THE MOTION OF A POINT ON A GIVEN LINE IN A VACUUM.

PROPOSITION 36. Problem.

328. With a uniform force g acting downwards everywhere, to find all the curves AMC (Fig. 41), upon which a body descends from rest at A in a given time to a straight line BC inclined at any angle to the horizontal. .

Solution.

The applied line BD of the curve AND expresses the time, in which the body from A reaches the line straight BC, and from any point M the straight line MQ is drawn parallel to the given line BC, on cutting the vertical AB at Q, and QN expresses the time in which the body has traversed the part AM . Whereby, if an infinitude of curves AND are considered, which all have the same applied line BD at B, all these generate curves AMC, upon which the body in the given time arrives at BC from A. Moreover the curves AND, as advised above, must concur at A with the vertical AB and as far as D must diverge from AB. Now putting the tangent of the angle ABC = k, and putting y y AP = x, PM = y, AQ = u, QN = t and AB = a; then PQ = k and thus x + k = u . Moreover since the speed at M must correspond to the height gx, the time to traverse AM

∫ dt = ( dx 2 + dy 2 ) gx , which has to be equal to QN = t ; therefore we have exactly ( dx 2 + dy 2 ) gx and gxdt 2 = dx 2 + dy 2 . But on account of the given curve AND, t is y given in terms of u, and since u is given by u = x + k , t is given by x and y ; on accountEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 239 of which the equation is obtained between x and y for the curve sought AMC. Or since we have y = ku − kx , then the equation becomes : gxdt 2 = ( k 2 + 1 )dx 2 − 2k 2dudx + k 2du 2 and [p. 152] it is allowed to find x in terms of u . In order to do this, let dt = pdu , [i.e. p is the gradient of the AND curve with the abscissa u vertical] then gxp 2du 2 = ( k 2 + 1 )dx 2 − 2k 2dudx + k 2du 2 and dx = k 2 du ± (( g ( k 2 +1 ) p 2 x − k 2 )du 2 ) . k 2 +1 Therefore the curve AND must be taken everywhere such that [the gradient] p is greater k than . Moreover that equation must be integrated thus, so that on making u = 2 gx( k +1 ) 0 it becomes x = 0. With which done, the equation between x and y for the curve sought can be obtained. Q.E.I. Corollary 1. 329. The curve AMC is a tangent to the straight line AB at A, if dy = 0 on putting x = 0; now it must then be the case that du = dx and 1 = ( g ( k 2 + 1 ) p 2 x − k 2 ) . Whereby this comes about, if on making ppx = 1g on making x = 0. Moreover since in this case y is infinitely smaller than x, in the beginning x = u; from which it follows that the curve AMC is a tangent to the vertical AB at A, if ppu = 1g with u = 0. [Note that this degenerates to the equation dx = gdt 2 in the neighbourhood of A, on setting du = dx, recognisable as an equation of motion; the customery 12 is missing as usual, as all such expressions involving free fall are compared against each other by Euler, rather than dealt with in an absolute manner, which could not be done at the time, due to the lack of an adequate set of units. The reader has no doubt observed that two curves are always used in these demonstrations : AND is the vertical displacement x versus time t, while AMD is the vertical displacement x versus horizontal displacement y. ] Corollary 2. y ydy 330. Then the normal to the curve AMC is along QM, if PQ = k = dx or 2 dx = kdy = k 2du − k 2dx or dx = k 2 du . Now this eventuates [from above] k +1 when p 2 x = kk . g ( k 2 +1 ) Corollary 3. 331. At the point A, the expression ppx either has a finite value and that is greater than kk the value it has on making x = 0 [one presumes that Euler has in mind by this 2 g ( k +1 )EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 240 statement, taking the limit as x tends towards zero], or it is infinitely great. In the latter dy 2 case [ dx = k 2 du ] becomes dx = ± ∝ du , and since du = dx + k , then dy = −kdx , [as k +1 du = 0 ; p. 153] In which cases, the tangent to the curve AMC at A is parallel to the line dx BC. Example. 332. Let the curve AND be any parabola, so that thus t = 2α u ; g then dt = αdu and p = α , gu hence this equation is obtained [from dx = gu k 2 du ± (( g ( k 2 +1 ) p 2 x − k 2 )du 2 ) ]: k 2 +1 The integral of this equation is : with with [Clearly a convenient way to demonstate this integral is to assume the final form, and to differentiate the logarithms of this product, at which point an attempt is made to generate the given differential equation by equating coefficients and making use of partial fractions. Note that the l.h.s. of the differential equation is the derivative of part of the numerator of the r.h.s., while the rest is du . Euler concerns himself with this aspect u below.] [If α < 1 , then B and hence also the number ρ is a positive number.] Therefore since the number π is a negative number, then where C denotes a constant, which is effected so that on putting x = 0, u = 0 also. Moreover it is clear, whatever the constant should be, that always u = 0 on making ρ or π to disappear. But it is not possible for π to vanish, and in this case ρ now vanishes, since α = 1 ; therefore in this case it must be that C =∝ and the equation becomesEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 241 or u = x and y = 0; whereby in this case the condition for the vertical line AB is satisfied. For the remaining cases, since C is now an arbitrary quantity, from one curve AND innumerable curves AMC can be found. Again the single case has to be treated on its own, if A = B or for then it becomes : with arising Consequently for this case this equation is obtained : [p. 154] where C too does not have to be found. If α > 1 , then B and hence ρ too becomes a negative number; then it follows that C =∝ . Hence in this case either or Which two equations are for straight lines inclined in a certain manner and passing through A and these also are satisfactory in general. So if we have α = 1 , then A = 1 and B = 0 and hence these equations which is a straight line perpendicular to BC; for this is traversed by a body in the same time as the vertical AB. Corollary 4. 332a. [Numbering error] Therefore other than α > 1 or α = 1 , innumerable curved lines can be found that satisfy the problem; hence all are completed in a smaller time than the perpendicular AB. Corollary 5. 333. Therefore since an infinitude of curves AMC are able to arise from the one curve AND, it is readily understood that infinitely more curves satisfying this question than in the preceeding.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 242 Corollary 6. [p. 155] 334. If α < 1 then the determination of the constant C can be brought about, so that the curve sought passes through a given point on the line BC. The for other assumed curves AND in a similar manner an infinitude of curves can be found, upon which the body ont only reaches the given line BC in a given time, but reaches some given point C on that line. Scholium 1. 335. In this example two cases occur in which α = 1 ; indeed in the first in turn only the vertical line satisfying the problem has been found, and in the other besides this line the other inclined straight line has been found, yet we have used both in the same way in the general equation. Moreover now more often cases of this kind are touched upon, in which the differential equations contain amongst themselves the integral equations, which nevertheless cannot be expressed by integration. As in the case α = 1 , this has the differential equation which on integration gives x = u. Yet meanwhile it is evident that this equation k 2u = ( k 2 + 1 )x is contained in that too, even if it cannot be produced by integration. And on this account the equation is equally satisfied by the latter integral as the former x = u. Hence it is permitted generally to gather together the differential equation into the form dt = Vdu , T in which T is such a function of t, which vanishes on putting t = 0, and V is some function of u, equally understood that this integration vanishes when t = 0 and this [p. 156] dt = Vdu , T ∫ ∫ is is elicited by integration. Indeed generally the case t = 0 can be disregarded if t a simple quantity; but if t is a composit quantity as in our case, it is wrongly omitted. We had a similar case above (300) in the equation where we observed the equation x = a to be contained in that, even if it could not be integrated by integration. For on putting a − x = t then − dx = dt and is a function of t, which becomes equal to zero, if we make t = 0 or x = a; in as much as Pdx is thus prescribed to be accepted, so that it vanishes on putting x = a. Hence on ∫ putting this function of t equal to T the equation becomes ds = dtT b , hence in which equation it is allowed to conclude without risk that the equation t = 0 or x = a is satisfied,EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 243 and thus the circle to satisfy that problem, as we agreed upon there (303). More generally , if T does not vanish on putting t = 0, it is understood that now in this equation Vdu = dt T the equation T = 0, from which t is equal to some constant quantity and thus dt = 0. . And Hence it is understood that T = 0 is contained in the proposed equation Vdu = dt T hence, if t should be a composite quantity for argument’s sake from u and x, then the equation of the integral is not likely to be had at once by integration. Scholium 2. 