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PROPOSITION 1. Theorem.

  1. Every body, that is carried from one place to another place by relative or absolute motion, passes through all the intermediate positions, and it is not able to suddenly arrive at the final place. Demonstration.

For absolute motion: if the body were suddenly able to arrive at the final place from the first place, it would be necessary for the body to be annihilated at the first place and suddenly produced in the final place, and which cannot be done according to the laws of nature, except by agreeing to a miracle. Therefore it proceeds from place to place until finally it arrives at the final place. For relative motion : if the body truly remains at rest on being substituted from the infinite space into the relative space, then the above reasoning prevails (10). But if it should move in the relative space, then it must pass through the individual intermediate places, and therefore the relative motion will be successive, and it will again be made through the intermediate positions. Q. E. D.

Corollary 1.

  1. From these it follows that a motion cannot be immediate [in time], but to need time for the body to arrive at some place from some previous place [p. 6.]. Since indeed it has to pass through the individual intermediate places, this cannot be consistent with instantaneous motion.

Corollary 2.

  1. Therefore it is necessary for the path to be assigned, by which the body moves, and with that known, there will be no point on that path from the first to the last that the body does not pass through in its progression. Moreover this is usually called the path that the body runs through or traverses.

Scholium.

  1. This view is readily adapted too for bodies rotating about an axis. Since indeed the position of the body itself does not change: yet the motion belonging to the parts of this motion are to be understood, if the individual parts of the motion are considered to arise from just as many different parts. Indeed the individual parts with respect to infinite space are taken to change their position, and no parts are at rest except those which are placed on the axis itself. And in a like manner it is necessary for bodies to be considered in every respect, so that not only a whole body, but also the situation of the individual parts of the body, needs to be examined for change.

DEFINITION 4

  1. A body is said to be moving equably or uniformly, that in equal intervals of time traverses equal distances. A motion is truly not equable that in equal times travels through unequal distances, or in traveling through equal distances are resolved by unequal intervals of time. [p. 7; Euler uses the word ’equable’ throughout his descriptions, where the modern equivalent word is ‘uniform’, which we use here for convenience, and which Euler also uses as a synonym.]

Corollary 1.

  1. Therefore a body with a uniform motion extended by twice the time completes twice the distance, three times as long results in three times as far, and in general the distances traveled through are in proportion to the times, and in turn the times with the distances. As a ship at sea begins with an uniform motion, if in one hour it travels through two miles, then in two hours the distance completed is four miles, in three hours six miles, and in n hours, 2n miles.

Corollary 2.

  1. On account of which, if a uniform motion is given, then from that there is obtained an accurate measurement of the time, [in circumstances where that] cannot be known except from the motion. For the ratio of the distances measured that the body traverses with a uniform motion, is noted to be the same as of the times in the same intervals.EULER’S MECHANICA VOL. 1. Chapter one.

Scholium

  1. Neither truly do we have from elsewhere the division of time into years, days, and hours other than from a motion that we consider as being regular. For with the earth taken as being at rest, the ancients believed that the sun was carried round in a regular motion, and the time it took to revolve around the earth they called a day. Again, they had taken the motion of the fixed stars around the earth to be regular too, and the time it took for the sunrise to return to its original position with respect to the fixed stars they called a year. Then these times were divided into equal parts, and in this way the hours, minutes, and seconds were arrived at. [p.8]. But it is readily apparent, if the uniform motion there, as they believed, was not really uniform, then there would be an error in the measurement of time too. And recently astronomers have detected an irregularity in these motions, and they have found that not all days are of the same length, on account of which they are accustomed to give a correction from other more regular motions; that they call the equalisation of times, from which the unequal lengths of the days are known.

DEFINITION 5.

  1. Every body that is moving is said to have a speed or velocity, and this is measured by the distance that body traverses in equal intervals of time, with a uniform motion. Clearly when body B travels twice the distance at a uniform speed that body A travels through in the same time, also moving uniformly, then body B is said to have a speed twice as great as body A.

Corollary 1.

