = 2 gax ∫ likewise thus it is understood that the curve is a cycloid convex downwards, clearly congruent with the brachistochrone line under the same hypotenuse of the force. Moreover since all the descents are made in the same time on the cycloid, that we have shown above. (187). Example 2. 418. Let the force acting be as some power of x ; the force becomes if indeed n + 1 is a positive number; but if indeed n + 1 is a negative number, then the integral Pdx =∝ . Therefore we have ∫ From which equation it is understood that the curve is a straight line inclined at some angle to the horizontal if n = 1. [p. 205] But if n > 1, the curve at the starting point A becomes imaginary, clearly until f n begins to be less than ( n + 1 )ax n −1 . Scholium 1. 419. From the general equation it is apparent that the line AP is the diameter of the curve. Whereby when it is in a vacuum, the ascents are similar to the descents, all the semi oscillations on the curve MA and on the other part as far as produced are also isochrones and consequently also the whole oscillations. Then since on account of the arbitrary a, there is an infinite number of tautochronous curves AM, and two which are joined at the point A so that they have a common horizontal tangent there, so they produce semi-EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 304 oscillations as well as whole isochronal oscillations, as if a pendulum is permitted thus to be accommodated, so that by oscillating it completes curves of this kind. Scholium 2. 420. Also it is understood from the solution that these curves we have found are the unique ones which satisfy the question. For in place of p another function of z cannot be substituted, as in the integral, if v is put equal to 0, clearly from the formula b or h emerge. That which by other methods, in which tautochrones are found, is not clear enough. Scholium 3. ∫ 421. Since the arc is given by s = 2 a Pdx , then it arises that P = 2sds . From which it adx is apparent, that the force acting must be of this kind, in order that the given curve is a tautochrone [p. 206]. Clearly the force acting downwards must be proportional to sds dx from the given curve taken. Whereby, unless the curve is rectifiable, the value of the force acting cannot be shown to be algebraic. PROPOSITION 48. Problem. 422. If the body is always drawn towards the centre of forces C (Fig. 52) by some force, to find the tautochrone line BMA, upon which the body completes all the descend as far as the point A in the same time. Solution. Call CA = c, CM = y and the centripetal force acting at M is equal to P. Again let the speed at A correspond to the height b and that at M to the height v. The integral Pdy is taken ∫ thus, so that it vanishes on putting y = c, with which done, the equation becomes v = b − Pdy . Therefore by taking b for h ∫ ∫ and Pdy for z , there is a function of h and z, which is equal to v, of one dimension ; whereby m is equal to 1. If now the arc AM is said to be ∫ equal to s, it gives s = 2 az = 2 a Pdy and hence ds = aPdy . If now at M the tangent is drawn, and to the same line the a ∫ PdyEULER’S MECHANICA VOL. 2. Chapter 2f. page 305 Translated and annotated by Ian Bruce. perpendicular CT is sent from the centre C, which is called p, then ∫ ydy ( y2 − p2 ) Whereby the equation is obtained : y 2 Pdy = ( y 2 − p 2 )aP 2 or p 2 = y 2 − = ds . y 2 ∫ Pdy , the aP 2 equation for the curve sought. Q.E.I. Corollary 1. [p. 207] 423. At the point A, where ∫ Pdy vanishes, there p = y, or the right line CA is normal to the curve , and on that account the speed of the body is a maximum since A is the point of the curve closest to the centre C. Corollary 2. 424. If we put p = 0, the point of the curve B is obtained, at which the right line CB is a tangent to the curve. And at that point, which is the maximum, the curve has a cusp. Now ∫ the point B is found from this equation : aP 2 = Pdy ; and y cannot be greater than the value found from this equation. Corollary 3. ∫ 425. Also it is apparent from the equation found : s = 2 a Pdy , since an ambiguous sign is involved with the square root, that the curve has two branches AB and AD similar and equal to each other, and on this account the oscillations which occur on the curve BAD are equal to each other. Corollary 4. 426. The radius of osculation at the point A is equal to c2−acP . And since AC = c, the 2 aP time in which one infinitely small descent is completed on the portion of the curve AB, is equal to π a (207); therefore all the times are equal to this time. [p. 208] On this account, since the oscillations, which are made on the curve BAD, are isochronous with the oscillations of a pendulum under the hypothesis of gravity equal to 1, of which the length is equal to 2a. Example 1. 427. Let the centripetal force be directly proportional to the distances from the centre, so y that it is given by P = f ; the integral becomes ∫ y 2 −c 2 Pdy = 2 f . Hence the are therefore becomes : AM = s = 2 a( y 2 − c 2 ) f ( yy − c 2 ) 2 and p