336. Here the case remains, which requires a particular resolution, [p. 157] when the straight line BC meets the line BA above the point A, and when it is parallel. But we consider only the case in the following problem, in which BC is made parallel to the vertical line AB and put at a given distance ; from which in a similar manner the case of the line inclined at some angle can be deduced. PROPOSITION 37. Problem. 337. Let the body be always acted on by a constant uniform downwards force g ; to find the innumerable curves upon which the body, starting to move from rest at A (Fig. 42), arrives at the vertical lineEC in a given time. Solution. Let AMC be any of the curves sought and the vertical line AB is taken for the axis; calling AP = x, PM = AQ = y; then the speed at M corresponds to the height gx and the time for AM is equal to ( dx 2 + dy 2 ) ∫ gx Again let AND be a curve, of which any applied line QN expressed the time for AM, and let QN = t be a function of y; from the given curve AND it is possible to find the curve AMC. Whereby if an infinity of curves AND are considered, which all have the common point of apllication DE at E, all produce the curves AMC, upon which in a given time expressed by DE arrive at CE from A. And thus the equation becomes ; t= and on putting dt = pdy it becomes : ∫ ( dx 2 + dy 2 ) gxEULER’S MECHANICA VOL. 2. Chapter 2e. page 244 Translated and annotated by Ian Bruce. gp 2 xdy 2 = dx 2 + dy 2 and dx = dy ( gp 2 x − 1 ) [p. 158] Which thus on integration, since putting x = 0 makes y = 0, gives the curves sought AMC. Let R be some function of y and Rdy thus taken, so that it vanishes on putting y = 0. ∫ Then it becomes ∫ Rdy = A with y = AE = a and with DE = b arising, taking t= b ∫ Rdy ; A there becomes : Which equation, whatever is substituted for R, gives innumerable curves satisfying the question. Q.E.I. Corollary 1. 338. Therefore in the simplest case, if we put R = 1; then indeed a separable equation is y b produced. Now it becomes t = a becomes as A = a. hence on this account the equation But which is of no use as the value of y is imaginary. Corollary 2. 339. Moreover since ( gbR 2 x − A2 ) cannot be an imaginary quantity, it is required that 2 R2 x > A ; even if x = 0. Whereby R neither can be a constant quantity nor a function of gb y, which vanishes on making y = 0. Hence on this account R must be such a function of y, which becomes infinite, if y = 0. Yet besides it must be a function of the kind, such that Rdy is not made infinite, which comes about if we should make R = 1y or 12 etc. ∫ y Example. [p. 159] 340. Therefore we put R = 1 y ; and the integral becomes ∫ Rdy = 2 y and A = 2 a . Hence this equation is obtained: Which equation, since it is homogeneous, on putting x = qy; there is foundEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 245 or gb On putting 4a = n or (nq - 1) = r then we have Which latter formula depends on the quadrature of the circle, if n < 2. Now in the other case n > 2 the integral is : Which since becomes equal to this : Where any constant is allowed to be accepted for C, since the equation thus is comparable, as on putting x = 0 makes y = 0. Moreover by the method treated above (335) this integral is at once had from the differential equation : hence there arises : which gives two straight lines, unless gb < 8a , in which case the equation is imaginary. In the case in which n = 2, it becomes hence it becomes : or Also there any quantity can be taken for C. If n < 2, then the construction of the integal depends in part on logarithms, and in part on the quadrature of the circle[p. 160] ; indeed on account of the imaginaryEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 246 ( n 2 − 4 ) imaginary logarithms are found. Therefore in this case it is expedient for the construction to be carried out analytically. PROPOSITION 38. Problem. 341. Let the body always be acted on by a uniform downwards force g and any curve is given BSC (Fig. 43) ; to find all the curves AMC, upon which the body by descending from A in a given time arrives at the given curve BSC. Solution. Let any of the given curves sought be AMC, through some point M of this, the curve MQ similar to the [known]curve BSC with respect to the fixed point A is drawn, and the applied line NQ expresses the time for the arc AM of the curve AND to be completed; hence the applied line BD sets out the time for the whole curve AMC to be completed. With which done, the curve AMC can be found in turn from the given curve AND. Whereby if an infinity of curves AND is considered, which all have the common line of application BD at B, these generate the infinity of curves AMC, upon all of which the body arrives at the curve BSC by descending from A in the given time. If now AB = a, BD = b and for the arc QM a similar arc BS is cut off from the given curve BSC; then we have : AQ : AB = PM : RS = AP : AR = PQ : BR. [p. 161] Again putting AP = x , PM = y , AQ = u , QN = t , AR = r , RS = s; on account of the given curve BSC there is an equation given between r and s and on account of the given curve AND there is an equation given between t and u. But on account of the similitude there is and x = ur . the ratio u : a = y : s = x : r , hence this becomes : y = us a aEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 247 Then the speed of the body at M corresponds to the height gx, from which the time to pass along AM is equal to : ∫ ( dx 2 + dy 2 ) gx , that must be put equal to t; hence this equation arises gxdt 2 = dx 2 + dy 2 . Moreover since t is given through u, let dt = pdu and p is a function of u; and because dx = udr +a rdu and dy = uds +a sdu this equation goes over into this : garup 2du 2 = ( udr + rdu )2 + ( uds + sdu )2 But since the curve BSC is given, then s is a function of r and it becomes ds = qdr with q present some function of r. With these put in place, an equation is obtained between u and r: garup 2du 2 = ( udr + rdu )2 + ( uqdr + sdu )2 Which with the square root extracted gives : From which if the equation between r and u can be found, then likewise the equation between x and y for the curve sought is obtained. Moreover since it is restricted to the curve AND, let P be some function of u and thus Pdu integrated, so that it vanishes on making u = 0 and it becomes equal to A on puttng ∫ u = a; then we can take t = b ∫ Pdu for the equation of the curve AND. Hence A p = PA b , where any allowed function of u can be put in place for the function P. Q.E.I. Corollary 1. [p. 162] 342. If u is put equal to 0, for that both x and y vanish, unless perhaps r or s become infinite. Therefore in that case in the integration of differential equation some constant found can be added, since there is no need that r should have a given value, if u is made equal to 0. Corollary 2. 343. Therefore then on account of the arbitrary constant to be added, from one given curve AND, innumerable curves AMC can be found satisfying what is sought.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 248 Corollary 3. 344. If the curve BSC has thus been prepared, so that neither r nor s is able to become infinitely great, an infinite number of curves sought AMC are given always from a single curve AND. Not only do they have this property so that bodies descending on these arrive at the same time at the given line BSC, but also likewise they can reach any given similar curve QM . Corollary 4. 345. Therefore since in the integration of the equation found it is allowed to add some constant[p. 163], thus that can always be assumed, so that the curve AMC is directed towards a given point C of the given BSC. And in this way an infinite number of curves AMC can be found, which all come together at a given point C. Scholium 1. 