  1. Therefore since a body in uniform motion will traverse equal distances in equal intervals of time (17), the body that has moved uniformly will have the same speed or velocity always. Truly in non-uniform motion, the body adopts one speed after another in succession.

Corollary 2.

  1. Moreover the [variable] speed that a body has at some point in the distance traversed with a non-uniform motion, [p.9] is to be measured by the distance that the body would traverse in a given time with that uniform speed.

Corollary 3.

23 [a; (There are two sections 23)]. Again the speed of a body with absolute uniform motion can be measured from the distance that the body traverses, for example, in a single second. And the speed of the same perfectly well-known body is agreed upon, which prevails to be defined as the distance which that body travels through in a time of one second.

Scholium.

  1. This ratio is also especially useful in the measurement of speed. For we see sailors finding the speed of the ship by measuring the distance that the ship travels in a given time. Commonly they take a quarter of an hour interval, and they find how many miles the ship has traveled in this time. From which is understood at the same time, how many feet the ship has traversed in the same time, if indeed the ship should progress with a uniform motion.

PROPOSITION 2. Theorem. 25. For two bodies progressing with uniform motion, the speeds vary directly with the distances traveled, and inversely with the time with which these distances are traversed.

Demonstration.

Let the two bodies be A and a, and the speeds of these are C and c ; that body A
travels through a distance S in time T, truly the body a a distance s in time t. Now since
they are in uniform motion, the distances are in proportion to the times (18), the distance
sT
that body a completes in time T can be determined from the proportion t : T = s : t :
sT
therefore in a time T, the body a will move through a distance t . But body A in the
same time T moves through a distance S. The speeds of the bodies ought to be measured
from the distances the bodies travel in the same time (18). On account of which C : c = S
sT
t , or C : c = TS : st . From which it follows that the speeds are directly as the distances and inversely as the times, with which these are traversed. Q. E. D. Corollary 1. CT ct
  1. The equation S = s is produced from the final ratio. In any uniform motion therefore the product of the speed and the time, if divided by the distance traveled in that time, always gives the same quotient.

Corollary 2

  1. Also there is the ratio T : t = CS : cs . From which it follows that the times are in the ratio of the direct proportions of the distances and in the inverse ratio of the speeds, or to be as the distance divided by the speed. Corollary 3

  2. Then the proportion found can also be changed into this S : s = CT : ct. From which the distances traversed with the uniform motion are gathered together to be in a ratio composed from the ratio of the speeds and from the ratio of the times.

Corollary 4

  1. Therefore with the speed of the body given, moving uniformly, together with some distance described, the time can be noted in which this distance has been traversed; clearly by dividing the distance by the speed. Since indeed we have shown that this amount is always in proportion to the time, we can take the same as a measure of time. Corollary 5.

  2. Similarly the speed can be expressed by the distance traveled through divided by the time, and the distance also by the product of the time and the speed. Scholium I.

  3. For if the speed is such that, in order that the moving body completes three feet in one second, and therefore we can call the speed by the number 3; we will try to find the time in which for example 60 feet in the same motion are completed. For 60 divided by 3 and the quotient is 20 will show that it takes 20 seconds for the motion to be traversed. And if the distance is sought for the time of 12 seconds to be traversed, the product will be 36 feet. And also for a body traveling 4.8 feet in 6 seconds, the speed that arises is 8, which shows that this body travels 8 in one second. Scholium 2.

  4. And hence the times, the distances, and the speeds are to be measured in the following ratios [i. e. units] that we will always adhere to.[p. 12] For we will always express the time in seconds, and the distance in feet, and these will be Rhenish feet. The speed truly, as it is now made, we will specify by the number of feet traversed per second. Below indeed it is more convenient for the speed to be determined from the units that it will encounter, which henceforth we will be using, but these units arises from those, and the former can easily be recalled.

PROPOSITION 3. Theorem.

