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

PROPOSITION 12. Problem.

83. A body which is moving on the curve AM (Fig.12) is acted on everywhere by a force MF, the direction of which is parallel to the axis AP; to determine the speed of the body at individual points and the time taken for any part of the curve to be described by the body, with the force due to the curve acting on the body at individual points.

Solution.

The body describes the arc AM and the speed of the body at A corresponds to the height b while the speed at M corresponds to the height v. Now with AP = x, PM = y and with the arc AM = s the force MF which is called p is resolved into the sides, clearly the normal MN and the tangent MT; there is ds : dx = MF : MT and ds : dy = MF : MN . Therefore there is hence produced the tangential force pdx pdy MT = ds and the normal force MN = ds . It is evident ∫ ∫ here that dv = − pdx and v = C − pdx (42). Moreover with the integral thus taken pdx , ∫ in order that it vanishes with x = 0, we have v = b − pdx ; [p. 40] from which equation the speed of the body is known at individual points. From the same equation it is found also the time in which the arc AM is completed; for with the time put as t, there results pdy The normal force MN = ds is completely devoted to pressing the body to the curve along MN (39), therefore it increases the pressing force arising from the centrifugal force, since MN falls in the region opposite to the radius of osculation MO. Whereby, with the radius of osculation put as MO = r , the centrifugal force is equal to 2rv (20), and the total pdy force pressing on the curve next to MN = ds + 2rv . Q.E.I.

Corollary 1.

84. The speed at M is therefore of such a size as it would be at P, if the body rises with the same initial speed b along AP to the particular heights acted on by the same force.

Corollary 2.

85. Therefore the speed does not depend on the nature of the curve, but only on the height that the body traverses. Clearly if the height of the element were dx, then dv = – pdx or dv = pdx, as the body either rises or falls. ∫

Corollary 3.

86. Since we have v = b − pdx , if the abscissa x to be used is taken as far as AC, for which ∫ pdx = b , then the speed of the body corresponding to that height B is equal to zero. Therefore the body rises as far as B and there it is at rest ; now continuing by falling from B along BMA.

Corollary 4.

87. If the ascent along AMB is compared with the rectilinear ascent along APC, the time to pass along the element Mm to the time for Pp is as Mm to Pp, i. e. as ds to dx. Corollary 5.

88. Whereby if the line AMB is straight, as the ratio Mm to Pp is constant, the time to pass through AM to the time to pass along AP is in a constant ratio, surely that which the total sine has to the cosine of the angle A, or which the length AB has to AC.

Corollary 6.

− ds and thus the 89. With the element Pp placed constant, the radius of osculation r = dxddy 3 centrifugal force is equal to Whereby the total force pressing on the curve is equal to

Scholium 1.

90. As in this problem, from the given curve and force acting, the speed at individual points, the time to pass through any arc, and the pressing force on any point of the curve, are found : thus from these five things with any two, the remaining three can be found. From which ten problems arise, which all have a solution from the solution of this problem.

Scholium 2.

91. Similarly there can be ten questions of this kind, if the directions of the forces acting are not parallel, but either converge to a centre of force or have their directions determined in some other way. But if also the direction between these sought is put in place, then from the six quantities taken in the computation, from any three the other three can be found; and hence twenty problems are to be found. [One direction is fixed, and there are directions for the other four quantities, leaving 6 variables.]

Scholium 3.

92. Again indeterminate problems may arise, as if in place of the time through which some part of the curve is traversed can only be given by an integral along AMB ; for then an infinite number of solutions can be put in place. Besides if more ascents or descents are considered to be integrated upon various parts of the same curve, and the ratio of these is given, then the number of questions is increased much more. To this kind belongs the question of finding the curve, upon which all the ascents and descents to the same point are made in the same time, as these are the most difficult we will handle finally. Moreover now we take the first curve and force acting as given and we solve the problems pertaining to this. Next, we will show how from different given quantities, in what way the others are to be found.

PROPOSITION 13. Problem.

93. If the force acting is uniform and acting downwards everywhere, to determine the descent of the body on a given curve AM (Fig.13), beginning from A at rest, and to find the force pressing on the curve at individual points M.

Solution.