346. We have put the curves QM similar to the curve BSC, so that curve itself raised to A becomes infinitely small and all the points of the curve BSC come together at A and both x and y vanish on putting u = 0. Moreover in the same manner we have put the curves QM either to be congruent with BSC, or disagreeing according to some condition. As Q is some function of u vanishing on setting u = 0 and that goes into B on making u = a, the curve QM thus is able to depend on the curve BSC, so that it becomes Qr Qs x = B et y = B ; in as much as on making u = a the curve QM is changed into BSC itself and at A the curve is changed into the point, unless the curve BSC should become infinite. But also in this case a funtion can be accepted for Q, in order that, even if on making r =∝ , yet Qr and Qs become equal to 0, if u = 0. Moreover on putting dQ = Vdu the following general equation is obtained : Which equation appears to be the most wide, and from a single curve AND an infinite number of curves AMC can be furnished, and it can also be supposed that an infinite number of curves pass through the point C. Scholium 2. [p. 164] 347. Moreover, however general this equation is, yet the curve QM is similar to the curve BSC, since it satisfies x : y = r : s. Whereby thus the more general solution can be shown, in which curves QM dissimilar in some manner can be put in place of the curve BSC, yet of the same kind, such that QM becomes BSC on making u = a. Now this solution is obtained, if some function R of u is taken vanishing on making u = 0 and R is evaluated at D on putting u = a, and it is given by dR = Wdu. Indeed takingEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. Qr x = B and y = Rs ; D page 249 it changes to x at r and y at s, on putting u = a, and with u vanishing as long as x and y vanish, for whatever r may be. Moreover this most following general equation arises : In this equation in place of dr, ds can be introduced by putting ds in place of dr, or even q in place of r, and then in place of r , x can be introduced by putting the value of this Bx Q the equation is obtained between u and x. But it should be noted, since Q vanishes on making u = 0, then P must be such a function of u that P 2Q , on putting u = 0, either becomes a finite quantity or it becomes infinitely great; but yet one is required to beware, lest Pdu on being assumed in this manner, becomes infinitely great. ∫ Corollary 5. 348. The equation found in the solution is separable if P = 1 ; it becomes u A = 2 a and p = b . [p. 165] For there is obtained : 2 au for s and q given in terms of r. Corollary 6. 349. In a similar manner the equation of scholium 1 can be separated, if P 2Q = V 2 or P = V . Q For there is obtained : in which the indeterminates u and r are between themselves in turn separated.EULER’S MECHANICA VOL. 2. Chapter 2e. page 250 Translated and annotated by Ian Bruce. Example 1. 2 350. On retaining P Q = V 2 or ∫ Pdu = 2 Q on account of Vdu = dQ, on putting n = a , A = 2 B. Hence the equation becomes : Therefore ∫ Pdu has the required property, so that it vanishes on making u = 0; for Q vanishes. Now let the curve BSC be a circle described with the diameter AB ; it becomes s = ( ar − r 2 ) and q = a − 2r 2 ( ar − r 2 ) and 1 + qq = a2 ; 4( ar − r 2 ) with these values put in place of s and q this equation is produced : [p. 166] Which equation not only has the indeterminates separated from each other, but also in general it can be integrated by logarithms ; for indeed in the equation : with the irrational part to be effected with a rational one. Moreover, this integral is produced : Here it is to be noted the case is in agreement, in which gb = 4a or b = 2 a , in which g the time for some curve AMC is made equal to the time of descent along the vertical line AB; for then it becomes : Therefore if the + sign prevales, then dr = 0 and r = const. = c, hence the equation becomes :EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 251 s = ( ac − c 2 ) and which equation gives gives all the chords in the semicircle drawn from A, as we have now thus shown(102) the times for individual chords are equal to each other. Should the – sign prevale, then the equation becomes and hence Hence the equation becomes and since s = ( ac − c 2 ) there is obtained therefore with r eliminated, this equaton is produced (on puting CB = m ) : y 2 + x 2 = ma xy . Therefore these curves have this property, so that the arc of these cut by a semicircle are completed by descending in the same time, clearly in that time in which the chords of the semicircle are traversed. Corollary 7. [p. 167] 351. The figure of this is curve is AMFA (Fig. 44), of which the equation is y 2 + x 2 = ma xy ; clearly the curve has the diameter AF making a 45 degree angle to the vertical AP, and having a node at A. But now all these curves are similar to each other and all can be applied to all circles. Corollary 8. 352. Therefore if some point M is taken on this curve and through this point and A a circle is considered crossing the curve, and having the centre on the vertical AP, the body traverses the arc AM in the same time in which the diameter of the circle or in which the chord AM is traversed. Whereby this curve has the property that any arc AM is completed by a body descending from A in the same time as the subtangent chord AM.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 252 Corollary 9. 353. Therefore in this case, in which P 2Q = V 2 (349), likewise, either Q = u or otherwise; for the same equation is produced between x and y, as this is understood either from the example or from the equation. Example 2. 354. With keeping P 2Q = V 2 , so that the equation becomes : [p. 168] let the curve BSC be a circle described with centre A and with radius AB = a ; then s = ( a2 − r 2 ) and With which put in place the following equation is produced : where gb must be greater than 4a, since r cannot exceed a. Hence at once that radius becomes known which satisfies the question, on putting in which case the equation becomes Hence which is the tangent of the angle of that radius, upon which the body in the given time b arrives at the periphery with the vertical AB. In addition, algebraic curves are not given, since the differential formula cannot be made rational.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 253 Example 3. 355. With the most general equation taken from (347) the straight line BSC is put horizontal ; then r = a and dr = 0. On account of this in place of dr the value of ds is q introduced, where q is infinitely great. Therefore with the terms deleted, which vanish before q, this equation arises : Which equation, since Wdu = dR , and P, Q and V are given in terms of u, can be integrated. [p. 169] Now it becomes , but y met vanish From which equation s can therefore be found. Then, on setting y = Rs D on making u = 0, it must be that C = 0, if indeed the integral is thus taken, in order that it disappears on putting u = 0. Therefore it then becomes : Which is the general equation for all the curves, upon which the body descents from A to the given horizontal. Example 4. 356. The most general equation found above (347) is considere, and the straight line BSC is put parallel to the vertical AB and at a distance f from that; then s = f and q = 0. Whereby this equatin is obtained : Moreover when we set Qr = Bx, with this substituted there is produced : fR thus on finding x, y = D . But since in that equation the indeterminates x and u cannot be separated from each other in turn, it is not possible to derive much from that equation.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 254 Scholium 3. 357. From the general solution of this problem, when a single curve AND gives an infinity of curves AMC, it is allowed to collect the solution of this problem, when an infinity of curves are required[p. 170] on all of which the body arrives at a given point from A. Indeed some curve AND gives a single curve crossing through a given point of the curve BSC and in this way innumerable curves of this kind are found. But when the solution in this manner should be exceedingly difficult and laborious, it is agreed that another more fundamental method should be produced. But the manner in which we use this is thus prepared, so that it is necessary to know one curve, from which we show how to deduce innumerable other curves. Therefore this curve, which must be known, is elicited from one method of treatment, as from (350), where the curve can be found crossing through some point of the semicircle in a time to be described. PROPOSITION 39. Problem. 