  1. In motion with any non-uniformity, the smallest elements of the distance are considered to be traversed by uniform motions. Demonstration. As indeed in geometry the elements of curved lines are considered to be the elements of small straight lines, thus also in a similar way in mechanics non-uniform motion is resolved into an infinite number of uniform motions. For either the elements are actually traversed in a uniform motion, or with the change of the speed by an element of this kind is so small, that the increment or decrement can be ignored without error. In either case the truth of the proposition is therefore apparent. Q. E. D.

Corollary 1

  1. Therefore all the change of the speed of the non-uniform motion is considered to occur upon entering the individual elements, since the whole elements are placed to be traversed with a uniform motion. [p. 13] Corollary 2.

  2. Whereby after the manner of the analysis of the very small to be observed, if the speed in the first element were c, the speed in the following element will be c + dc, in the third c + 2dc + ddc, and thus henceforth. [Thus, each new speed is the old speed plus the increment of the old speed.] Scholium.

  3. This corollary depends on the basis of the demonstration of the given force, that the change of the speed possible, while an infinitely small element is traversed, ought to be infinitely small and that this force should vanish at the start of the interval, with the body having attained its speed ; except for this to be the case, a finite motion would have to be generated immediately, which would be absurd. From which it is seen that the proposition is admitted not to be possible, nevertheless if the motion and the speed itself is infinitely small, in which case the momentary increase or decrease can have a finite ratio to that initial speed. But we will say more about that below, when the motion to be produced will examined.

PROPOSITION 4. PROBLEM.

  1. A body moves with some kind of motion along the line AM (Fig. 1), with the speed of the body given at some point; it is necessary to determine the time in which the arc AM is completed.

SOLUTION.

AM shall be the distance, either of the straight line or the curve equal to s, and the speed, that the body has at M shall be c, which will be some function of s. The element Mm is taken from M, that has been considered to be traversed with the uniform speed c [p.14]. With the element Mm to be called ds, ds the time in which the element is traversed is equal to c (29). Therefore by integration, the time is obtained in which the ds whole arc is completed, to equal ∫ c . To the integral a constant of such a kind should be added, which restores this time to zero, when s = 0, following the known rules of integration. Q. E. I.

EXAMPLE 1.

  1. Let the speed at M now be described in terms of some power of the distance AM, n then we have ds = s1− n . To which there is no need to add on a ; ∫ c 1− n constant, if n < 1 or n has a negative value : for the time in which the arc AM is traversed clearly c = s s1−n is itself given by 1− n [or, the constant of integration is zero]. But if the number 1 – n is ds negative, then there is given ∫ c = To which the infinite quantity −1 . ( n −1) s n−1 1 would have to be added for the time to go the ( n −1)0 n −1 distance AM. Therefore with the time in this case necessary to be infinite, for which the body remains at A , nor in any finite time to progress from that place. Truly this shall be the case, when n is a positive number greater than one. If truly n = 1, then the time cannot ds be shown by an algebraic expression ; for it comes about that ∫ c = ls, [i. e. log s ] to that also it is necessary to add an infinite amount, for the time spent in passing along AM. Corollary 1.

  2. Therefore in the general case, bodies are unable to stop, except when the speeds of the body, even from the beginning, are traversed as powers of the distance of an exponent that is less than one. Corollary 2.

  3. The body is to progress along the straight line AM (Fig. 2),and the speed of the body at any point shall correspond to the line MN connected to the curve AN, which meets the straight line AM at A, thus, in order that the speed of the body at the beginning shall be zero. It is seen from the preceding, where the time to travel along AM is made finite, that it is necessary for the tangent AB at A to be perpendicular to AM. For with M coinciding with A, MN should be made equal to AM n and with the number n smaller than one, a fraction as it were, from which the normality of the tangent follows. [For a small increment AM, the tangent to the curve is approx. MN/AM = AM n −1 , and as n < 1, this quantity diverges; hence AB is perpendicular to AM]. For if the tangent AB makes an acute angle or an angle infinitely small with AM, then the time to pass along will be made infinite.