With the vertical AP or with the parallel in the direction of the force drawn MF and with the connected line MP at right angles, let AP = x, PM = y, with the curve AM = s. The force MF is put equal to g with the force of gravity present equal to 1 and the speed at M corresponding to the gdy height v. With these in place, the normal force = ds and the gdx tangential force = ds (83). Because in this case the tangential force accelerates, we have dv = gdx and v = gx as the speed at A = 0. Hence since the radius of osculation

• ds with dx placed along the direction MO is equal to dxddy 3 constant, the centrifugal force =
• dvdxddy , the direction of which is MN. Now the ds 3 gdy normal force ds acts along the same direction. whereby the total force pressing on the curve along MN at M is equal to as v = gx. Truly the time in which the body traverses the arc AM is equal to ∫ dsgx . Q.E.I. [p. 44]

Corollary 1.

94. Therefore the speed at M only depends on the height AP through which the body descends, and the body acquires the same amount as falling from A to P and acted on by the same force g.

Corollary 2.

95. Therefore, whatever the curve the body falls along from rest, acted on by the constant force g, the speeds are as the square roots from the squares [of the speeds] proportional to the heights traversed; that is as v , i. e. as gx .

Corollary 3.

96. The time, in which the first element Aa is traversed, is ∫ dsgx with x evanescent. If therefore the angle PAa is less than a right angle, or s = nx, then the time to pass through Aa is infinitely small and thus the time to pass along AM is finite, unless the curve either rises between A and M beyond A or it progresses to infinity. But if the angle PAa is right, for the point A, s n = ax with a number n present greater than one and thus Whereby if n is less than two, the time to pass along Aa is infinitely small and the time to traverse AM is finite. But if n is greater than or equal to 2, the time to pass through the first element Aa is infinitely great or the body never escapes from A. [p. 45] Corollary 4.
97. But whenever n < 2, to the radius of osculation at A is indefinitely small. Whereby in this case, in which the tangent to the curve at A is normal to AP, the body does not descend in this case, unless the radius of osculation at A is infinitely small. Scholium 1.
98. From which, as the first element is traversed in an infinitely short time, it is correctly concluded that the time to traverse the arc AM is finite; for since the body descends along AM with an accelerated motion, the following elements are described more quickly and on this account the corresponding time must be finite. Moreover everything is illustrated in the following. Example 1.
99. Let the line AM (Fig. 14) be straight and inclined at some angle to the vertical AP and the cosine of the angle A = n; then x = ns. Therefore the time, in which the body descends along AM, is equal to or the time to travel along the line at any inclination varies directly as the root of the length of the line and inversely as the root of the cosine of the angle of inclination MAP. Moreover the centrifugal force is equal to 0 [i.e. infinite radius of curvature], whereby the line AM is only acted on by the normal force, which is equal to

Corollary 5.

100. Therefore the time to pass along AM is to the time to pass along AK as AM to AK . But the time to pass along AM is to the time along AP as AM to AP (88). [p. 46] Whereby if it is the case that AM : AP = AM : AK or AM : AP = AP : AK , which comes about if PK is perpendicular to AM , then the descent time along AK is equal to the descent time along AP.

Corollary 6.