yy − . f 2aEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. The radius of osculation of this curve at the point M which is 2a 2a − f page 306 ydy , is found to be dp ( 2a − f ) y 2 + fc 2 ) . 2a From which it follows, if 2a < f, that the curve has the convex side turned towards the centre C, as the figure shows. But if 2a = f, the curve becomes a right line normal at A to the line AC. Moreover the point B, where CB is a tangent to the curves, is found from this equation ( f − 2a ) yy = ccf , from which it formed : BC = c f ( f − 2a ) . Therefore as often as it happens that f > 2a or the curve is convex towards C, the curve has a cusp at B. And in these cases the curve is a hypocycloid, which is generated by the rotation of a circle, the diameter of which is equal to c f −c ( f − 2a ) ( f − 2a ) radius equal to , upon the concave part of a circle with centre C described by the c f ( f − 2a ) . Therefore in this case the tautochrone curves agree with the brachistochrone curves found above (390). But if 2a > f, in which case the curve is concave towards C, then BC becomes imaginary and the curve AM no further is a hypocycloid. [p. 209] Moreover then we have p2 = ( 2a − f ) yy + ccf , hence p everywhere besides A is greater than AC. Let c = 0; then it 2a becomes p = y 2a − f , whereby in this case the tautochrone curves are logarithmic 2a spirals described around the centre C. Clearly a body on logarithmic spirals always arrives at the centre C in the same time, from wherever the descent started. Therefore the tautochrones are under this hypothesis of the centripetal force : first all the hypocycloids, then all the right lines drawn in some manner, thirdly all the logarithmic spirals and in the fourth place all the infinitude of other curves contained in this equation : p2 = ( 2a − f ) yy + ccf , if indeed it should be that 2a > f and c is not equal to 0. Moreover 2a under this hypothesis for tautochrones Newton in the Principia [Book I, Prop. LI, theorem XVIII.] and Hermann in Phoronomia and in the Comment. Acad. Petrop. A 1727, [p. 139, see in particular p.150; P.St.] have given only hypocycloids, although here our equally general equation can be had. Example 2. 428. The centripetal force is put inversely proportional to the square of the distances from ff the centre, so that P = yy ; the integral becomes: Therefore the curve AM is equal toEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 307 hence the radius of osculation is found: Therefore where p vanishes, there also the radius of osculation is equal to 0. Moreover the value of BC is itself found from this equation : y 4 = cy3 + acff . [p. 210] Therefore if − ck , that which can be assumed, since k is an arbitrary we put BC = k, then a = k cff 4 3 quantity. Moreover it is apparent that this curve between A and B must have a turning point, which arises if the curve at A is concave towards C, clearly if 2aff > c3. Example 3. 429. Let the centripetal force be constant everywhere or P = 1; then we have Pdy = y − c. Whereby if from the centre C with radius CM the arc MP is drawn, it ∫ becomes ∫ Pdy = AP and the arc AM = 2 a( y − c ) = 2 a AP . But again it becomes • hence BC = a + c is found and the curve AB = 2a = 2( BC − AC ) . Now the radius of osculation at the point M is equal to Therefore at the lowest point A the radius of osculating is equal to 22aac . Therefore the −c curve at A is concave towards C, if 2a > c, but convex if c > 2a, and the radius of osculation at A is infinitely large, if 2a = c. In the first case, in which at A the curve is concave towards C, the curve has a turning point, where If c = 0 thus so that the point A falls on the centre C, y becomes the chord of this curve and AM = 2 ay , the radius of osculation of this curve at the centre C is infinitely small, and p = y a− y ; moreover the curve can itself be constructed from the quadrature of the a circle. [p. 211]EULER’S MECHANICA VOL. 2. Chapter 2f. page 308 Translated and annotated by Ian Bruce. CAPUT SECUNDUM DE MOTU PUNCTI SUPER DATA LINEA IN VACUO. [p. 178] PROPOSITIO 41. Problema. 367. Si corpus perpetuo deorsum trahatur vi quacunque, invenire lineam brachystochronam AMC (Fig. 46),super qua corpus citissime ex A ad C descendit. Solutio. [p. 179] Positis AP = x, PM = y et arcu AM = s sit vis, quae corpus in M deorsum urget, = P; erit v = Pdx hoc integrali ita accepto, ut ∫ evanescat posito x = 0, si quidem corpus motum in A ex quiete inchoare ponitur, atque dv = Pdx. Erit ergo du = Pdx = dv et ddw = 0, quia du invariatum manet eunte m in n. Quocirca habebitur ista aequatio dy dydv 2vd . ds = ds , cuius integralis est dy l av = 2l ds seu vds 2 = ady 2 , hincque ∫ ∫ dx 2 Pdx = ady 2 − dy 2 Pdx. Quamobrem pro lineae brachystochrona quaesita habebitur ista aequatio dy = dx ∫ Pdx , ( a − ∫ Pdx ) in qua indeterminatae x et y sunt a se invicem separatae. Curvae autem longitudo habetur ex hac aequatione ds = dx a . ( a − ∫ Pdx )EULER’S MECHANICA VOL. 2. Chapter 2f. page 309 Translated and annotated by Ian Bruce. Q.E.I. Corollarium 1. ∫ Pdx = 0 , erit dy = 0, seu tangens curvae in A erit verticalis incidens in AP. At ubi fit ∫ Pdx = a , ibi tangens curvae erit 368. In A igitur, ubi celeritas corporis evanescit seu horizontalis. Corollarium 2. 369. Quia ddw = 0 et du = Pdx, erit 2v = Pdy r ds Pdy (363). Est vero ds vis normalis, [p. 180] qua curva in M secundum normalem versus axes AP ductam premetur. Consequenter vis normalis est aequalis vi centrifugae et eandem plagam tendens. Quocirca linea brachystochrona hanc habet proprietatem, ut tota pressio, qua curva premitur, sit duplo maior quam vis normalis sola. In sequentibus vero demonstrabimus hanc proprietatem in omnibus lineis brachystochronis sive in vacuo sive in medio resistente locum habere. Corollary 3. 370. Propter arbitrariam a dantur infinitae curvae brachystochronae omnes in A initium habentes. Atque hac litera a effici potest, ut curva ex A per datum punctum C transeat, quae erit linea inter A et C, super qua tempus est minimum. Corollarium 4. 371. Quia curva AMC (Fig. 47) alicubi habet tangentem horizontalem, sit ea BC et in C sumatur alius axis verticalis CQ. Sit CQ = X, QM = Y et CM = S; erit dX = −dx , dY = −dy , et dS = −dS atque ∫ Pdx = a − ∫ PdX integrali ∫ PdX ita accepto, ut evanescat posito X = 0. Ad hunc ergo axem CQ si curva referatur, habebitur ista aequation dY = dX ( a − ∫ PdX ) ∫ PdX seu dS = dX a . ∫ PdX Corollarium 5. 