358. Let the body be acted on always by a uniform force g downwards and let the curve AMC (Fig. 45) be given, upon which the body released from A arrives at some given point C; to find all the curves ANC upon which the body in the same time descends in the same time to the point C from A. Solution. With the vertical line AB taken for the axis of all the curves, let the abscissa of the given curve AMC be AP = t, with the applied line PM = u and let the curve ANC be one of the sought curves ; in that the arc AN is taken, [p. 171] which is completed in the same time as the arc AM; the points M and N are joined by the straight line MN and the curve ALB is constructed such that the applied line PL is equal to the line MN. Hence this curve ALB crosses the axis AB at the points A and B; for with the point M falling on A the point N also falls on A and on putting M at C the point N is also at C, since the arcs AMC and ANC are put to be transversed in the same time. Moreover, it is understood from the curve ALB that it is possible to find the curve ANC; whereby if an infinitude of curves of this kind ALB are considered at A and B coinciding with AB, a certain one of these gives the curve ANC and in this manner an innumerable number of curves ANC satisfying the question are produced. Now let PL = r; then r is some function of AP = t; moreover putting the abscissa of the curve ANC, AQ = x and QN = y. With these in place we have :EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 255 MN = (( x − t )2 + ( u − y )2 ) = r and thus y = u ± ( r 2 − ( x − t )2 ) . Again, since the time for AM is put equal to the time for AN, there arises or xdt 2 + xdu 2 = tdx 2 + tdy 2 . But Let du = pdt and dr = qdt ; where r, p and q are functions of t; again for brevity put x − t = z or x = t + z ; then the equation arises : and Hence this equation is obtained : from which z and t can be determined ; an equation can be obtained between x and y. [p. 172] Moreover where it is apparent, a function of t of this kind ought to be accepted in place of r, so that r vanishes on putting t = 0 since t = AB = a, let P be some function of t vanishing on putting t = 0 and Q is also such a function vanishing on putting t = 0; Now Q can be changed at A, if on making t = a; it is hence able to put r = P( A − Q ). And with this value whatever substituted in place of P and Q, the equation for the curve sought is obtained. Q.E.I. Scholium 1. 359. Indeed from this equation it can be concluded that it is a little complex to be resolved, even if it is considered to be fundamental. Moreover often the equation found must be reductio ad absurdum, if the curve given AMC were the line of shortest descent, in which case it is not possible to give other curves, upon which the descent can be made in the same time. Therefore it is seen to be in agreement with our intention that the lines of quickest descent are to be treated, and this problem resolved, in which between all the curves either of the same length or having some other common property that is sought the line which is completed in the minimum time; and also as between all the lines upon which the descent is made in the same time, that line is to be found, which has some given property provided. For even if it is most difficult, all the lines having the same timeEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 256 of descent can be shown, yet from these [p. 173] any can be found with the property it possesses before all the rest. Moreover in order to carry out this program, it is required to deal with the method of isoperimetrics, that as it is set out everywhere, we will not explain here. Scholium 2. 360. Moreover the solution of this problem is obtained in the following manner by(348). There it was found that : Let C be the point(Fig. 43), at which it is agreed that all the curves must meet, and put AE = f and EC = h ; if therefore on putting r = f, then it is also the case that s = h. According to this, let S be some function of r, which goes over into F on putting r = f, . This value is substituted into the above equation and that is with which done, put s = Sh F thus integrated, so that on putting u = a , it gives r = f. Then from that equation there is produced an equation between the coordinates of the curve AMC sought, clearly AP = x and PM = y, from which, that is x = ur and y = us . a a And the value of the arbitrary S gives an infinity of curves AMC joining the points A and C an on which the body descending in a given time equal to b arrives at C from A. Moreover let dS = Tdr; then q = hT and [p. 174] F or Which equation can thus be integrated, as on putting u = a makes r = f; with which done x = ur and y = hSu a Fa and an equation is obtained between x and y for an infinite number of curves AMC satisfying the equation.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 257 PROPOSITION 40. Problem. 361. To find the general rule, following which a curve ought to be disposed, in order that a body descending on that curve arrives at any point on the curve in the shortest possible time. Solution. Let AMC (Fig. 45) be a curve of this kind, upon which a body arrives at C from A in a shorter time than on any other curve crossing from A to C. Therefore for any two point M and μ between these, the curve intercepted must be compared, in order that its motion along the arc AMC is completed in a shorter time than any other, if it should be intercepted. Now the nearby points M and μ are joined by the two elements Mm, mμ , and the time to pass through Mmμ is a minimum, or by the rules of the method of maxima and minima, equal to the time for the nearby elements Mn, nμ . [p. 175] The applied lines MP, mp, and μπ are drawn to the axis AP and with the elements Pp, p π equal to each, or also MG = mH and pm, if there is a need, is produced to n and mn in made infinitely small with respect to the elements Mm and nμ . Therefore it must be : t .Mm + t .mμ = t .Mn + t .nμ . [The time to traverse the elements Mm and mμ is equal to the time to traverse the infinitesimally close elements Mn and nμ , the condition for the time variable to be a stationary point w.r.t. the independent variable x, for which Pp = pπ = dx. Meanwhile the curve is described at these points M , m , and μ by y , y + dy , and y + dy + ddy ] Let the speed that the body has at M correspond to the height v, with which therefore both the elements Mm and Mn are traversed. Moreover the speed that the body has at m, corresponds to the height v + du and the speed that it has at n, corresponds to the height v + du + ddw [note that du includes contributions from the vertical force, as well as from a general horizontal force and some tangential force acting along the curve; the final speed includes second order contributions from the horizontal and tangential forces, as we are shown eventually in the corollaries. Recall that in Euler’s analysis, an increment is always added at the start of an element, and holds a constant value within theEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 258 increment : thus, the speed at M corresponds to v throughout Mm, the speed at m corresponds to v + du, and is held constant throughout mμ , etc]; moreover with that [former] speed the element mμ is traversed, as now with this [latter] speed so the element nμ is traversed. Hence therefore this equation is obtained : now hence with the centres M and μ and with the arclets mg and nh drawn, there arises : Again, From which, with the negligible parts ignored, on substituting there arises Now on account of the [four] similar triangles nmg, mMG and nmh, μmH , it follow that, and On account of which the above equation is given by : Which equation is homogeneous and determines the nature of the curve AMC, the so- called brachistochrones, [p. 176] upon which the body in the shortest time arrives at C from A. Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 259 Corollary 1. 362. Therefore, if the lengths are called : then With these substituted, there is obtained : In which, if from the forces acting v, du, and ddw can be determined, and the equation for the brachystochrone curve can be obtained. But always ddw thus involves mn, as from the calculation it exceeds mn. Corollary 2. 363. Let the radius of osculation of the curve Mmμ = r ; and this lies in the region directed away from the axis AP, given by : but dy as the differential of ds ;] [as dx2 + dy2 = ds2, ddx = 0, and dsdds = dyddy, giving dx r hence this equation is produced : Where it is to be noted that 2rv is the centrifugal force, which is sent along the normal to the curve at M . Corollary 3. 364. If, from the forces acting, it follows that [we now interpret this equation as the sum of the works done per unit mass for increments dx with the force P, dy with force Q, and force R along ds ] : thenEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 260 since with the point m translated to n the small distance dy is increased by mn and ds by dy .mn the small amount ng. [p. 177] Moreover since ng = ds , then with which substituted this equation is obtained : Scholium 1. 