EXAMPLE 2

  1. The body is to move along the straight line AB (Fig. 3) thus, in such a manner that, with the semi-circle ANB described upon this line, the speed at any point M is in proportion to the line MN attached to the circle at this place. It is understood that the speed at the point M is of such a size that the body can traverse the distance m.MN in one second. [i. e. the speed of the body at M is proportional to the length MN, where m is a constant of proportionality for a particular motion, and which has the dimensions of inverse time. Obviously, the speed is zero at A and B, while it is a maximum at C.] The radius of this semi-circle is put as AC = a, now the distance traversed will be AM = s; and MN = (2as − ss ) . [From elementary geometry, the length MN is the geometric mean of the lengths AM or s, and MB or 2a – s.] Therefore the speed at M, that we have put previously as c, in this case will be equal to m (2as − ss ) . [p. 16] Therefore the time, in which the distance AM is traveled through, will be equal to ∫ ∫ ds ∫ ( 2as −ss ) specifies the arc AN of the 1 ds = ma m ( 2 as − ss ) ads . ( 2 as − ss ) But circle. [For according to Fig. 3’. due to the translator, the angle dθ is the increment in the arc length/radius, and sin θ = MN/a = ds/NN’, where NN’ is an increment of the order of ds.] In accordance with this, the time in which the body travels through the distance AM will be AN seconds. Thus the time, in m. AC sec. Indeed since ANB = 22 which the body moves from A to B will be mANB . AC AC 7 22 approximately, the time will therefore be equal to seconds. From which it is 7.m understood, that whatever the size of the line AB, that it is always traversed in the same time. [Thus, we have the rudiments of simple harmonic motion, for which the period is independent of the amplitude, set out in this way; also, the constant m for a particular motion is seen to be the angular velocity ω.]EULER’S MECHANICA VOL. 1. Chapter one.

Corollary 3

  1. It appears from the solution of the problem that the body arrives at A from M in the same time (Fig. 1), from the backwards motion from M to A, but only if the motion has the same speeds at the same locations. Corollary 4.

  2. The applied line MN [that is, the y co- ordinate of a point on the curve for n > 1] of the curve AN represent the speed (Fig. 4), which the motion of the body has at the individual point M on the line AM, and moreover the curve has an angle less than a right angle at A with the straight line AM. From these arguments put in place already, the time has now been shown, in which the body arrives at M from A, to be of an infinite size. Whereby also with a retrograde motion of the body from M towards A , the length of time afterwards is finally infinite, i. e. the body never reaches A, even though at all points other than A the body has a finite speed.

PROPOSITION 5.

Theorem.

  1. Two bodies are moving on the right lines AM and am (Fig. 5), and the speeds of these bodies are to be expressed by the similar lines attached to the curves AN and an[i. e. the y co-ordinates; Euler always refers to them in this way.] I say that these bodies travel homologous distances AM and am in the same time. [i. e. in the same ratio as the speeds.]

DEMONSTRATION

Since AM and am are homologous distances, these therefore have the same ratio as the attached lines MN and mn; thus if the ratio is m : n, on putting AM = s et mn = nc . and MN = c, then am = ns m m The time to travel AM is given by ∫ ds c (37) ; moreover, the time to travel am will be obtained by putting nds m in place of ds and nc in place of c in m ∫ . From this product there is again ds c

∫ ds c : whereby each time to travel through AM and am will be therefore these are equal. Q. E. D. produced ∫ ds c , and Corollary 1.

  1. Hence it is also understood that (41) is included; for all the curves of circles are similar and the diameters homologous distances. Corollary 2.

  2. Let a be the [distance] parameter of the curve AN, which if taken either greater or smaller can change AN into another curve similar to itself. Truly for this to eventuate, the equation for the curve AN should be such that the applied line [for the speed] c is equal to a function of a and s of only one dimension. Moreover, for the different values of the distance a itself, s expresses the homologous distances, whether are taken to equal either a or na. Therefore according to what factor c is increased or decreased [p. 18] by an equation of this kind, so also are the proportional distances na, or how much larger or smaller a value is put in place for a, resulting in equal times for the distances traversed. Scholium.