101. It is also apparent that the descent time along the perpendicular PK is equal to the . Whereby, since the time descent time along AP. For the cosine of the angle APK = PK AP along AP to the time along KP is as AP to 1 KP : PK , this is an equal ratio. AP Corollary 7.
102. From this it is evident in the circle APPB (Fig. 15), that all the descents along the chords AP drawn from the uppermost point A and all the descents along the chords drawn to the lowest point B are to be made in equal times, clearly each in that time in which the body falls freely perpendicularly along the diameter AB. Example 2.
103. If the curve AMB (Fig. 16) is a circle and the radius BC = a and AP is a tangent to the circle, then Hence we have ds = adx and hence v = gx ( a2 − x2 ) and r = a , and the centrifugal force is equal to 2 gx and on this account dy = a xdx , and the ( a2 − x2 ) total force sustained by the circle at M is equal to 3 gx . a Therefore the total force pressing on the curve is three times as great as the normal force.EULER’S MECHANICA VOL. 2. Chapter 2a. page 62 Translated and annotated by Ian Bruce. The time then, in which the arc AM is traversed, is equal to ∫ g( aadxx− x ) , [p. 47] 2 3 the integration of which neither depends on circular nor hyperbolic quadrature, but with the help of rectification of the elastic curve it is able to be constructed. Meanwhile the time along the quadrant AB is equal to Corollary 8.
104. When the body reaches the lowest point B, there it has the speed corresponding to the height ga. Therefore this ascends in the other quadrant BD and reaches D, where its speed vanishes, and thus again it descends to B and then it re-ascends to A along BA. Now in a similar way is the ascent and descent along the sides of a square, since the body either in the ascent or in the descent, has the same speed at the same points. Scholium 2.
105. We will not offer other examples now, as in what follows, where we consider more descents on a given line, we are to report on more examples. Now truly we disclose in the first place these questions which pertain to the motion on a given line starting from rest at a given point, and the following problem is of this kind.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 63 PROPOSITION 14. Problem.
106. If there is an infinite family of similar curves AM, AM, etc. (Fig.17) beginning from the fixed starting point A, to find a curve CMM from these other curves cutting the arcs AM, AM, etc, which are traversed in equal times by a body descending along these arcs, as before, with a uniform force present acting downwards everywhere. [p. 48] Solution. From the infinite number of curves given one is taken AM, the parameter of which is a. Putting in place AP = x, PM = y and with the arc AM = s and as before the force acting being equal to g , the body descends on the given curve AM; the speed at M corresponds to the height gx. Hence the descent time on AM is equal to ds . Hence ∫ gx from all these curves AM, AM, etc., so many arcs are to be cut, in order that from these the quantity ds is constant. ∫ gx But ∫ dsgx is referring to other curves, if besides s and x, the parameter a also is made a variable. Therefore with a made variable as well in ds , the quantity ds is indeed constant for that time in ∫ gx ∫ gx which all the descents are to become the same. Let this time be equal to k, and k = ds for individual curves. Whereby if ds is thus differentiated, as also a is placed ∫ gx ∫ gx variable, this differential is put equal to zero. In order that the differential of this integral can be found, let ds = pdx and p is a function in which a and x likewise make numbers with zero dimensions [such as a function of x/a]. [p. 49] Hence we have the integral ∫ gx ; this differential with the variable a in place also, gives pdx which must be equal to 0. Now the quantity q is to be found in the following way. Since k= ∫ gx , in the quantity k [or in dk], the variables a and x produce a number having the pdx dimensions 12 . Moreover this is shown elsewhere, in Vol. VII Comment.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 64 [E044 : Concerning an infinite number of curves of the same family. Or a method for finding an infinite number of curves of the same kind, Commen. acad. sc. Petrop. 7 (1734/5) 1740. We now present §23 of this paper, in the original and in translation : ‘Sin vero fuerit u functio m dimensionum ipsarum a et x, atque du = Rdx + Sda; erit um x functio ipsarum a et x nullius dimensionis. Differentietur igitur um et prodibit x xdu − mudx seu Rxdx − mudx + Sxda . Quod cum sit differentiale functionis nullius dimensionis x m+1 x m+1 2 erit Rx − mux + Sax = 0 , seu Rx + Sa = mu . Quare si fuerit u functio m dimensionum ipsarum a et x; atque ponatur du = Rdx + Sda; erit Rx + Sa = mu ideoque du = Rdx + da ( mu − Rx ) seu adu = Radx − Rxda + muda . ' a ‘However, if u is a function of m dimensions of a and x, and du = Rdx + Sda; then um x is a function of a and x zero dimension. Therefore um is differentiated, and there is x xdu mudx Rxdx mudx Sxda − −