333. Hae ergo omnes curvae ad utramque partem axis CQ duos arcus habent similes et aequales. [p. 181] Simili modo ad utramque partem axis AB curva aequaliter est disposita. Quamobrem huiusmodi curvae infinitas diametros habebunt inter se parallelas et ad distantiam BC positis, nisi forte potentia sollicitans ita accipiatur, ut supra A sit negativa, quo casu curva CMA sursum tendere poterit et partem concavam deorsum convertere.EULER’S MECHANICA VOL. 2. Chapter 2f. page 310 Translated and annotated by Ian Bruce. Exemplum 1. 373. Sit potentia sollicitans uniformis seu P = g; erit ∫ Pdx = gx; unde loco a posito gb pro brachystochrona in hac potentiae sollicitantis hypothesi habebitur ista aequatio dy = dx x ( b− x ) seu ds = dx b . ( b− x ) At si aequatio ad axem CQ referatur, erit dY = dX ( b − X ) X seu dS = dX b , X cuius integralis est S = 2 bX . Ex qua aequatione patet curvam esse cycloidem super basi horizontali a circulo diametri b descriptam et deorsum conversam, quemadmodum hoc a Cel. Ioh. Bernoulli aliisque eximiis geometris iam pridem est inventum. Si itaque dentur duo quaecunque puncta A et M, linea, super qua corpus ex A citissime ad M descendit, invenitur, si describatur cyclois cuspidem in A et basem horizontamem habens atque per punctum M transiens; id quod ex eo, quod omnes cycloides sunt curvae similes, ex unica descripta cycloide facile efficitur. Tempus autem, quo corpus ex A ad M pertingit quodque est minimum, erit = ∫ g( bx − x ) dx b 2 et curvae AM longitudo erit = ∫ ( b− x ) = 2b − 2 b( b − x ). dx b Cum autem sit PM = y = ∫ ( bxdx− xx ) , [p. 182] erit tempus per AM = 2 y + 2 ( bx − xx ) gb = arcui in circulo diametri b, cuius sinus versus est = x, ducto in 2 . gb [G.G. Leibniz, Cummunicatio suae pariter duarumque alienarum ad edendum sibi primum a Dn. Io. Bernoullio, deinde a Dn. Marchione Hospitalio communcatarum solutionum problematis curvis celerrimi descensus a Dn. Io. Bernoullio geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea, Acta erud. 1697, p. 201; Mathematische Schriften, herausgegeben von C. I. Gerhardt, 2. Abteilung, Band 1, Halle 1858, p. 301. Iac. Bernoulli, Solutio problematum fraternorum ………….una cum propositione aliorum, Acta erud. 1697, p. 211; Opera, Genevae 1744. p. 768. G. De L’Hospital, Solutio problematis de linea celerrimi descensus, Acta erud. 1697, p. 217. I. Newton, Epistola missa ad praenobilem virum D. Carolum Montague, in qua solvuntur duo problema mathematicis a Johanne Bernoulli math. cel. proposita, Phil. trans.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 311 (London) 1697; Acta erud. 1697, p. 223; Opuscula, Tom. I, Lausannae et Genevae 1744, p. 280. R. Sault, Analytical investigation of the curve of quickest descent, Phil. trans. (London) 1698, p. 425. I. Craig, The curve of quickest descent, Phil. trans. (London) 1701, p. 746. P. St.] Exemplum 2. n 374. Si potentia sollicitans P fuerit ut potestas quaecunque abscissae CQ, nempe P = X n , f erit ∫ PdX = ( nX+1 ) f . Consequenter curva brachystochrona AMC exprimetur hac n+1 n aequatione ita ut sit Quare si fuerit vel n = 1 vel n > 1, curva CM erit infinite magna seu ipsa recta BC. Cuspis autem curvae A seu locis, in quo motus incipit, habetur sumendo Curvae prodibunt algebraicae, si fuerit n = 11+− 22m m denotante m numerum integrum affirmativum quemcunque. His igitur casibus erit n numerus negativus unitate minor, ita tamen, ut n + 1 sit numerus affirmativus. Sit m = 1, erit n = − 13 . Quare fiet cuius integralis est [p. 183] Quae aequatio ab irrationalitate liberata fit ordinis sexti. Simili modo aliae curvae algebraicea invenientur, quae in certis hypothesibus sunt brachystochronae.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 312 Scholion 1. 375. Ex data problematis solutione sequitur simul solutio problematis inversi, quo quaeritur potentia sollicitans deorsum directa talis, ut data curva sit brachystochrona. Debet autem haec curva in puncto infimo C habere tangentem horizontalem et alicubi in A, ubi est motus initium, tangentem horizontalem et alicubi in A, ubi est motus initium, tangentem verticalem. Ut si fuerit aequatio pro curva data haec dY = RdX , erit Unde invenitur 3 adY . dS Si ergo radius osculi in M ponatur r, propter r = − dXddY habebitur P = 2rdS Quare problema hac unica solventur analogia : ut radius osculi curvae in M ad lineam datam, ita sinus anguli, quem tangens curvae in M cum verticali facit, ad potentiam sollicitantem, quae quaeritur. Altutudo vero debita celeritati, quam corpus in M habet, est ex quo sequitur celeritatem corporis esse illi ipsi sinui anguli, [p. 184] quem tangens curvae cum verticali constituit, proportionalem. Ut si sit curva CMA circulus radio c descriptus, erit r = c et ex quibus fit Vis ergo corpus deorsum trahens proportionalis esse debet abscissae AP, cui etiam celeritas est proportionalis. Scholion 2. 376. Inventa linea brachystochrona pro hypothesi potentiae sollicitantis deorsum tendentis ordo requireret, ut linaes brachystochronas in hypothesi virium centripetarum determinaremus. At proposito fundamentalis (361) ita est comparata, ut elementa curvae Mm et mμ (Fig. 46) ad axem AP et ordinatas orthogonales MP, mp referantur, quod ad casum virium centripetarum non commode quadrat. Videntur quidem elementa MG et mH ut convergentia ad centrum virium considerari posse; sed hic ipse error, qui ex hoc oritur, quod elementa MG et mH non essent parallela, ut propositio fundamentalis requirit, perperam negligitur. Perspicuum hoc reddi potest determinando radio osculi, qui, mG ; quae autem expressio non locum si MG et mH fuerint inter se parallela, est = MG : d . Mm habet, si MG et mH ad centrum virium convergunt. Quare antequam ad brachystochronasEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 313 in hypothesi virium centripetarum accedamus, [p. 185] ex propositione fundamentali generalem derivabimus proprietatem cuicunque potentiarum sollicitantium hypothesi accommodatam. Ex quibus perspicietur Cel. Hermannum in Phorononia aliosque, qui brachystochronas pro viribus centripetis dederunt, esse deceptos, dum usi sunt principio cum veritate non consentaneo, ut mox indicabitur. [Iac. Hermann, Phoronomia, seu de viribus et motibus corporum solidorum et fluidorum, Amstelodami 1716, p. 81. Ioh. Machin, Inventio curvae, quam corpus descendens brevissimo tempore describeret, urgente vi centripeta ad datum punctum tendente, quae crescat vel decrescat iuxta quamvis potentiam distantiae a centro; dato nempe imo curvae puncto et altitudine in principio casus, Phil. trans. (London) 1718, p. 860. P. St. ] PROPOSITIO 42. Theorema. 