  3. For this to be so, if c is equal to a function of a and s of one dimension, then the times for a or na are all equal, whatever the value of a : thus also if we were to consider c to be equal to a function of a and s which had m dimensions, then the times for a or na, for c some a, would be in the ratio a1−m . For a m −1 will be the function of one dimension of a and s, which can be put as k. Hence, therefore, c = a m−1k and ∫ = a ∫ . But ds c 1−m ds k ∫ will give, for the position s = a or na, a constant quantity, in whatever manner a may be varied (46) . On account of this, a ∫ will ds k 1− m 1−m give some power of a . Consequently the time to 1− m traverse na will be as a ds k

DEFINITION 6

  1. The extent [or scale] of the speed is represented by a curve, the attached lines [y co-ordinates] of which represent the speeds, which the body has at places corresponding to the distances that the body has traversed. Thus of the body moving along the right line AM (Fig. 6) the scale of the motion is the curve AN, and of which the attached lines MN sets forth the speed of the body at some individual point

DEFINITION 7

  1. The scale of the times is a curve, of which the applied lines [y co-ordinates] represent the times at which the corresponding parts of the distance have been completed in the motion. Thus if the curve AT (Fig. 6) were of this kind, [p. 19] so that any attached line of this MT [y co-ordinate] presents the time in which the distance AM is traversed, then AT will be the curve with the scale of the times. Corollary.

  2. Since the curve of the scale of the times ought to be found from the given scale of the speeds AN, it is now apparent from the preceding problem (37). Clearly if the given distance AM = s, the speed at M, i. e. MN, = c and the time, in which AM is traversed .

Therefore from the given curve AN , with the quadrature i. e. MT = t, will be t = ∫ ds c permitted, the curve AT can be constructed.

PROPOSITION 6. PROBLEM.

  1. From the given scale of the times AT (Fig. 6) to find and construct the scale of the speed AN. SOLUTION.

As before, put AM = s, MN = c and MT = t, it is required from the given equation between s and t to find the equation between s and c. Truly, this is easily effected from the above rule found : t = ∫ . For by differentiation it becomes dt = dsc and c = . ds c ds dt MT . [For the ratio Therefore the normal TO to the curve is drawn at T, and it will be ds = MO dt MT/MO is the tangent of the angle MOT the normal makes to the axis AO; this is the complement of the angle the gradient of the curve AT makes to the same axis, which is also dt/ds; from which the result follows.] Therefore as MO is to MT, thus a certain line is made to equal unity, which indicates the seconds for the fourth proportional, which is equal to MN. Therefore from M the interval is taken MQ = 1, and QN is drawn parallel to the normal TO, will be the point N on the scale of the speeds sought. Q. E. I.

EXAMPLE 1.

  1. Let the scale of the times be a right line inclined at some angle to AM; then t = ms 1 =m and dt = mds. Therefore c = ds is produced. The scale of the speed is therefore a dt right line parallel to AM, and the body is carried forwards uniformly.

EXAMPLE 2. 53. The times are as some power of the distances described, or t = s m and thus 1− m dt = ms m−1 ds . From which c = ms1m−1 = m1 s . Whereby, if the curve AT were the 1 1 parabola of APOLLONIUS, i. e. t = s 2 , then m = 12 and c = 2s 2 . From which it is apparent in this case that the scale of the speeds is of the same kind of parabola also. Corollary. 54. It is also understood, if the equation is given between c and t, in the same way the distance traversed s and both the scales of the speeds and of the times can be found. For because c = ds , then ds = cdt et s = cdt . dt ∫

Scholium.

  1. Here it is reminded for these, that as far as scales of speeds and times have been described, not only are they seen to apply to absolute motion, but also to pertain to relative motion. For the nature of this motion has not yet been considered, neither has anything been assumed, that is a special property of absolute motion. Now indeed we will produce some propositions, that are peculiar to absolute motion, and from which in a certain way, the difference between absolute motion and relative motion will be made apparent.

PROPOSITION 7. THEOREM.