produced . Because it arises from the differentiation of a or x m+1 x m+1 2 function of zero dimensions, Rx − mux + Sax = 0 , or Rx + Sa = mu . Whereby if u is a function of dimension m of a and x themselves; and the equation du = Rdx + Sda is put in place, then Rx + Sa = mu and thus du = Rdx + da ( mu − Rx ) or a adu = Radx − Rxda + muda . ’ Thus, in the present case, R = p , S = q, m = 12 , and gx u = k . In addition, the establishment of functions of zero dimensions is discussed in E012 in these translations.], then the equation arises From which it is found that [there is a misprint present here in the O. O., but not in the original text.] Therefore we have which is the equation for the curve sought. But if the equation between the coordinates x and y for the curve CMM is desired, from the equation for each of the curves AM, the value of a found must be substituted into the equation between x and y. Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 65 Corollary 1. 107. Also the equation found at first, is sufficient for the curve CMM to be found. As for any given abscissa AP = x , from that the parameter a of this curve AM is found, of which the corresponding point M of the assumed abscissa x lies on the curve CMM sought. Corollary 2. 108. Moreover since this is a differential equation, and thus to which more curves pertain according to the added constant, it is to be noted that with the addition of the constants, only that solution is to be agreed upon for which the given curve is completed in the time of descent k, for the given value of a that gives the abscissa x only of the required arc AM to be cut off. [p. 50] Corollary 3. 109. If the time of descent k must be equal to the time of descent through the vertical distance AC = b, then k = 2 b . With which value put in place, we have the equation : g In the integration of this equation, it has to be arranged that the curve passes through the point C. Scholium 1. 110. Moreover it is always the case that the vertical line AC arises as a kind of curve AM, if the parameter a is taken to be indefinitely large or small. Whereby the constant time k is most conveniently expressed by the descent through the vertical AC, clearly a kind of curve AM. And in the construction of the equation found a constant of such a size is to be added, as by putting x = b, a is made infinite or zero, as one or the other value corresponds to the value of the line AC. Scholium 2. 111. If it is possible to integrate ds itself, without the aid of any equation, by means of gx which q can be found. For if the integral ds is itself again differentiated with respect so gx some variable a also, q is again found; and it is only necessary to put this differential equal to zero. Now most conveniently in these cases the problem is solved, if the integral of ds is at once put equal to k or to 2 b and in place of a the value is substituted in gx gEULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 66 terms of x and y from the given equation for the curve. [p. 51] And in this way the solution is established not only for similar curves, but also for dissimilar ones, but only if the descent times can be expressed by a finite quantity. Exemplum 1. 112. If all these curves AM are straight lines inclined in different ways to the vertical AC, then where n is to be considered as a parameter. Hence the time becomes which must be placed equal to 2 b itself. Thus there becomes g y Moreover since n is a variable quantity, put the value x for that from the equation y = nx; with which accomplished, this equation is produced for the curve CMM between the orthogonal coordinates x and y : which is the equation for the circle, the diameter of which is the line AC = b. Scholium 3. 113. This case is the one examined before (102); indeed there it was shown that the body descends in equal intervals of time by all the chords drawn in the circle from the uppermost point. Here indeed the case does not concern similar curves, but we report on this example as a case illustrating scholium 2, because for straight lines and for these, the descent times are expressed by finite quantities. Now the following examples will include similar curves, as the proposition postulates. [p. 52] Example 2. 114. Let all the curves AM, AM be circles tangent to the vertical AC at A. The radius of each of these is equal to a, and it is given by Now these circles are all similar curves, because a, y and x in the equation keep a number of the same dimension, or they complete the homogeneity alone. Therefore the radius a must be handled as a variable parameter. Moreover, there is obtained from that equation adx a ds = , whereby we have p = and thus it has the prescribed property, 2 2 2 2 ( a −x ) ( a −x )EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 67 as the dimension of the number formed from a and x is zero. On this account we have this equation for the curve CMM : or this Which equation can be solved; for on putting x = au there is produced in which the indeterminates are separated from each other. Moreover, in which the equation is obtained for the curve CMM between the coordinates x and y, in place of a is put the value y2 + x2 y 2 dy + 2 yxdx − x 2 dy , and in place of da the differential of this . With 2y 2 y2 which put in place the following differential equation is obtained Which thus must be integrated, so that with x = b it makes y = 0, which curve must pass through the point C. Corollary 4. 115. From this equation the tangent of the curve CMM is known at individual points and from the position of the tangent the angle AMM is known, [p. 53] in which whatever of the given curves is divided. Clearly the tangent of the angle AMM = y x − bx . Therefore here the angle at C is right on account of x = b, or the curve CMM is normal at C to AC. Corollary 5. 116. If b is taken either greater or less, the curve CMM is different also and in this way an infinity of isochronous curves arise being cut from the circular arc. And these curves are all similar between themselves on account of the parameter b, which constitute a homogeneous equation with x and y. Hence with one given curve CMM innumerable others can be constructed from that, clearly with the x and y coordinates of the curve CMM augmented or diminished in the same ratio as AC or b is augmented or diminished.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 68 Example 3. 117. Let all the curves AM, AM be cycloids having cusps at A and vertical tangents AC at A. With the parameter of any cycloid AM put in place, or with twice the diameter of the generating circle equal to a, from the nature of the cycloid, we have s = a − ( a 2 − 2ax ) and ds = adx and ( a 2 − 2ax ) hence dy = dx2 2ax . Hence in this case we have the ( a − 2 ax ) function of a and x of zero dimension, p = a , as ( a − 2ax ) 2 required. Whereby for the curve CMM, this equation is required or [p. 54] If the equation between the orthogonal coordinates x and y is desired, from the equation or with the differential of this likewise, for the variable a, the value of a itself must be substituted. Now this differential equation with the variable q in place gives or Which goes into this : Now the above multiplies by 1 4 a provides this :EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 69 These two added equations give an integrable equation, the integral of which is : From which the value of a elicited becomes : and With which values in the equation : which arises from the two differential equations with da eliminated, on substitution gives the equation for the curve sought CMM. Corollary 6. 