377. Quaecunque fuerint potentiae sollicitantes, ea linea erit brachystochrona, quam corpus super ea motum premit vi duplo maiore, quam est vel sola vis centrifuga vel sola vis normalis. Demonstratio. Quaecunque et quotcunque fuerint potentiae sollicitantes, eae omnes in binas resolvi possunt, quarum altera trahat secundum MG (Fig. 46), altera secundum MP. Sit illa secundum MG trahens = P et, quae secundum MP trahit, = Q et dicantur AP = x, PM = y et AM = s itemque altitudo celeritati in M debita = v. Erit ex his duabus viribus vis tangentialis = normalis = Pdx−Qdy et vis ds Pdy+ Qdx . Hanc ob rem erit ds dv = Pdx − Qdy . Cum hac expressione comparetur, quod supra (364) est allatum, ubi posuimus dv = Pdx + Qdy + Rds ; [p. 186] erit Q negativum et R = 0. Sequitur Pdy + Qdx . At est ergo exinde fore 2rv = ds 2v vis centrifuga, qua curva in M premitur, et Pdy+ Qdx est vis normalis. Quare cum vis r ds centrifuga sit aequalis vi normali, tota pressio, quam curva sustinet, duplo maior est quam vel sola vis centrifuga vel sola vis normalis. Q.E.D.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 314 Scholion 1. 378. In sequente capite demonstrabimus hanc eandem propositionem locum etiam habere in medio quocunque resistente; id quod quidem eadem opera hic demonstrare potuissemus ; sed quia resistentiae sequens caput est destinatum, eo potius hoc theorema transferre visum est. Corollarium 1. 379. Ex hac igitur propositione facile erit in quacunque potentiarum sollicitantium hypothesi brachystochronas determinare. Hocque ipsum iam ex aliqua parte supra praestitimus, ubi curvas determinavimus, in quibus pressio totalis datam habeat rationem ad vim centrifugam. Corollarium 2. ∫ ∫ 380. Cum sit dv = Pdx − Qdy , erit v = Pdx − Qdy his integralibus ita accipiendis, ut evanescant factis x et y = 0, [p. 187] si quidem motus in A ex quiete incipere debet. Corollarium 3. 381. Si ergo hic pro v inventus valor substituatur, habebitur aequatio pro curva brachystochrona haec 2 ∫ Pdx − 2 ∫ Qdy Pdy + Qdx

. r ds 3 ds (363) sumto dx pro constante, quia in parem oppositam axi AP cadere Est vero r = dxddy r ponitur; unde habetur haec aequatio Corollarium 4. 382. Quia haec aequatio est differentialis secundi gradus atque ideo duplicem integrationem requirit, altera integratione constans quaevis poterit adiici, altera effici debet, ut facto x = 0 fiat quoque y = 0. Infinitae ergo prodeunt curvae brachystochronae pro eadem potentiarum sollicitantium hypothesi. Atque constante arbitraria effici poterit, ut curva per datum punctum transeat. Corollarium 5. 383. Tempus, quo corpus ex A ad M pervenit, est = [p. 188] quae quidem expressio prius ex aequatione curvae est investiganda; minimum vero hoc tempus esse debet inter omnia alia tempora motuum per curvas omnes puncta A et M iungentes.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 315 Scholion 2. 384. Quemadmodum porro in quacunque potentiarum sollicitantium hypothesi eae curvae libere describuntur, in quibus vis centrifuga aequalis est et contraria vi normali, ita eae curvae erunt brachystochronae, in quibus vis normalis quoque aequalis est vi centrifugae, sed in eandem plagam tendens. Atque quemadmodum illa proprietas communis est omnium curvarum libere descriptarum etiam in medio resistente, ita haec quoque propriets ad omnes lineas brachystochronas in medio resistente extenditur. PROPOSITIO 43. Problema. 385. Si corpus in quacunque perpetuo trahatur ad centrum virium (Fig. 48), invenire lineam brachystochronam AM, super qua corpus ex A citissime ad M pertinget. Solutio. A puncto A, in quo motus initium ponitur, ad centrum virium C ducatur recta AC, [p. 189] item MC et in tangentem MT ex C perpendiculum CT. Ponatur AC = a, CM = y, CT = p, vis centripeta in M = P et celeritas in M debita altitudini v. His positis erit dv = − Pdy et v = − Pdy hoc integrali ∫ ita accepto, ut evanescat posito y = a. Pp Vis normalis autem erit = y ; cui aequalis esse debet vis centrifuga et in eandem plagam tendens; tum enim proveniet curva brachystochrona, ut propositione praecedente demonstravimus. Curva igitur debebit esse convexa versus centrum C et radius osculi in partem aversam a centro C cadet. Quare, cum haec ydy expressio dp exhibeat radium osculi, quatenus versus centrum cadit, erit vera − ydy radii osculi expressio in nostro casu dp . Vis igitur centrifuga erit =EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 316 − 2vdp 2dp ∫ Pdy Pp = ydy , cui aequalis poni debet vis normalis y ; ex quo oritur haec aequatio ydy 2dp Pdy = Pdy , cuius integralis est p ∫ quae est aequatio pro curva quaesita inter y et p. At si centro C ducatur arcus MP hicque ys dicatur a , erit Hinc fiet atque valore ipsius p ex superiore aequatione substituto habebitur quae est aequatio inter y et arcum circuli s radio a descriptum, qui metitur angulum ACM, ex qua fluit constructio curvae quaesitae. Q.E.I. [p. 190] Corollarium 1. ∫ pp 386. Quia altitudo celeritati debita est v = − Pdy = b , celeritas corporis in quovis loco erit ut perpendiculum ex C in tangentem demissum, simili modo, quo in motu libero celeritas est huic perpendiculo reciproce proportionalis. Corollarium 2. Pp 387. Sit radius osculi in M = r; erit 2rv = b ex conditione problematis. Hinc ergo 2 yv 2 py habebitur r = Pp = bP . Quia autem in initio curvae in A est p = 0 seu AC tangens curvae, erit radius osculi quoque in A = 0, nisi forte simul vis centripeta P in A evanescat. Corollarium 3. 