  1. A body remains in a state of absolute rest, unless it is disturbed to move by some external cause.

DEMONSTRATION

We consider this body to exist in infinite and empty space, and it is seen that there is no reason why the body should be made to move from one place to another rather than stay at rest. Consequently on account of the lack of sufficient reason why it should move, it must remain at rest for ever. Nor indeed does this reason ever change in the universe; although it is possible to object that in the universe, there is sufficient reason that it might fall in one place or another. And indeed it cannot be believed that in that empty infinite space, the failure of sufficient reason for a single motion to occur [by this mechanism], is taken as the cause of the body remaining in a single place ; as there is no doubt that the nature of the body itself is the cause of this phenomenon [i. e., of staying at rest; Euler is touching on the idea of inertia : the innate tendency of a body to remain in its present state]. Clearly, the failure to move due to any insufficient reason, cannot give the true and essential cause of the event, but rather it rigorously demonstrates the true cause; and that likewise it shows that the hidden nature of the thing is the true essential cause, and this cause does not cease, from the failure of that other insufficient cause to move [the first explanation that Euler rejects]. Thus, the demonstration of ARCHIMEDES illustrates the principle with the equilibrium of two balances each similar to the other, that the truth of the matter can be shown not only in empty space, but also in the world. [Here two bodies cancel out each other’s turning effects in the earth’s gravity.] Moreover each is given a natural reason for this equilibrium [p. 22], and which is located on the earth also. In empty space a body should therefore be able to remain in a state of rest, since it is in the nature of the body for the reason given, and on that account for a body on the earth too, that once it is at rest, except by some other cause acting on it, it can be considered to remain at rest. Q. E. D.

Corollary 1.

  1. It is therefore a law in the established nature of things, that every body shall remain in a state of rest, except that by some external cause it is disturbed to move. Corollary 2.

  2. As the fundamentals of this demonstration has been found from the nature of absolute rest, it is hence in error to extend this law to relative rest. Scholium.

  3. We are well taught from trials that this law does not extend to relative rest. For we see bodies on ships at relative rest, but if the ship is suddenly shaken violently, the bodies do not remain in a state of rest, but also are shaken and move from their own positions, and even if before they were at rest, nothing approached to cause this motion. [The modern physicist would take great exception to these assertions, as the bodies try to maintain their previous state.]

Corollary 3. 60. In like manner, from what we have asserted, that a body once at rest must continue to be at rest, except in the circumstance of being affected by some external cause, then the body can be shown, which is now in an state of absolute rest [p. 23], that previously it was always also in a state of rest; if indeed it were left to itself. Since there is no reason, whereby the state of its present position should have come about from one region or another, thus it is also concluded that the body was always present before in its present situation.

Corollary 4

  1. Therefore the body, since once it is at rest, if no external cause had neither acted on it or taken it away, then not only henceforth will it always remain at rest in the same place, but also previously it had to be for ever in a state of rest in that place.

Corollary 5.

  1. It follows from this, that a body once set in a state of absolute motion is never able left to itself, to come to a state of absolute rest. For if finally it should come to rest, as before it is necessary that it was always at rest, which is contrary to hypothesis. [One might regard this proposition and others of the same nature as Euler’s musings. The idea of separating absolute and relative motion seems now to be rather naive, but as always, one must suspend judgment until the whole story unfolds in the future propositions. The ideas of absolute time and space were of course set out by Newton; but none of these agree completely with modern physics, which regards all inertial frames as equivalent.]

PROPOSITION 8. THEOREM 63. A body having uniform absolute motion will always be moving, and with the same speed now that it had at an any earlier time, unless an external cause should act on it or have acted on it. DEMONSTRATION. If indeed the speed with which a body moved was not kept the same always, then the speed might either increase or decrease. Moreover from this cause it might tend towards rest, which, since it is never allowed to do this as a consequence of (22), cannot happen.

Likewise for the body to have emerged into its present state from rest is considered to be absurd. Besides if this body situated in infinite empty space should be considered in this way, which is that it has speeded up or slowed down, there is nor reason why it should have a greater or lesser speed than what it already has, on account of which it must always be moving with the same speed. Q. E. D. Corollary.

  1. Therefore how much we should see the speed of the moving body to have either increased or decreased, we must attribute the change to some external cause.