118. From this equation the tangent of the angle is found that the curve CM makes with the applied line PM, truly Then also the tangent of the angle becomes known, [p. 55] that the cycloid AM makes with the applied line PM. From the equation of the cycloid is without doubt Now on eliminating a the tangent is equal to Whereby, since either of these given angles is the complement of the other, hence with this taken into account, the angle that the curve CMM makes with any of the given AM is right. Consequently the curve CMM is the orthogonal trajectory of all the given cycloids AM, AM, etc. Corollary 7. 119. With AC taken of another size also other curves CMM are produced and thus an infinity of orthogonal trajectories are found, which are all similar to each other. Hence from one easy given, it is possible to construct any number you please.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 70 Scholium 4. 120. All these isochronous curves being cut from the arc, whatever the curves cut, can always be constructed, even if it is not apparent from the equation. For by quadrature the arcs which are completed in a given time of descent can be removed from the given curves, and in this way any points on the curve sought can be found. If certain curves cut are algebraic, then the equation for the curve cut can always thus be compared, as by making a substitution of the indeterminate, which can then be separated in turn from each other. But if the curves cut are expressed by a differential equation, [p. 56] the differential equation for the curve cut most rarely admits to being separable in terms of the indeterminates. Because, in a singular manner, as I have used in the case of the cycloid, the parameter a can be eliminated and there the substitution cannot deduce a separation. Scholium 5. 121. Then it is necessary to observe that all the isochronous curves cut by an arc, the number of which is infinite, are similar to each other, according to the value of the variable b, if indeed the curves cut are of such a kind. This is gathered from the general equation in which, since p is a function of zero dimensions of a and x, and the quantities a, b and x constitute a homogeneous equation. But from the equation of the curve cut, since in that equation a, x, and y are put in place everywhere to make a number of the same dimensions, the value of a is a function of x and y of one dimension. Whereby with that substituted in place of a, the equation is obtained for the curve cut, in which b, x, and y everywhere constitute a number of the same dimensions. Consequently, for the variable b put in place, there arises an infinite number of curves similar to each other with respect to the point A. Hence with a single curve given, the rest can easily be described by reason of the similitudes. Scholium 6. 122. Now this material concerned with the cutting of isochronous arcs was published in the past [p. 57] in the Act. Erud. Lips. A. 1697 [p.206] [The radius of curvature in translucent media ….and concerning synchronous curves, or the construction of rays from waves; Opera Omnia, Book I, p. 187] by the Celebrated Johan. Bernoulli and later in the Comment. Acad. Paris by the Cel. Saurino, [1709, p. 257 and 1710, p. 208 ; General Solution of the problem, …..] who indeed used another method. Now I have used that method that I have treated in our Comment. pro A. 1734, as the most convenient for the solving of this kind of problems. Now in their works these celebrated men only considered similar curves as I have done, since without doubt for dissimilar curves the solution can be exceedingly difficult and often also too hard to solve. Now these curves are called synchronous in the places they are cited, since arcs traversed in the same time are cut off.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 71 [Note that there was a delay of several years between the writing and the eventual publishing of Euler’s works, even at the start of his career at St. Petersburg; thus occasionally he was able to extend his observations from volumes to be published later into earlier ones that had not yet been published either, as below. Thus, one must take the Enestrom Index with a pinch of salt as regards the chronological order of the works, as there was some coming and going, and of course a number of papers did not make it into the index at all in the original assessment, which should be looked at by somebody with an interest. The original Euler Archive is of course at St. Petersburg, and access does not seem to be that simple, and neither is it free.] Scholium 7. 123. It is apparent, as shown from my dissertation in Vol.VII Comment. Acad. Petrop. that these synchronous curves can be found in a like manner, also if the given curves are not similar, but yet of such a kind that, as with ds = pdx in place, in p the quantities a and x constitute a number of given dimensions; for then it is equally easy to find the value of the letter q ; as if the number of the dimensions of a and x in p is n, this equation for the curve is found : Whereby if n = − 12 , so that we can make p = ac , then we have dx = da and thus x a ( a − x2 ) 2 x = ma, or x can be taken in a given ratio to the parameter a ; therefore in which case the construction of the isochrones is most easy. [p. 58] But if p does not have a value of this kind, from my paper cited above it is understood how the required equation has to be sought.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 72 PROPOSITION 15. Problem. 124. If as before there is an infinite family of similar curves AM, AM, etc. (Fig.18) and the straight line DE in the position given, to find that curve AMN, upon which the body arrives at the line DE in the shortest possible time from A. Solution. By the preceding proposition, with some curve CMM described cutting the isochronous arcs AM, the tangent GMH is drawn parallel to the given line DE. It is evident that the body is to arrive in the shortest time along the curve AM, which touches the line GMH at the point of contact M, since all other points of the line GMH fall beyond the curve CMM and thus a longer time is needed for the body to reach that line. Now, since all the curves [such as CMM] cutting isochronous arcs from the curves AM, AM, are similar to each other, (121), one is taken from these , which the line DE touches; I say that the point of contact is to be at the point N, in which the line AM drawn through the previous point of contact M crosses the line DE. From this it then follows from the nature of the similarity of the curves CMM with respect to the point A, that also it follows, as the arc AMN is similar to the arc AM and crossed the line DE at the same angle that the curve AM crosses the line GH. Whereby, [p. 59] since the body arrives in the shortest time along AM to GH, it is necessary that it also arrives in the shortest time on the curve AMN to the line DE. Q.E.I. Corollary 1. 125. From this it is understood, that if the line DE is horizontal, with the descent along the vertical AC , then the body arrives in the shortest possible time, on account of the horizontal tangent to the curve CMM at C ; which is indeed evident by itself.