388. Maximum corpus habebit celeritatem in loco, ubi dp = 0; ibi autem ex aequatione pro curva fit dy = 0. Quare in eo loco corpus celerrime movetur, ubi recta CM in curvam est normalis. Curva ergo ultra hoc punctum a centro C recedit.EULER’S MECHANICA VOL. 2. Chapter 2f. page 317 Translated and annotated by Ian Bruce. Scholion 1. 389. Celeritas ergo corporis in singulis brachystochronae punctis non est proportionalis sinui anguli, quem tangens curvae cum directione vis centripetae constituit; [p. 191] huius p enim anguli TMC sinus est y , celeritas vero ipsi p inventa est proportionalis. Haec quidem proprietas locum habet, si centrum virium infinitae distat et directiones vis dy sollicitantis sunt inter se parallelae, ut ex prop. 41 intelligitur, ubi celeritas erat ut ds , i. e. ut sinus anguli, quem elementum curvae cum directione potentiae sollicitantis constituit. Hanc autem proprietatem Cel. Hermannus in Comm. Acad. Petrop. A 1727 omnibus brachystochronis tam in vacuo quam in medio resistente commumem esse est arbitratus. Atque hanc ob rem non solum eae lineae, quas in medio resistente pro brachystochronis dedit, tales non sunt, sed etiam quas in vacuo pro viribus centripetis − Pdy p2 invenit. Hoc autem casu invenit hanc aequationem ∫ b = 2 a nostra atque vera y aequatione prorsus discrepantem. Exemplum 1. y 390. Sit vis centripeta ipsis distantiis corporis a centro proportionalis; fiet P = f . Quare erit Quae est aequatio pro brachystochrona in hac vis centripetae hypothesi inter p et y. Altero vero aequatio inter arcum s radio a descriptum, que est mensura anguli ACM, et y est haec Huius curvae punctum infimum seu centro proximum habetur [p. 192] ponendo vel dy = 0 vel p = y; tum autem erit haec ergo est minima curvae a centro C distantia. Radius osculi huius curvae in quovis puncto est = In puncto ergo centro proximo radius osculi est maximus, quippe = Ponatur tangens anguli ACM = t posito sinu toto = 1; erit ds = 1+dttt ; ponatur porro aEULER’S MECHANICA VOL. 2. Chapter 2f. page 318 Translated and annotated by Ian Bruce. habebitur ista aequatio Ex quo intellititur curvam toties esse algebraicam, quoties est b +b2 f numerus quadratus. Longitudo curvae AM est porro generaliter = hoc ergo casu erit Ex qua aequatione sequitur curvam brachystochronam AM esse hypocycloidem, quae generatur rotatione circuli, cuius diameter est = super concava parte peripheriae AE centro C radio AC descriptae. Cum igitur b pro lubitu accipere liceat, apparet omnes hypocycloides super peripheria AE natas esse brachystochronas. Exemplum 2. 391. Sit vis centripeta proportionalis quadratis distantiarum, ut sit P = y2 ; unde erit f2 [p. 193] Quae est aequatio pro brachystochrona in hac vis centripetae hypothesi. Altero vero aequatio inter arcum s et y erit ista Huius ergo curvae punctum infimum posito dy = 0 determinabitur ope huius aequationis cubicae ay 3 + bf 2 y = abf 2 . Ceterum ista aequatio inter s et y sufficit ad curvam quaesitam construendam. Scholion 2. 392. Ex his igitur, quae in hac et praecedentibus propositionibus allata sunt, intelligitur, quomodo in quacunque potentiarum sollicitantium hypothesi ea linea sit invenienda super qua corpus ex dato puncto ad datum punctum citissime perveniat. Nunc ergo etiam determinari oportet eam lineam, super qua corpus a dato puncto citissime non ad datum punctum, set ad datam lineam perveniat. quae sane curva una erit infinitis brachystochronis; at quaenam ea sit, in sequente propositione declaribimus.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 319 PROPOSITIO 44. Theorema. 393. Corpus a dato puncto A (Fig. 49) ad quamvis lineam datam BM celerimme pervenit super linea brachystochrona AM, quae datae linea BM ad angulos rectos occurrit, hocque in quacunque potentiarum sollicitantium hypothesi. [p. 194] Demonstratio. Sit AM ea linea super qua corpus ex A citissime ad lineam BM perveniat; perspicuum est primo hanc lineam fore brachystochronam; nam si daretur linea, super qua corpus citius ab A ad M pervenerit, ea potius quaesito satisfaceret. Praeterea haec linea AM ad angulos rectos in M curvae BM occurrit; nisi enim ad angulos rectos occurret ducta minima normali mn ob mn < mM corpus citius per Amn ad curvam BM perveniret quam per AmM. Quare ne haec exceptio locum invenire possit, necesse est, ut curva AM datae curvae normaliter insistat. Consequenter corpus super ea infinitarum brachystochronarum ex A ad curvam BM ductarum citissime ad curvam BM pervenit, quae curvae BM ad angulos rectos occurrit. Q.E.D. Corollarium 1. 394. Si ergo infinitae curvae quaerantur, super quibus corpus dato tempore ab A ad BM perveniat, oportet, ut datum tempus sit maius quam tempus per brachystochronam AM; alias enim problem fieret impossibile. Corollarium 2. 395. Si accidat, ut plures curvae brachystochronae sint normales in curvam BM, [p. 195] plura quoque prodibunt tempora minima vel maxima. Haec enim methodus tam minima quam maxima declarat. Corollarium 3. 396. Quia tempus per curvam brachystochronam AM est minimum, intelligitur ex methodo maximorum et minimorum, si duae brachystochronae proximae concipiantur normaliter insistentes curvae BM, tempora per eas esse inter se aequalia. Corollarium 4. 397. Hinc porro perspicitur, si curva BM fuerit eiusmodi, ut omnes brachystochronas ex puncto A ductas secet ad angulos rectos, tempora per omnes brachystochronas ad curvam BM usque ductas fore inter se aequalia.