365. From the solution, it is understood that the formula found extends the widest and can be extended to any forces acting, and also including resistance. For whatever forces should be acting, so du as well as ddw can be determined, and which values put in place gives the equation for the brachystochrone sought. But yet these are only in place, if the directions of the forces are in the same plane; for the curve found is situated in the same plane. Yet no less, if the forces should not be acting in the same plane, a brachystochrone curve can be found in a given plane with the help of this formula. Indeed in any given particular plane it pleases, a brachystochrone curve is given, whatever the forces acting should be. Now the other question is, if a brachystochrone curve is sought from among all the curves joining two given points, also not situated in one given plane. Now as often as the directions of the forces acting in the same plane are put in place, there is no doubt that the brachystochrone curve is in the same plane. For if the curves are not in the same plane, oblique forces are acting and therefore they do not give the body as great an acceleration as is possible. [p. 178] Therefore from this solution, so the brachystochrone curve is found completely, if the directions of the forces acting are in the same plane, and also a brachystochrone is present in a given plane, whatever should be the forces acting. [However, we understand that the time will be larger for such curves where the forces are not coplanar.] Scholium 2. 366. This question about brachystochrone curves or lines of quickest descent was first set out by the most celebrated Joh. Bernoulli [actually Leibniz and Jas. Bernoulli] and many solutions of this are extant, as in the Act. Lips., so also in the English Transactions, and in the Comm. Acad. Paris, and in others, where the solution for this problem, under the hypothesis of forces acting directed downwards and for centripetal forces, have been given. But no one has solved the fundamental problem that we have given here, set out so widely, so that it can also be extended to include forces of any kind, as well as resistance. For all have supposed that ddw = 0 , because that is always wrong, unless the direction of the force is MG or mH. And the most celebrated Hermann stumbled over this, while he used such a proposition in finding the brachystochrones in a resistive medium in theEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 261 Comm. Acad. Petrop. A 1727; and I have corrected that very problem, in the same Comm. A. 1734. [Ioh. Bernoulli, Curvatura radii in diaphanis non uniformis solutionque problematis a se in Actis 1696, p. 269, propositi de invenienda linea brachystochrona, Acta erud. 1697, p. 206; Opera Omnia, Tom. I, p. 187; Lettre de Mr. Jean Bernoulli a Mr. Basnage, sur le problème de isopérimètres, Histoire des Ouvrages des Sçavans, Paris 1697, p. 452; Opera Omnia, Tom. I, p. 194; Remarques sur ce qu’on a donné jusqu’ici de solutions des problèmes isopérimètres, Mém. de l’acad.d. sc. de Paris 1718, p. 100; Opera omnia, Tom. II, p. 225. Iac. Hermann, Theoria generalis motuum, qui nascuntur a potentiis quibusvis in corpora indesinenter agentibus, Comment. acad. sc. Petrop. 2 (1727), 1729, p. 139. L. Euleri Commentatio (E042) : De linea celerrimi descensus in medio quocunque resistente. Comment. acad. sc. Petrop. 2 (1734/5), 1740, p. 135. L. Euleri Opera Omnia, series I, vol. 25. Commentatio (E056) : Curvarum maximi minimive proprietate gaudentium inventio nova et facilis, Comment. acad. sc. Petrop. 2 (1736), 1741, p. 172. L. Euleri Opera Omnia, series I, vol. 25. Notes by P.St.]EULER’S MECHANICA VOL. 2. Chapter 2e. page 262 Translated and annotated by Ian Bruce. CAPUT SECUNDUM DE MOTU PUNCTI SUPER DATA LINEA IN VACUO. [p. 151] PROPOSITIO 36. Problema. 328. Posita potentia sollicitante uniformi g et ubique deorsum directa invenire omnes curvas AMC (Fig. 41), super quibus corpus ex A dato tempore ad rectam BC ad horizontalem utcunque inclinationam descendat. Solutio. Exprimat curvae AND applicata BD tempus, quo corpus ex A ad rectam BC pertingit, et ducta per quodvis punctum M recta MQ parallela rectae datae BC secante verticalem AB in Q exprimat applicata QN tempus, quo corpus partem AM percurrit. Quare si infinitae curvae AND concipiantur, quae omnes in B eandem habeant applicatam BD, hae omnes generabunt curvas AMC, super quibus corpus dato tempore ab A ad rectam BC pervenit. Curvae autem AND, ut supra monitum, concurrere debent in A cum verticali AB et usque ad D divergere debent ab AB. Ponatur nunc tangens anguli ABC = k sitque AP = x, PM = y, AQ = u, QN = t et AB = a; erit y y PQ = k ideoque x + k = u . Quia autem celeritas in M debita est altutudini gx, erit tempus per AM = quod aequale est debet ipsi QN = t ; erit adeo dt = ( dx 2 + dy 2 ) gx y ∫ ( dx 2 + dy 2 ) gx , et gxdt 2 = dx 2 + dy 2 . At ob curvam AND datam dabitur t in u, et cum sit u = x + k , dabitur t per x et y ;EULER’S MECHANICA VOL. 2. Chapter 2e. page 263 Translated and annotated by Ian Bruce. quamobrem habebitur aequatio inter x et y pro curva quaesita AMC. Vel cum sit y = ku − kx , erit gxdt 2du 2 = ( k 2 + 1 )dx 2 − 2k 2dudx + k 2du 2 , ex qua aequatione x per u invenire licebit. Sit ad hoc dt = pdu, erit gxp 2du 2 = ( k 2 + 1 )dx 2 − 2k 2dudx + k 2du 2 atque [p. 152] dx = k 2 du ± (( g ( k 2 +1 ) p 2 x − k 2 )du 2 ) . k 2 +1 Curva igitur AND talis accipi debet, ut ubique p maius sit quam k . gx( k 2 +1 ) Aequatio illa autem ita debet integrari, ut facto u = 0 fiat x = 0. Quo facto quoque eruetur aequatio inter x et y pro curva quaesita. Q.E.I. Corollarium 1. 329. Curva AMC tanget in A rectam AB, si fit dy = 0 posito x = 0; tum vero debebit esse du = dx atque 1 = ( g ( k 2 + 1 ) p 2 x − k 2 ) . Quare hoc eveniet, si fit ppu = 1g facto x = 0. Quia autem hoc casu est y infinities minor quam x, erit in ipso initio x = u; ex quo sequitur curvam AMC in A tangere verticalem AB, si fuerit ppx = 1g positu u = 0. Corollarium 2. y ydy 330. Deinde curva AMC normalis erit in QM, si furerit PQ = k = dx seu 2 dx = kdy = k 2du − k 2dx sive dx = k 2 du . Hoc vero eveniet, ubi erit p 2 x = k +1 kk . g ( k 2 +1 ) Corollary 3. 331. In ipso puncto A expressio ppx vel finitum valorem eumque maiorem quam kk habebit facto x = 0 vel infinite magnum. In posteriore casu erit dx = ± ∝ du , et 2 g ( k +1 ) dy cum sit du = dx + k , erit dy = − kdx. [p. 153] Quibus casibus tangens curvae AMC in A parallela erit rectae BC. Exemplum. 332. Sit curva AND parabola quaecunque, ita ut sit t = 2α u ; g eritEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. dt = αdu et p = α , gu gu page 264 unde habebitur ista aequatio Huius aequationis integralis est existente et [Si est α < 1 , fit B et hinc quoque ρ numerus positivus.] Cum igitur sit π numerus negativus, erit ubi C denotat constantem, quae efficiat, ut posito x = 0 fiat u = 0. Manifestum autem est, quaecunque fuerit constans, semper fieri u = 0 posito ρ vel π evanescit. At π evanescere non potest, ρ vero evanescit casu, quo α = 1 ; hoc igitur casu debet esse C =∝ fietque seu u = x et y = 0; quare satisfacit hoc casu recta verticalis AB. Reliquis casibus vero ob C arbitrariam quantitatem ex unica curva AND innumerabiles curvae AMC inveniuntur. Unicus porro casu est seorsim tractandus, si A = B seu tum enim erit existente Consequenter pro hoc casu habebitur haec aequatio [p. 154] ubi C quoque determinatione non opus habet. Si est α > 1 , fit B et hinc quoque ρ numerus negativus; tum ergo debet esse C =∝ . Hoc ergo casu erit velEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 265 vel Quae duae aequationes sunt pro lineis rectis certo modo inclinitis et per A transeuntibus haeque etiam generaliter satisfaciunt. Ut si fuerit α = 1 , erit A = 1 et B = 0 hincque hae aequationes quae est linea recta perpendicularis in BC; haec enim a corpore eodem tempore percurritur quo verticalis AB. Corollarium 4. 332a. Nisi igitur sit α > 1 vel α = 1 , innumerabiles lineae curvae inveniuntur problemati satisfacientes; quae ergo omnes minore tempore absolventur quam perpendicularis AB. Corollarium 5. 333. Cum ergo ex unica curva AND infinitae oriri queant curvae AMC, facile intelligitur infinities plures curvas huic quaestioni satisfacere quam praecedenti. Corollarium 6. [p. 155] 334. Si α < 1 , effici potest determinanda constante C, ut curva quaesita per datum punctum rectae BC transeat. Deinde aliis assumendis curvis AND simili modo infinitae curvae poterunt inveniri, super quibus corpus non solum dato tempore ad rectam BC perveniat, sed ad quodvis in ea punctum datum C. Scholion 1. 