Corollary 2.

126. If therefore the curves AM, AM are cycloids, as we put in example 3 of the preceding proposition, the body on that cycloid arrives at the line DE the fastest which crosses this line at N at right angles, since the angle that any cycloid makes with the curve CM is right. Corollary 3.
127. If therefore the line DE is vertical or parallel to AC, the portion of the cycloid AMM is half the cycloid. Whereby the horizontal motion on half the cycloid is the fastest. Corollary 4.
128. If the curves AM, AM are straight lines drawn from the point A to the given line position DE, the body on that line AM (Fig. 19) arrives at DE, which is the chord of the circle passing through A and having the centre in the vertical line AB, and having the tangent line DE (112) [p. 60] Corollary 5.
129. If therefore the angle DEA is n degrees, the angle BAM is 902+ n degrees and the angle AMG is 902− n . Or with AGH drawn the horizontal, and with the angle DGH bisected by the line GF, then the sought line AM is parallel to GF. Corollary 6.
130. Whereby if the line DE is vertical, the body arrives at that line the fastest by descending on the line inclined at 45 degrees to the horizontal [on setting n = 0]. Therefore a body inclined at this angle to the horizontal advances the quickest. Scholium.
131. In a like manner it can be found too, which of an infinite number of similar curves AM, AM (Fig. 18), a body can arrive at a given curve by descending the fastest. For if the line GMH should be any kind of curve touching the curve CMM at M, the body on this curve AM arrives the quickest at the curve GMH, if indeed the whole of the curve GMH is placed beyond the curve CMM. Also in the same way it might be possible to be determined, if the curves AM, AM are not similar, how the above body can arrive the fastest at a given line GH. Indeed from the infinity of curves CMM the isochronous arc from those being cut is the one sought, which touches the given curve GMH, and it is on that curve AM, which passes through that point of contact, it is that point which is sought. But since in these cases generally the curves CMM are to be found with difficulty and it is much more difficult to determine that curve which is a tangent to the given line, [p. 61] we have restricted the question to similar curves only.

PROPOSITION 16. Theorem.

132. The times of the descent, by which a body placed on the curves traverses the curves AM and Am, etc. (Fig.20) similar and similarly from the point A, are in the ratio of the square root of the homologous sides.

Demonstration.