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 320 Corollarium 5. 398. Quamobrem curva, quae ab omnibus curvis brachystochronis ex puncto A ductis arcus isochronos seu eodem tempore percursos abscindit, ea quoque omnes brachystochronas ad angulos rectos secabit seu erit illarum traiectoria orthogonalis. Corollarium 6. 399. Atque vicissim quoque perspicitur, si curva, [p. 196] quae ab infinitis curvis arcus isochronos abscindit, fuerit earum traiectoria orthogonalis, eas infinitas curvas omnes esse brachystochronas. Scholion. 400. Facile intelligitur hanc propositionem locum quoque habere in medio resistente; simili enim modo apparet tempus per elementum mn normale in curvam BM minus esse quam tempus per elementum mM, quod non est perpendiculare; in hoc autem totius demonstrationis vis est sita. Quare si infinitis curvis ex puncto A eductis lex potentiarum sollicitantium et resistentiae poterit inveniri, in qua eae curvae omnes sint brachystochronae, simul harum curvarum traiectoria orthogonalis poterit exhiberin quaerendo tantum curvam ab iis curvis arcus isochronos abscindentem. Atque hanc ipsam traiectorias orthogonales inveniendi methodum iam adhibuit Cel. Ioh. Bernoulli in Act. Lips. A 1697. [vide notam post (366)] PROPOSITIO 45. Problema. 401. Inter omnes curvas puncta A et C (Fig. 50) iungentes et aequaliter longas eam determinare AMC, super qua corpus celerrime ex A ad C perveniat, in hypothesi potentiae sollicitantis unformis g et deorsum directae. [p. 197] Solutio. Ducta verticali AP et horizontali PM dicatur AP = x, PM = y et AM = s eritque tempus, quo arcus AM absolvitur, = ds . Iam per methodum ∫ gx isoperimetricorum, de qua peculiarem dedi dessertationem cum formulis generalibus, ex quibus quaevis problemata facile resolvi possunt, in Comment. Acad. Petr. 1733, [Problematis isoperimetrici in latissimo sensu accepti solutio generalis, p. 123; Opera Omnia, series I, vol. 25; E027], duae quantitates considerandae sunt; arcus AM = s = ds et tempus per AM = ds , ∫ ∫ gx quarum altera alterius respectu debet esse minima vel maxima. Eodem enim redit, sive inter omnes curvas aeque longas quaeratur ea, quae brevissimum habeat descensum, siveEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 321 inter omnes, super quibus descensus fiunt eodem tempore, ea, quae est brevissima. Per formulas autem meas dat ∫ ds hanc quantitatem diff . ds et ∫ dsgx dat diff . ds gx , quarum dy dy altera alterius multiplicato cuicunque aequalis est ponenda. Habetur ergo integrando seu Sumendis vero quadratis erit unde erit Ex quo curva quaesita determinabitur. Q.E.I. Corollarium 1. 402. In aequatione inventa duae insunt quantitates a et m arbitrariae, quibus effici potest, ut curva per datum punctum C transeat et ut simul sit datae longitudinis [p. 198]. Atque tum haec curva celerrime absolvetur inter omnes alias curvas eiusdem longitudinis per A et C transeuntes. Corollarium 2. 403. Si ponantur a et m infinite magna, prodibit cyclois, quae non solum inter omnes curvas eiusdem longitudinis, sed inter omnes omnino citissime absolvitur. Corollarium 3. 404. Si ponantur m = 0, prodit dy = 0 seu recta verticalis. At si fiat a = 0, oritur recta quaecunque per punctum A ducta. Est enim recta linea inter omnes lineas, quae eodem tempore absolvuntur, minima seu brevissima. Corollarium 4. 405. Si ponantur m = 1, prodit curva algebraica; erit enim cuius integralis est Haec curva etiam est rectificabilis; namque eritEULER’S MECHANICA VOL. 2. Chapter 2f. page 322 Translated and annotated by Ian Bruce. Quin etiam tempus per arcum AM algebraice poterit exprimi; erit enim Corollarium 5. [p. 199] 406. Si in hac curva sumatur x = a4 , ibi tangens erit horizontalis atque recta verticalis in 4a et longitudo curvae ad eo curvae puncto erit diameter curvae. Illo autem loco fit y = 15 hoc punctum erit = 25a . Atque tempus, quo hic arcus absolvitur, est = 4 a . Eodem ergo 3 g tempore corpus recta descendet per altitudinem 49a . Scholion. 407. Missis nunc hisce de celerrimo descensu progredimur ad eas curvas considerandas, super quibus plures descensus inter se comparati datam teneant relationem. Huc maxime pertinet quaesito de curva tautochronis, super quibus vel omnes descensus ad punctum infimum curvae fiunt eodem tempore vel integrae oscillationes. Ad haec deinde aliae accedere possunt quaestiones cum difficiles tum vim methodi, qua utemur, illustrantes. PROPOSITIO 46. Problema. 408. Invenire legem generalem curvarum tautochronarum, super quibus omnes descensus ad punctum A, initio descensus ubicunque in curva AM (Fig. 51) accepto, absolvantur eodem tempore. [p. 200] Solutio. Sumta recta AP pro axe dicatur curvae portio AM = s sitque altitudo celeritati in A debita = b et altitudo celeritati in M debita = v ; erit tempus, quo arcus AM absolvitur, = ds , quod integrale ∫ v ita est capiendum, ut evanescat posito s = 0. Tum, si in eo integrali ponatur v = 0, habebitur tempus descensus a loco, in quo celeritas erat nulla, usque ad punctum A seu totam descensus tempus; id quod eadem quantitate expressum esse debet, quaecunque fuerit quantitas b. Haec igiturEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 323 quantitas b neque in expressione temporis inesse debet neque in expressione pro curva AM, quia haec eadem curva idem descensus tempus producere debet, utcunque varietur b. Sit iam v quantitas composita ex littera z, ad curvam pertinente et a curva AM cum abscissa AP et applicata PM tantum pendente neque b involvente, atque ex littera h, quae ex b et quantitatibus constantibus sit composita. Sit autem v talis ipsarum h et z functio, ut evanescat posito z = h atque ut fiat = b, si sit z = αh existente α numero quocunque. Ponatur porro ds = pdz pro aequatione curvae quaesitae; debebit p talis esse quantitas, in qua non contineatur b vel h, quia hae litterae in aequationem curvae ingredi nequeunt. Habebimus ergo pro expressione temporis per AM ∫ v hoc integrali ita accepto, ut pdz evanescat posito v = b seu z = αh . Deinde hoc integrale, si in eo ponatur z = h, [p. 201] dabit tempus totius descensus, in quo h inesse non poterit. Hoc vero evenit, si fuerit functio ipsarum h et z nullius dimensionis seu si ∫ v pdz pdz fuerit functio nullus v dimensionis. Sit v functio m dimensionum ipsarum h et z; debebit esse m−2 p = Cz 2 denotante C quantitatem constantem a b non pendentem. Quoties ergo pro v talis functio fuerit comperta, habebitur pro curva quaesita haec aequatio si opus est, quo s evanescat, si in z evanescat vel x vel y. Q.E.I. Corollarium 1. 