335. In hoc exemplo casus, quo α = 1 , bis occurrit; prima enim vice linea recta verticalis tantum satisfaciens est inventa, altera vice praeter hanc rectam alia inclinata, utroque tamen modo eadem aequatione generali sumus usi. Saepius autem iam huiusmodi casus obtigerunt, in quibus aequationes differentiales continent in se aequationes integrales, quae nihilhominus per integrationes non eruuntur. Ut in casu α = 1 haec habetur aequatio differentialis quae integrata dat x = u. Interim tamen perspicuum est hanc aequationem k 2u = ( k 2 + 1 )x in illla quoque contineri, etiamsi per integrationem non prodeat. Et hanc ob rem posterior aequatio integralis aeque satisfacit ac prior x = u. Hinc generaliter colligere licet aequationem differentialem dt = Vdu , T in qua T est talis functio ipsius t, quae evanescat posito t = 0, et V functio quaecunque ipsius u, aeque comprehendere hanc integralem t = 0 ac hanc [p. 156]EULER’S MECHANICA VOL. 2. Chapter 2e. page 266 Translated and annotated by Ian Bruce. dt = Vdu , T ∫ ∫ quae per integrationem elicitur. Plerumque quidem casus t = 0, si t est simplex quantitas, negligi potest; at si t est quantitas composit ut in nostro casu, perperam omittitur. Similem casum supra habuimus (300) in aequatione ubi observavimus aequationem x = a in ea contineri, etiamsi integratio nequidem possit perfici. Nam posito a − x = t erit − dx = dt et erit functio ipsius t, quae fit = 0, si fit t = 0 seu x = a; namque ∫ Pdx ita accipi iubebatur, ut evanescat posito x = a. Posita ergo hac ipsius t functione = T erit ds = dtT b , ex qua aequatione ergo tuto concludi licet satisfacere aequationem t = 0 seu x = a ideoque problemati illi satisfacere circulum, ut ibi innuimus (303). Magis universaliter vero in hac aequatione Vdu = dt , si T non evanescit posito t = 0, comprehendetur ista aequatio T = 0, T ex qua erit t = constanti quantitati ideoque dt = 0. Unde intelligitur T = 0 contineri in aequatione proposita Vdu = dt . Atque hinc, si t fuerit quantitas composita v.g. ex u et x, T statim habetur aequatio integralis per integrationem vix eruenda. Scholion 2. 336. Casus hic superest, qui peculiarem resolutionem requirit, [p. 157] quando recta BC supra punctum A cum BA concurrit et quando est parallela. Considerabimus autem sequente problemate tantum casum, quo fit BC parallela verticali AB et in data distantia posita; ex quo simili modo casum rectae BC utcunque inclinatae deducere licebit.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 267 PROPOSITIO 37. Problema. 337. Sollicitetur corpus perpetuo deorsum vi uniformi g ; invenire curvas innumerabiles, super quibus corpus ex A (Fig. 42) motum a quiete incipiendo dato tempore ad rectam verticalem EC perveniat. Solutio. Sit AMC curva quaecunque quaesitarum et pro axe sumatur recta verticalis AB; dicatur AP = x, PM = AQ = y; erit celeritas in M debita altitudini gx et tempus per AM = ( dx 2 + dy 2 ) ∫ gx Sit porro AND curva, cuius quaevis applicat QN exprimat tempus per AM, et sit QN = t functioni ipsius y; poterit ex curva AND data curva AMC inveniri. Quare si infinitae curvae AND concipiantur, quae omnes in E communem habeant applicatam DE, omnes producent curvas AMC, super quibus corpus dato tempore per DE expresso ex A ad CE perveniet. Erit itaque t= ∫ ( dx 2 + dy 2 ) gx et posito dt = pdy erit gp 2 xdy 2 = dx 2 + dy 2 atque dx = dy ( gp 2 x − 1 ) [p. 158] Quae ita integrata, ut posito x = 0 fiat y = 0, dabit curvas AMC quaesitas. Sit R functio quaecunque ipsius y et Rdy ita capiatur, ut evanescat posito y = 0. Tum fiat ∫ ∫ Rdy = A posito y = AE = a et existente DE = b sumatur t= b ∫ Rdy ; A erit Quae aequatio, quicquid pro R substituatur, dabit innumeras curvas quaesito satisfacientes. Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 2e. page 268 Translated and annotated by Ian Bruce. Corollarium 1. 338. Casus ergo habetur simplicissimus, si fuerit R = 1; tum enim aequatio separabilis y b prodit. Erit vero t = a ob A = a. Hanc ob rem fit Qui autem nullius est utilitatis ob valorem ipsius y imaginarium. Corollarium 2. 339. Quia autem ( gbR 2 x − A2 ) non potest esse quantitas imaginaria, oportet sit 2 R2 x > A ; etiamsi x = 0. Quare R neque quantitas constans esse potest neque functio gb ipsius y, quae evanescat facto y = 0. Hanc ob rem R talis esse debet functio ipsius y, quae fiat =∝ , si ponatur y = 0. Praeterea tamen eiusmodi esse debet, ut Rdy non fiat ∫ infinitum, quod evenerit, si esset R = 1y vel 12 etc. y Exemplum. [p. 159] 340. Ponamus ergo esse R = 1 ; erit y ∫ Rdy = 2 y et A = 2 a . Hinc habebitur ista aequatio Quae aequatio quia est homogenea, ponatur x = qy; erit seu gb Posito 4a = n et (nq - 1) = r erit Quae posterior formula a quadratura circuli pendebit, si n < 2. Altereo vero casu n > 2 erit integrale Quae ob abit in hancEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 269 Ubi pro C constantem quamcunque accipere licet, quia ipsa aequatio ita est comparata, ut posito x = 0 fiat y = 0. Per methodum autem supra traditam (335) ex aequatione differentilai statim habetur haec integralis unde oritur quae dat duas lineas rectas, nisi sit gb < 8a , quo casu aequatio est imaginaria. In casu, quo n = 2, erit unde fit seu Ubi etiam pro C quantitatem quamcunque accipere licet. Si n < 2, tum constructio curvae partim a logarithmis, [p. 160] partim a quadratura circuli pendet; fiunt enim ob ( n 2 − 4 ) imaginarium logarithmi inventi imaginarii. Hoc igitur casu expedit constructionem perficere prae expressione analytica.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 270 PROPOSITIO 38. Problema. 341. Sollicitetur corpus perpetuo deorsum vi uniformi g dataque sit curva quaecunque BSC (Fig. 43) ; invenire omnes curvas AMC, super quibus corpus descendo ex A dato tempore ad curvam BSC perveniat. Solutio. Sit curvarum quaesitarum quaecunque AMC, per cuius quodvis punctum M ducatur curva MQ similis curvae BSC respectu puncti fixi A, et exprimat curvae AND applicatae NQ tempus per arcum AM; exponet ergo applicata BD tempus per totam curvam AMC. Quo facto poterit vicissim ex data curva AND curva AMC inveniri. Quare si infinitae curvae AND concipiantur, quae omnes in B habeant applicatam BD communem, eae generabunt infinitas curvas AMC, super quibus omnibus corpus ex A descendendo dato tempore per BD expresso ad curvam BSC perveniat. Si nunc AB = a, BD = b et arcui QM abscindatur arcus similis BS ex curva data BSC; erit AQ : AB = PM : RS = AP : AR = PQ : BR. [p. 161] Ponatur porro AP = x , PM = y , AQ = u , QN = t , AR = r , RS = s ; dabitur ob curvam BSC datam aequatio inter r et s atque ob curvam AND datam dabitur aequatio inter t et u. At ob similitudinem erit u : a = y : s = x : r , unde erit y = us et x = ur . a a Celeritas deinde corporis in M debita est altitudini gx, ex qua tempus per AM erit = ∫ ( dx 2 + dy 2 ) gx , quod aequae poni debet ipsi t; unde oritur ista aequatio gxdt 2 = dx 2 + dy 2 .EULER’S MECHANICA VOL. 2. Chapter 2e. page 271 Translated and annotated by Ian Bruce. Quia autem t per u datur, sit dt = pdu et p sit functio ipsius u; atque ob dx = udr +a rdu et dy = uds +a sdu transibit illa aequatio in hanc garup 2du 2 = ( udr + rdu )2 + ( uds + sdu )2 At quia curva BSC datur, erit s functio ipsius r sitque ds = qdr existente q functione ipsius r quacunque. His substitutis habebitur aequatio inter u et r ista garup 2du 2 = ( udr + rdu )2 + ( uqdr + sdu )2 Quae radice quadrata extracta dat Ex qua si aequatio inter r et u inveniatur, inde habebitur simul aequatio inter x et y pro curva quaesita. Quod autem ad curvam AND attinet, sit P functio quaecunque ipsius n et Pdu ita ∫ integratum, ut evanescat facto u = 0 et fiat = A posito u = a; tum sumatur t = b ∫ Pdu pro A aequatio curvae AND. Erit ergo p = PA b , ubi pro P functionem quamvis ipsius u ponere licet. Q.E.I. Corollarium 1. [p. 162] 342. Si u ponatur = 0, eo ipso quoque x et y evanescunt, nisi forte fiat r vel s infinitum. Illo igitur casu in integratione aequationis differentialis inventae constantem quamcunque addere licet, quia non opus est, ut r datum habeat valorem, si u fit = 0. Corollarium 2. 343. Tum igitur ob constantem arbitrariam addendam ex unica curva AND data innumerabiles inveniuntur curvae AMC quaesito satisfacientes. Corollarium 3. 344. Si curva BSC ita est comparata, ut nusquam neque r neque s fieri queat infinite magnum, semper unica curva AND infinitas dabit curvas quaesitas AMC. Quae non solum hanc habebunt proprietatem, ut corpora super iis descendentia simul ad datam BSC perveniant, sed quoque simul ad quamvis datae similem curvam QM pertingent. Corollarium 4. 