Since the curves AM, Am are similar, AM:Am, AP:Ap and PM:pm are in a given ratio, clearly that, which the homologous sides hold; let the ratio of the homologous sides be N:n. Because the speed at M is to the speed at m as AP to Ap , the speeds at M and m are in the square root ratio of the homologous sides. Now similar elements are taken from M and m, clearly holding the ratio N to n, are the times in which these two elements are traversed, in the ratio composed from the direction of the elements, i. e. N to n, and to the reciprocals of the speeds, i. e. N : n . From which it follows that the times, in which the homologous elements of the curves AM, Am are traversed, are in the square root ratio of the homologous sides. Whereby, since this ratio is constant, the times in which the whole curves AM and Am are traversed, keep this same ratio. Q.E.D. [p. 62] Corollary 1. 133. Therefore the times, in which similar and similarly circular arcs put in place for the descent, are in the square root ratio of the radii.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 75 Corollary 2. 134. Therefore pendulums, which describe similar arcs, complete oscillations in times which are in the square root ratio maintained by the lengths of the pendulums. Corollary 3. 135. The same ratio of the times is kept in place, if the pendulum bodies do not describe circular orbits, but other curves, provided these are similar to each other and they complete similar arcs. Scholium. 136. Moreover in these all the forces acting we put to be uniform and to be pulling downwards, even if we have disregarded this condition. For we have put this hypothesis in place to be acted on previously, as we are now about to progress to others. PROPOSITION 17. Problem. 137. With the force present acting uniformly downwards, a body moves on some curve AM (Fig.21) with an given initial speed at A; to determine the motion of the body on this curve and the force pressing the body to the curve sustained at individual points. [p. 63] Solution. With the force acting put as g and with the initial speed at A corresponding to the height b and as well, AP = x, PM = y, AM = s and with the speed at M corresponding to the height v, with these in place, there is dv = gdx (93), and thus v = b + gx . And again the time in which the arc AM is completed is equal to ∫ ( bds+ gx ) . Furthermore, the total force experienced by the curve along the direction of the normal MN, is equal to (93) with the element dx present constant. For this solution only differs from that solution for proposition 13, because there it was v = gx , here it is v = b + gx . From these formulae therefore the motion as well as the pressing force are known. Q.E.I. [We note that this is not the reaction force, as that force acts on the body, and which is the force we would now calculate. For some reason, Euler persists with the force the body exerts on the curve throughout this book.]EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 76 Corollary 1. 138. If the line AM is straight, now from (88) it is understood that the time to traverse AM is to the time to traverse AP beginning with the same speed b , is as AM to AP. Now on account of the centrifugal force vanishing, the force pressing the body on the gdy curve is equal to ds or constant. Corollary 2. 139. Also it appears in this case, since the motion does not start from rest, that the speed only depends on the height. Whereby, whatever the curve AM shall be, the speed of the body at any point of this is known, also with the kind of curve unknown. [p. 64] Example 1. 140. Let the curve AM be a parabola, having the vertex at A and the axis AP vertical; hence with the parameter of this is put equal to a , y 2 = ax and Hence the time obtained for the body to pass along AM is equal to Hence, as with dx kept constant, then ddy = − adx , and we have 2 4 x ax Consequently the total pressing force is equal to Corollary 3. ga 141. Therefore if b = 4 , then the force on the curve vanishes. Thus in this case the body is free to move along this parabola, which is also the case treated in the preceding book (564).EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 77 Corollary 4. 142. Therefore with the present b = 14 ga , the time to traverse the arc AM is equal to Therefore this is equal to the time to descend beginning from rest along the abscissa AP . Corollary 5. 143. If b > 14 ga , then the force becomes negative; then the curve is therefore pressed in the direction away from the axis AP. But if b < 14 ga , the direction of the force is along MN. The size of the force pressing at the individual points of the curve varies inversely as the radius of osculation. [p. 65] Example 2. 144. If the curve AM is a circle, the radius of which is equal to a and the centre is placed on the vertical line AP, then it is given by y 2 = 2ax − x 2 , hence Hence the time in which the arc AM is traversed is equal to And since dxddy −1 = a , the force that the circle undergoes at a point is equal to ds 3 Corollary 6. 145. The time can be expressed by logarithms, if b = 0; moreover it is equal to infinity , or the body perpetually remains at A. That is apparent from the above treatment (97). For since the normal to the curve at A is AP and since neither is the radius of osculation infinitely small, then the body cannot descend. Corollary 7. ga 146. If b = 2 or the initial speed is the same size as the body acquires in falling from a height of half the radius of the circle, the total force acting with the centrifugal force is 3 gx equal to a ; and thus it is in proportion to the height travelled through.