409. Quo igitur haec methodus possit adhiberi, oportet, ut v quantitatibus finitis sit expressum atque ut ea expressio transmutari possit in functionem homogeneam ex h et z constantem. Corollarium 2. 410. Hanc ob rem necesse est, ut fiat v = b, si ponatur z = αh , quo in quantitate constante adiecta etiam non reperiatur h. Sufficit igitur, si viderimus fieri v = b facto z = αh , neque opus est, ut integratio absolvatur. Corollarium 3. 411. Intelligitur etiam curvam in A habere debere tangentem normalem in directionem vis sollicitantis; [p. 202] nisi etiam hoc fuerit, tempus per arcum descensus infinite parvum foret quoque parvum. Scholion. 412. Valet haec solutio non solum, si, uti figura indicat, curva exponatur per coordinatas orthogonales; nihil enim interest, quibusnam quantitatibus naturam curvae exponere velimus, dummodo in z non ingrediatur b. Potest autem z continere lineas et quantitates quascunque a curva pendentes. Hac igitur methodo in vacuo, in quacunque potentiarum sollicitantium hypothesi, lineae tautochronae poterunt inveniri, quia semper celeritas per quantitates finitas exprimi potest. At si, ut in mediis resistentibus fieri solet, celeritas non potest exhiberi finitis quantitatibus, haec methodus usum habere nequit, sed alia desideratur, quae succedit, etiam si celeritas per aequationem differentialem tantum detur.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 324 PROPOSITIO 47. Problema. 413. Si corpus deorsum sollicitetur vi quacunque, invenire lineam tautochronam super qua omnes descensus fiant eodem tempore. Solutio. [p. 203] Posito AP = x, PM = y et AM = s (Fig. 51) sit celeritas in A debita altitudini b et in M debita altitudini v. Sit porro vis sollicitans in M = P; erit v = b − Pdx integrali Pdx ita accepto, ut ∫ ∫ evanescat facto x = 0. Iam si ponatur b = h et Pdx = z , erit v functio unius dimensionis ∫ ipsarum h et z et evanescit facto z = h fitque v = b facto z = 0. Erit igitur m = 1 ideoque habebitur pro curva quaesita ista aequatio Si desideretur aequatio inter x et y, erit ob ds = aPdx a ∫ Pdx Q.E.I. Corollarium 1. 414. Quia evanescit ∫ Pdx facto x = 0, tangens curvae in A erit horizontalis, nisi in A evanescat P. Atque curva alicubi habebit tangentem verticalem ibique plerumque cuspidem; hoc evenit, ubi erit ∫ Pdx = aP . Ibi enim fit dy = 0. 2 Corollarium 2. ydy 415. Huius curvae in puncto infimo A radius osculi est aequalis subnormalis = dx = sds , dx quia in A s et y fiunt aequalia. Quare in A erit radius osculi = 2aP, ubi P denotat potentiam sollicitantem in puncto A.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 325 Corollarium 3. [p. 204] 416. Ex radio osculi in A et potentia sollicitante in A invenitur tempus ascensu vel descensus per infinite parvum arcum = (172). Huicque tempori tempus cuiusque descensus est aequale. In hypothesi ergo gravitatis = 1 pendulum longitudinis 2a descensus infinite parvos eodem tempore absolvet. Exemplum 1. 417. Sit potentia sollicitans ubique constans, nempe P = g; erit Pdx = gx atque s = 2 gax ∫ itemque unde intelligitur curvam esse cycloidem deorsum convexam, prorsus cum linea brachystochrona in eadem potentiae hypothesi congruentem. Quod autem omnes descensus fiant eodem tempore super cycloide, iam supra demonstravimus (187). Exemplum 2. 418. Sit potentia sollicitans ut potestas quaecunque ipsius x ; erit si quidem fuerit n + 1 numerus affirmativus; sin enim esset n + 1 numerus negativus, fieret Pdx =∝ . Erit igitur ∫ Ex qua aequatione intellititur curvam fore rectam utcunque inclinatam ad horizontalem si fuerit n = 1. [p. 205] At si n > 1, curva in initio A fit imaginaria, quo usque nimirum f n incipit minus esse quam ( n + 1 )ax n −1 .EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 326 Scholion 1. 419. Ex aequatione generali apparet rectam AP esse diametrum curvae. Quare cum in vacuo ascensus sint similes descensibus, semioscillationes omnes super curva MA ad alteram partem usque producta erunt quoque isochronae et consequenter etiam integrae oscillationes. Deinde cum ob arbitrariam a infinitae sint curvae tautochronae AM, duae quaequo in puncto A coniunctae, ut ibi habeant tangentem commumem horizontalem, producent tam semioscillationes quam integras oscillationes isochronas, si scilicet pendulum ita accomodetur, ut oscillando huiusmodi curvas absolvat. Scholion 2. 420. Intelligitur etiam ex solutione eas curvas, quas invenimus, esse solas, quae quaesito satisfaciunt. Nam loco p alia functio ipsius z substituti nequit, ut in integrali, si ponatur v = 0, prorsus ex formual exeant b seu h. Id quod aliis methodis, quibus tautochronae sunt inventae, non satis liquet. Scholion 3. ∫ 421. Quia est s = 2 a Pdx , erit P = 2sds . Ex quo apparet, cuiusmodi esse debeant adx potentia sollicitans, [p. 206] ut data curva sit tautochrona. Scilicet potentia deorsum tendens debet esse proportionalis ipsi sds ex data curva desumto. Quare, nisi curva sit dx rectificabilis, valor potentiae sollicitantis non potest algebraice exhiberi.EULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 327 PROPOSITIO 48. Problema. 