345. Cum igitur in integratione aequationis inventae constantem quamcunque addere liceat [p. 163], ea ita poterit assumi, ut curva AMC ad datum punctum C curvae datae BSC dirigatur. Hocque modo infinitae curvae AMC poterunt inveniri, quae omnes in dato puncto C conveniant.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 272 Scholion 1. 346. Posuimus curvas QM similes curvae BSC, ut curva ipsa in A erecta fiat infinite parva et omnia puncta curvae BSC in A conveniant et x et y evanescant posito u = 0. Potuissemus autem eodem modo curvas QM vel cum BSC congruentes ponere vel discrepantes lege quacunque. Ut sit Q functio ipsius u quaecunque evanescens posito u = 0 abeatque ea in B facto u = a, curva QM ita pendere poterit a curva BSC, ut sit Qr Qs x = B et y = B ; namque facto u = a curva QM transibit in ipsam BSC et in A curva in punctum transibit, nisi curva BSC in infinitum progrediatur. At etiam hoc casu pro Q talis accipi poterit functio, ut, etiamsi fiat r =∝ , tamen Qr et Qs fiat = 0, si u = 0. Posito autem dQ = Vdu habebitur aequatio generalis sequens Quae aequatio latissime patet et ex unica curva AND infinite infinitas curvas AMC suggeret, quae etiam infinitas suppeditabit, quae per datum punctum C transeunt. Scholion 2. [p. 164] 347. Quantumvis generalis autem est haec aequatio, tamen curva QM est similis curvae BSC, quia est x : y = r : s . Quare adhuc generalior solutio poterit exhiberi, in qua curvae QM utcunque dissimiles ponuntur curvae BSC, eiusmodi tamen, ut QM in BSC abeant facto u = a. Obtinebitur vero haec solutio, si R sumatur functio quaecunque ipsius u evanescens facto u = 0 abeatque R in D posito u = a sitque dR = Wdu. Sumatur enim Qr x = B et y = Rs ; D abibit x in r et y in s, si fiat u = a, atque evanescente u tam x quam y evanescent, quicquid sit r. Hinc autem sequens orietur aequatio generalissima loco dr, vel etiam loco r poterit In hac aequatione loco dr introduci potest ds ponendo ds q x introduci ponendo loco r eius valorem Bx et tum habebitur aequatio inter u et x. Q Notandum autem est, quia Q evanescit facto u = 0, P talem esse debere functionem ipsius u, ut P 2Q posito u = 0 vel fiat quantitas finita vel infinita magna; at tamen cavendum, ne ∫ Pdu debito modo sumtum fiat infinite magnum.EULER’S MECHANICA VOL. 2. Chapter 2e. page 273 Translated and annotated by Ian Bruce. Corollarium 5. 348. Aequatio in solutione inventa fit separabilis, si fuerit P = 1 ; erit u A = 2 a et p = b . [p. 165] Habebitur enim 2 au dantur enim s et q per r. Corollarium 6. 349. Simili modo aequatio scholii 1 separationem admittet, si furerit P 2Q = V 2 seu P = V . Q Habebitur enim in qua indeterminatae u er r sunt a se invicem separatae. Exemplum 1. 350. Manente P 2Q = V 2 seu ∫ Pdu = 2 Q ob Vdu = dQ, erit facto n = a A = 2 B. Unde fit Habebit ergo ∫ Pdu requisitam proprietatem, ut evanescat facto u = 0; evanescit enim Q. Sit nunc curva BSC circulus super diametro AB descriptus; erit s = ( ar − r 2 ) et q = a − 2r 2 ( ar − r 2 ) atque 1 + qq = a2 ; 4( ar − r 2 ) his valoribus loco s et q substitutis prodibit ista aequatio [p. 166] Quae aequatio non solum indeterminatas a se invicem habet separatas, sed etiam generaliter per logarithmos integrari potest; potest enim in aequationeEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 274 membrum irrationale rationale effici. Prodibit autem integralis haec Notari hic convenit casum, quo gb = 4a seu b = 2 a , quo tempus per quamvis curvam g AMC aequale ponitur tempori descensus per rectam verticalem AB; tum enim erit Si igitur signum + valeat, erit dr = 0 et r = const. = c, unde fit s = ( ac − c 2 ) et quae, aequatio omnes dat chordas in hoc semicirculo ex A ductas, quemadmodum iam demonstrvimus (102) tempora per singulas chordas esse inter se aequalia. Valeat signum – , erit atque hinc Erit ergo atque ob s = ( ac − c 2 ) habebitur eliminata ergo r prodibit ista aequatio (posito CB = m ) y 2 + x 2 = ma xy . Hae ergo curvae hanc habent proprietatem, ut arcus earum a semicirculo abscissi absolvantur descendendo eodem tempore, eo scilicet tempore, quo singulae semicirculi chordae percurruntur.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 275 Corollarium 7. [p. 167] 351. Huius autem curvae, cuius aequatio est y 2 + x 2 = ma xy , Figura est AMFA (Fig. 44); habet nimirum diametrum AF cum verticali AP angulum semirectum constituentem et in A nodum. At vero omnes hae curvae sunt inter se similes et omnes ad omnes circulos accomodari possunt. Corollarium 8. 352. Si ergo in hac curva sumatur quodcunque punctum M et per hoc et A circulus transiens concipiatur centrum habens in verticali AP, corpus arcum AM eodem tempore percurret, quo diametrum circuli seu quo chordam AM. Quare haec curva hanc habet proprietatem, ut quivis arcus AM a corpore ex A descendente absolvatur eadem tempore, quo subtensa AM. Corollarium 9. 353. Hoc ergo casu, quo P 2Q = V 2 (349), perinde est, sive sit Q = u sive secus; eadem enim prodit aequatio inter x et y, uti tam ex exemplo hoc quam ex aequatione intelligitur. Exemplum 2. 2 2 354. Manente P Q = V , ut sit [p. 168] sit curva BSC circulus centro A radio AB = a descriptus; erit s = ( a 2 − r 2 ) et Quibus substitutis prodibit sequens aequatio ubi gb maius esse debet quam 4a, quia r non excedere potest a. Hinc statim ille radius innotescet, qui quaesito satisfacit, ponendo quo casu eritEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 276 Unde erit quae est tangens anguli illius radii, super quo corpus dato tempore b ad peripheriam pervenit, cum verticali AB. Curvae praeterea algebraicae non dantur, quia formula differentialis non effici potest rationalis. Exemplum 3. 355. Sumta aequatione generalissima ex (347) ponatur linea BSC recta horizontalis; fiet r , ubi q erit infinite magnum. = a et dr = 0. Hanc ob rem loco dr introducatur eius valor ds q Deletis ergo terminis, qui prae q evanescunt, proveniet ista aequatio Quae aequatio ob Wdu = dR , et P, Q et V data per u integrationem admittit. [p. 169] Erit nempe Ex qua aequatione ergo invenitur s. Deinde, cum sit y = Rs , at y evanescere debeat facto D u = 0, debebit esse C = 0, si quidem integrale ita sumatur, ut evanescat posito u = 0. Tum ergo erit Quae est aequatio generalis pro omnibus curvis, super quibus corpus ex A ad horizontalem datam descendit. Exemplum 4. 356. Teneatur aequatio generalissima supra inventa (347) et ponatur linea BSC recta verticalis parallela ipsi AB et ad distantiam f ab ea posita; erit s = f et q = 0. Quare habebitur ista aequatio Cum autem sit Qr = Bx, hoc substituto prodibitEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 277 fR unde invento x erit y = D . At quia in illa aequatione indeterminatae x et u non sunt a se invicem separatae, non multum ex ea derivare licet. Scholion 3. 357. Ex generali huius problematis solutione, quando unica curva AND infinitas dat curvas AMC, colligere licet solutionem huius problematis, quo infinitae requiruntur curvae, [p. 170] super quibus omnibus corpus ex A ad datum punctum pervenit. Quaelibet enim curva AND unam dabit curvam per datum punctum curvae BSC transeuntem hocque modo innumerabiles huiusmodi curvae obtinebuntur. Sed cum hoc modo solutio nimis esset difficilis et operosa, aliam genuinam magis afferre convenit. Modus autem, quo utemur, ita est comparatus, ut unam curvam iam nosse oporteat, ex qua innumerabiles deducere docebimus. Haec ergo curva, quae nota esse debet, ex alterutra traditarum methodo eliciatur, ut ex (350), ubi curva per quodvis punctum semicirculi transiens inveniri potest dato tempore describenda. PROPOSITIO 39. Problema. 358. Sollicitetur corpus perpetuo deorsum vi uniformi g dataque sit curva AMC (Fig. 45); super qua corpus ex A ad punctum datum C pervenit; invenire omnes curvas AnC, super quibus corpus eodem tempore ad punctum C ex A descendit. Solutio. Sumta verticali AB pro axe omnium curvarum sit curvae datae AMC abscissa AP = t, applicata PM = u sitque curva ANC una quaesitarum; capiatur in ea arcus AN, [p. 171] qui eodem tempore absolvatur quo arcus AM; iungantur puncta M et N recta MN et construatur curva ALB talis, ut applicata PL aequalis sit rectae MN. Haec ergo curva ALB occurret axi AB in punctis A et B; nam incidente puncto M in A punctum N quoque in A incidet et posito M in C punctum N quoque erit in C, quia arcus AMC et ANC eodem tempore percurri ponuntur. Intelligitur autem ex curva ALB inveniri posse curvam ANC; quare si infinitae huiusmodi curvae ALB concipiantur in A et B incidentes in AB, earum quaeque dabit curvam ANC hocque modo innumerabiles prodibunt curvae ANC quaesito satisfacientes. Sit nunc PL =EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 278 r; erit r functio quaedam ipsius AP = t; curvae autem ANC ponatur abscissa AQ = x et QN = y. His positis erit MN = (( x − t )2 + ( u − y )2 ) = r ideoque y = u ± ( r 2 − ( x − t )2 ) . Porro, quia tempus per AM aequale ponitur tempori per AN, erit seu xdt 2 + xdu 2 = tdx 2 + tdy 2 . Est autem Sit du = pdt et dr = qdt ; erunt r, p et q functiones ipsius t; ponatur porro brevitatis gratia x − t = z seu x = t + z ; erit et Hinc obtinetur ista aequatio ex qua z et t determinetur; habebitur aequatio inter x et y. [p. 172] Quo autem appareat, cuiusmodi functio ipsius t loco r debeat accipi, ut r evanescat tam posito t = 0 quam t = AB = a, sit P functio quaecunque ipsius t evanescens posito t = 0 et Q sit etiam talis functio evanescens posito t = 0; abeat vero Q in A, si fiat t = a; poterit ergo poni r = P( A − Q ). Hocque valore substituto quicquid loco P et Q substituatur, habebitur aequatio pro curvis quaesitis. Q.E.I. Scholion 1. 