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 78 DEFINITION 3. 147. Oscillatory motion is reciprocal motion in which the body alternately approaches and recedes from the starting point of the motion M (Fig. 22). Thus if the body is moving on a given curve MAN [p. 66], first it descends on MA, then it ascends on AN, while it loses speed; then it descends from N again and ascends on the arc AM, with which done it again descends and this periodic motion continues. Such motion is called oscillatory. Corollary 1. 148. Oscillatory motion hence consists of alternate descents and ascents on a given line; and in the descending motion the body moves with an acceleration, and in the ascents the body truly loses the speeds acquired. Corollary 2. 149. Hence whatever the previous ascent touched upon, the descent is made on the same part of the curve. Whereby, since the speed of the body depends only on the height in a vacuum, the body at the same point on the curve either in the ascent or in the descent has the same speed. Corollary 3. 150. From which it follows that the time for the descent along MA is equal to the time of the descent along AM and likewise in the same manner the time of the ascent along AN is equal to the time of the descent along NA. Corollary 4. 151. The body ascending on the arc AN until it reaches the point N, since the height is equal to that of the point M from which it fell. The one follows from the other, since the speed is determined only by the height. [p. 67] Corollary 5. 152. If the curve AN is similar and equal to the curve AM, then the motion along AN is equal to the motion along AM. Whereby all the ascents and descents are made in equal s times.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 79 Corollary 6. 153. If the curves MA and AN are dissimilar, at least the time along MAN is equal to the time along NAM, or the times of approaching and receding are equal to each other. Corollary 7. 154. Since the body always reaches the same height, clearly this oscillatory motion must last indefinitely. Corollary 8. 155. Hence any curve is suitable for the production of oscillatory motion if it has two arcs such as MAN ascending from the lowest point A. Scholium 1. 156. Here we have set out the properties of oscillatory motion, such as follow from the exposition of the hypothesis of uniform forces acting, and always pulling downwards. Indeed the same is true also, if the forces depend in some manner on the height, or even if they are directed towards a fixed point, that becomes more apparent in what follows. In a medium with resistance, truly the matter is otherwise, for neither is the ascent along a given curve similar to descent along the same, nor in the ascent does the body reaches an height equal to that from which in the descent it had fallen. Scholium 2. 157. It is usual to call the motion along MAN the going movement, following the returning movement along NAM ; hence oscillatory motion consists of alternate goings and comings [we do not have such handy words as the Latin itus and reditus used here in the English language to express this notion, though to and fro’ might be our equivalent]. Truly an oscillation is called by others constant to and fro’ motion, as the [term] oscillation is called by others, to and fro’. Here we accept the name oscillation in the basic sense, so that thus one oscillation is agreed to be one to and fro’ motion. The to motion and indeed the fro’ motion each consists of one descent and one ascent, and thus the whole oscillation includes two descents and two ascents. Therefore since the time for the to motion is the same as the time for the fro’ motion, the time of one oscillation is double the time for one to or one fro’ motion. [This may sound pedantic, but that is not the case, as previous writers incl. Newton and Huygens, had timed pendulum swings from one extreme to the other, and not back to the starting point, as is the more logical thing to do. Thus again we see the hand of Euler gently guiding humanity in the right direction.] Corollary 9. 158. Therefore in this chapter, in which motion is a vacuum is undertaken, if we wish to examine the motion of oscillations, we have a need to consider only either the ascent or the descent upon the two parts of the curve AM, AN.EULER’S MECHANICA VOL. 2. Chapter 2a. Translated and annotated by Ian Bruce. page 80 Scholium 3. 159. Nothing matters, provided the arcs AM and AN in succession make one curve, but if they are different curves, then they are connected at A thus so that they have a common tangent [p. 69]; for otherwise the motion is disturbed. Whereby there is only the need in an inquiry about oscillatory motion to define the motion on the curves AM and AN themselves. This then is sufficient in the determination of oscillations, as the relation between larger and smaller oscillations can then be found. Moreover these oscillations are called larger which are completed by larger arcs, and the smaller by lesser arcs. Scholium 4. 160. It is evident from Proposition 6 (49), how oscillations are able to be effected with the help of pendulums, clearly with the aid of the evolute of the curves AM and AN, around which the thread is taken. Also the use of pendulums was adapted to oscillations by Huygens, as is apparent from his habit of applying that motion to the perfection of clocks. Truly the same difficulties that we have mentioned in the place cited, they have here in this place. On account of which we only investigate the motion of points upon given lines, and we lead the mind away from all the circumstances of pendulums which are able to disturb our intention.