422. Si corpus perpetuo trahatur ad centrum virium C (Fig. 52) vi quacunque, invenire lineam tautochronam BMA, super qua omnes corpus descensus ad punctum A usque eodem tempore absolvat. Solutio. Dicatur CA = c, CM = y et vis centripeta in M sollicitans = P. Sit porro celeritas in A debita altitudini b et ea in M altitudini v. Sumatur Pdy ita, ut evanescat posito y = c, quo facto ∫ ∫ erit v = b − Pdy . Sumtis ergo b pro h et ∫ Pdy pro z erit functio ipsarum h et z, cui v aequatur, unius dimensionis; quare erit m = 1. Si nunc arcus AM dicatur = s, erit ∫ s = 2 az = 2 a Pdy atque hinc ds = aPdy . a ∫ Pdy Si nunc in M ducatur tangens in eamque ex centro C demittatur per pendiculum CT, quod dicatur p, erit ydy ( y2 − p2 ) ∫ = ds . Quare habebitur y 2 Pdy = ( y 2 − p 2 )aP 2 seu p 2 = y 2 − y 2 ∫ Pdy , aequatio pro curva quaesita. Q.E.I. aP 2 Corollarium 1. [p. 207] 423. In puncto A, ubi ∫ Pdy evanescit, erit p = y, seu recta CA erit normalis in curvam in eoque propterea celeritas corporis erit maxima, quia A est punctum curvae centro C proximum. Corollarium 2. 424. Si ponatur p = 0, habebitur punctum curvae B, in quo recta CB curvam tangit. In eoque puncto, quod erit supremum, curva cuspidem habebit. Invenitur vero punctum B ex ∫ hac aequtione aP 2 = Pdy ; atque y non poterit esse maior quam valor ex hac aequatione inventus.EULER’S MECHANICA VOL. 2. Chapter 2f. page 328 Translated and annotated by Ian Bruce. Corollarium 3. ∫ 425. Apparet etiam ex aequatione inventa s = 2 a Pdy , quia signum radicale signum ambiguum involvit, curvam duos habere ramos AB et AD inter se similes et aequales et hanc ob rem oscillationes, quae super curva BAD fiunt, esse inter se aequales. Corollarium 4. 426. Radius osculi in puncto A est = c2−acP . Et quia est AC = c, erit tempus, quo unus 2 aP descensus super curvae AB portione infinite parva absolvitur, = π a (207); huic igitur tempori omnes descensus erunt aequales. [p. 208] Hanc ob rem oscillationes, quae fiunt super curva BAD, isochronae erunt cum oscillationibus penduli in hypothesi gravitatis = 1, cuius longitudo est = 2a. Exemplum 1. y 427. Sit vis centripeta directa proportionalis distantiis a centro, ut sit P = f ; erit y 2 −c 2 ∫ Pdy = 2 f . Hinc ergo erit AM = s = f ( yy − c 2 ) 2 a( y 2 − c 2 ) 2 atque p

yy- . f 2a ydy Huius curvae radius osculi in puncto M qui est dp , invenitur = 2a2−a f ( 2a − f ) y 2 + fc 2 ) . 2a Ex quo sequitur, si fuerit 2a < f, curvam versus centrum C fore convexam, ut exhibet figura. At si fuerit 2a = f, curva evadet linea recta in A normalis ad rectam AC. Punctum autem B, ubi CB tangit curvam, invenitur ex hac aequatione ( f − 2a ) yy = ccf , ex qua fit BC = c f ( f − 2a ) . Quoties ergo accidit, ut sit f > 2a seu curva convexa versus C, habebit curva cuspidem in B. Atque in his casibus curva erit hypocylois, quae generatur rotatione circuli, cuis diameter est = c f ( f − 2a ) c f −c ( f − 2a ) ( f − 2a ) , super concava parte circuli centro C radio = descripti. Hoc ergo casu curvae tautochronae conveniunt cum brachystochronis supra inventis (390). At si 2a > f, quo casu curva est concave versus C, fit BC imaginaria et curva AM non amplius est hypocyclois. [p. 209] Tum autem erit p 2 = praeter in A erit maior quam AC. Sit c = 0; fiet p = y ( 2a − f ) yy + ccf , unde p ubique 2a 2a − f , quare hoc casu curvae 2a tautochronae erunt spirales logarithmicae circa centrum C descriptae. Corpus scilicet super spirali logarithmicae perpetuo eodem tempore ad centrum C pertinget, ubicunque descensum inceperit. Hac igitur vis centripetae hypothesi tautochronae erunt: primi omnes hypocycloides, deinde omnes lineae rectae utcunque ductae, tertio omnes spiralesEULER’S MECHANICA VOL. 2. Chapter 2f. page 329 Translated and annotated by Ian Bruce. logarithmicae et quarto infinitae curvae aliae hac aequatione contentae p2 = ( 2a − f ) yy + ccf , si quidem fuerit 2a > f et c non = 0. In hac autem vis centripetae 2a hypothesi pro tautochronis tantum dederunt hypocycloides Newtonus in Princ. [Lib. I, Prop. LI, theorema XVIII.] et Hermannus in Phoronomia et Comment. Acad. Petrop. A 1727, [p. 139, vide praecipue p. 150; P.St.] quamvis hic aequationem aeque generalem ac nostram habuerit. Exemplum 2. 428. Ponatur vis centripeta reciproce proportionalis quadratis distantiarum a centro, ut sit ff P = yy ; erit Curva igitur AM erit = unde invenitur radius osculi Ubi ergo p evanescit, ibi etiam radius osculi fit = 0. Ipsius BC valor autem reperietur ex − ck , id hac aequatione y 4 = cy3 + acff . [p. 210] Si igitur ponatur BC = k, erit a = k cff 4 3 quod assumi potest, quia k est quantitas arbitratia. Ceterum apparet hanc curvam intra A et B habere posse punctum flexus contrarii, id quod evenit, si curva in A est concava versus C, nempe si fuerit 2aff > c3. Exemplum 3. 429. Sit vis centripeta ubique constans seu P = 1; erit radio CM ducatur arcus MP, erit ∫ Pdy = y − c. Quare si centro C ∫ Pdy = AP atque arcus AM = 2 a( y − c ) = 2 a AP . • At erit porro unde invenitur BC = a + c et curva AB = 2a = 2( BC − AC ) . Radius osculi vero in puncto M erit = In puncto ergo infimo A erit radius osculi = 22aac . Curva igitur in A erit concava versus −c C, si 2a > c, at convexa, si c > 2a, atque radius osculi in A erit infinite magnus, si 2a = c. Primo casu, quo curva in A est concave versus C, curva habebit punctum flexus contrarii, ubi estEULER’S MECHANICA VOL. 2. Chapter 2f. Translated and annotated by Ian Bruce. page 330 Si est c = 0 ita ut punctum A in centrum C cadat, fiet y chorda huius curvae eritque AM = 2 ay , cuius curvae radius osculi in ipso centro C est infinite parvus, atque est p=y a− y a ; curva autem ipsa per quadraturam circuli potest constui. [p. 211]

The Rectilinear Motion Of A Free Point Acted On By Absolute Forces

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