359. Ex hac quidem aequatione maxime intricata parum concludi potest ad propositum, etiamsi haec methodus genuina esse videatur. Saepe autem aequatio inventa ad absurdum deducere debet, ut si curva data AMC fuerit linea brevissimi descensus, quo casu non dari potest alia curva, super qua descensus fiat eodem tempore. Ad nostrum ergo institutum conveniens videtur de lineis celerrimi descensus tractare eaque problemata resolvere, in quibus inter omnes curvas vel eiusdem longitudinis vel aliam proprietatem communem habentes ea quaeritur, quae minimo tempore absolvatur, atque etiam quemadmodum inter omnes lineas, super quibus descensus fit eodem tempore, ea sit invenienda, quae data quapiam proprietate sit praedita. Etiamsi enim difficillimum sit omnes lineas idem [p. 173] descensus tempus habentes exhibere, tamen ex iis quaelibet potest inveniri exEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 279 proprietate, quam prae reliquis omnibus possidet. Requiritur autem ad hanc rem pertractatandam methodus isoperimetricorum, quam ut passim expositam hic non explicabimus. Scholion 2. 360. Huius autem problematis solutio per (348) sequenti modo habetur. Erat ibi Sit punctum C (Fig. 43), ad quod omnes curvae convenire debent, ponaturque AE = f et EC = h ; si ergo fit r = f, fieri debet s = h. Ad hoc sit S functio quaecunque ipsius r, quae abeat in F posito r = f, quo facto ponatur s = Sh . Substituatur hic valor in superiore F aequatione eaque ita integretur, ut posito u = a fiat r = f. Deinde ex ea aequatione prodibit aequatio inter coordinas curvae quaesitae AMC, nempe AP = x et PM = y, ex quo, quod est x = ur et y = us . a a Atque arbitrarius valor ipsius S dabit infinitas curvas AMC puncta A et C iungentes et super quibus corpus descendens tempore dato = b perveniet ex A ad C. Sit autem atque [p. 174] dS = Tdr; erit q = hT F seu Quae aequatio ita integretur, ut posito u = a fiat r = f; quo facto ponatur x = ur et y = hSu a Fa atque habebitur aequatio inter x et y pro infinitis curvis AMC quaesito satisfacientibus.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 280 PROPOSITIO 40. Problema. 361. Invenire legem generalem, secundum quam curva disposita esse debet, ut corpus super ea descendens citissime perveniat ad quodvis curvae punctum. Solutio. Sit AMC (Fig. 45) curva huiusmodi, super qua corpus ex A ad C tempore breviore perveniat quam supe quavis alia curva per puncta A et C transeunte. Sumtis ergo in ea duobus quibusque punctis M et μ curva inter ea intercepta ita debet esse comparata, ut corpus in motu suo per AMC arcum breviore tempore absolvat quam quemvis alium, si esset interceptus. Sint nunc puncta M et μ proxima iuncta duobus elementis Mm, mμ , et debebit tempus per Mmμ esse minimum seu per regulas methodi maximorum et minimorum aequale tempori per elementa proxima Mn, n μ . [p. 175]Ducantur ad axem AP applicatae MP, mp, μπ sumtisque elementis Pp, p π inter se aequalibus seu quoque MG = mH et pm, si opus est, ad n producta erit mn infinite parvum respectu elementorum Mm et m μ . Debebit ergo esse t .Mm + t .mμ = t .Mn + t .nμ . Sit celeritas, quam corpus in M habet, debita altitudini v, qua ergo tam elementum Mm quam Mn percurret. Celeritas autem, quam in m habebit, debita sit altitudini v + du et celeritas, quam in n habebit, debita sit altitudini v + du + ddw ; illa autem celeritate percurret elementum m μ , hac vero elementum n μ . Hinc ergo habebitur ista aequatio est vero unde ductis centris M et μ arculis mg et nh erit Porro estEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 281 Quibus, neglectis negligendis, substitutis oritur Est vero propter triangula simila nmg, mMG et nmh, μ mH, ut sequitur, et Quamobrem erit Quae aequatio est homogenae et determinat naturam curvae AMC brachystochronae vocatae, [p. 176] super qua corpus tempore brevissimo ex A ad C pervenit. Q.E.I. Corollarium 1. 362. Si ergo dicatur erit His substitutis habebitur In qua si ex potentiis sollicitantibus determinatur v, du et ddw, habebitur aequatio pro curva brachystochrona. At semper ddw ita involvet mn, ut mn ex calculo excedat. Corollarium 2. 363. Sit radius osculi curvae Mmμ = r ; isque in plagam aversam ab axe AP directus erit at est hinc prodibit ista aequatioEULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 282 Ubi notandum est esse 2rv vim centrifugam, qua curva in M secundum normalem ad curvam premitur. Corollarium 3. 364. Si ex potantiis sollicitanibus fluat erit quia puncto m in n translato crescit dy particula mn et ds particula ng. [p. 177] Quia dy .mn autem est ng = ds , erit quibus substitutis habebitur ista aequatio Scholion 1. 365. Ex solutione intelligitur formulam inventam latissime patere atque ad potentias sollicitantes quascunque extendi, etiam resistentia non excepta. Quaecunque enim fuerint potentiae sollicitantes, determinari potest tam du quam ddw, qui valores substituti dabunt aequationem pro brachystochrona quaesita. Attamen haec tantum locum habent, si potentiarum directiones sint in eodem plano; curva enim inventa est in eodem plano sita. Nihilo tamen minus, si potentiae non fuerint in eodem plano, curva brachystochrona in dato plano ope formulae huius poterit inveniri. In quolibet enim plano dato peculiaris erit curva brachystochrona, quaecunque fuerint potentiae sollicitantes. Alia vero quaestio est, si quaeritur linea brachystochrona inter omnes omnino lineas data duo puncta iungentes, etiam non in uno plano sitas. Quoties vero potentiarum sollicitatium directiones in eodem plano sunt positae, dubium non est lineam brachystochronam in eodem positam esse plano. Nam si curvae non essent in eodem plano, potentiae oblique agerent et propterea corpus non tantum, quantum fieri potest, accelerarent. [p. 178] Ex hac igitur solutione tam linea absolute brachystochrona, si potentiarum sollicitantium directiones sunt in eodem plano, invenitur quam linea, quae in dato plano est brachystochrona, quaecunque fuerint potentiae sollicitantes.EULER’S MECHANICA VOL. 2. Chapter 2e. Translated and annotated by Ian Bruce. page 283 Scholion 2. 366. Questionem hanc de linea brachystochrona seu celerrimi descensus primus produxit Cel. Ioh. Bernoulli atque plures eius solutiones extant tam in Act. Lips. quam Transactionibus Angl. et Comm. Acad. Paris. et alibi, ubi hoc problema tam in hypothesi potentiae sollicitantis deorsum directae quam pro viribus centripetis solution dederunt. Nemo autem problema fundamentale, quae hic dedimus, tam late patens praemisit, ut ad potentias quascunque et resistantiam etiam extendi posset. Sumserunt enim omnes ddw = 0 , quod semper perperam fit, nisi directio potentiae sit MG vel mH. Et hanc ob rem Cel. Hermannus cespitavit, dum tali propositione ad brachystochronas in medio resistente inveniandas est usus in Comm. Acad. Petrop. A 1727; quasque correctas dedi in iisdem Comm. A. 1734 ex hoc ipso problemate. [Ioh. Bernoulli, Curvatura radii in diaphanis non uniformis solutionque problematis a se in Actis 1696, p. 269, propositi de invenienda linea brachystochrona, Acta erud. 1697, p. 206; Opera Omnia, Tom. I, p. 187; Lettre de Mr. Jean Bernoulli a Mr. Basnage, sur le problème de isopérimètres, Histoire des Ouvrages des Sçavans, Paris 1697, p. 452; Opera Omnia, Tom. I, p. 194; Remarques sur ce qu’on a donné jusqu’ici de solutions des problèmes isopérimètres, Mém. de l’acad.d. sc. de Paris 1718, p. 100; Opera omnia, Tom. II, p. 225. Iac. Hermann, Theoria generalis motuum, qui nascuntur a potentiis quibusvis in corpora indesinenter agentibus, Comment. acad. sc. Petrop. 2 (1727), 1729, p. 139. L. Euleri Commentatio (E042) : De linea celerrimi descensus in medio quocunque resistente. Comment. acad. sc. Petrop. 2 (1734/5), 1740, p. 135. L. Euleri Opera Omnia, series I, vol. 25. Commentatio (E056) : Curvarum maximi minimive proprietate gaudentium inventio nova et facilis, Comment. acad. sc. Petrop. 2 (1736), 1741, p. 172. L. Euleri Opera Omnia, series I, vol. 25. Relati per P.St.]