PROPOSITION 9. THEOREM.

  1. The body with a given absolute motion shall progress in a straight line, or the distance that it describes shall be a straight line.

DEMONSTRATION

There is indeed no reason, if this body is considered to be placed in infinite and empty space, why it should depart from moving in a straight line from one region to another. And from which it is to be concluded, that it is from the nature of the body itself that it moves in a straight line. On account of which also in the world, where indeed with sufficient reason for this principle to cease being applicable, it is nevertheless observed that any body moving progresses along a straight line, unless obviously it is prevented. Q. E. D. [p. 25]

Corollary 1

  1. From these two propositions that universal law can be set out : every body provided with motion progresses uniformly in a straight line. Corollary 2

  2. Therefore a body, that was compelled by some external cause to progress in a curve AM (Fig. 7), if, when it arrived at M, these causes suddenly ceased, then it would progress with the speed which it had at M, uniformly following the direction it would have from that time of being freed. Truly it is the tangent MT of no element of the curve other than the direction produced at M that is relinquished, on account of which the body at M progresses along the tangent MT with that uniform speed that the body had.

Scholium I

  1. These laws regarding absolute rest and motion have been gathered together under one authority. And this is the law NEWTON thus proposed in the Principia, as he has stated : Every body persists in its own state of being at rest or of moving uniformly in a direction, unless in as much as it is forced to change that state by impressed forces. Corollary 3

  2. Moreover these laws are concerned with the continuation of motion as applied to absolute motion, and not these in relative motion that retain their force. As indeed it can be possible, that a body in a state of relative rest not to be maintained at rest, if also by no external cause it should be agitated (59), thus also the motions bodies had relative to others, are not always to be moved in the same direction uniformly.

[Euler’s agitated ship example; p. 26]

Corollary 4

  1. Therefore when a body has been disturbed, by no external cause, that may move in some non-uniform relative way, and yet either it is required by consideration to remain at rest or to move in some uniform manner. From this it is to be understood in some manner, how much the relative state differs from the absolute state.

Scholium 2

  1. In the foundations of theoretical astronomy, as have been set out by Newton, the sun and the fixed stars are established as being entirely non-affected by external causes, or the effect is so small as to be negligible. Though however we can neither see the sun to be in uniform motion nor to be progressing in a fixed direction with respect to the earth; , yet it is certain that the sun is either at rest or is moving uniformly in a fixed direction. It is necessary that irregularities in the motion of the sun should be observed from the position of the earth.

DEFINITION 8.

  1. The direction of the motion is to be determined by a straight line, along which the moving body is trying to progress, and according to this the body progresses, unless it is impeded by some external cause. Corollary.
  2. Therefore a body having an absolute motion, unless it is affected by other causes, always keeps the same speed and moves in the same direction.

DEFINITION 9.

  1. The force of inertia is in all bodies is that in situ faculty of the body to maintain its state of rest or of continuing in its present state of motion in a straight line. [This is another supposed Newtonian idea; by whom inertia was regarded as a force. See, e. g. Cohn’s translation of the Principia, p. 96, § 4.7. However, as Euler points out, it had its origins with Kepler.] Corollary.

  2. Though indeed we have demonstrated with sufficient reasons that a body remains in a state of rest or of a continuation of uniform motion in a straight line, yet now we will note that this is not the effecting cause of the phenomenon, for that is situated in the nature of the body itself. This depends on the nature of the bodies themselves itself and is the reason for the conservation of the state of a body, and it is called the force of inertia.

Scholium.

  1. KEPLER, who first formed this notion, and attributed to it a force, that all bodies have, all of these are resistant to that which is trying to disturb their state; and calling this inertia is better than calling it resistance, [Curiously enough, some engineering texts on physics still treat the inertial force on the same footing as other forces, so that the sum of the forces on any body is always zero; and likewise with the sources of e.m.f. in circuits.] agreeing with the idea that there is a perseverance, that we have connected to with our ideas. But it is easily understood these definitions are not different from what we have already asserted ; for it is the same force of continuing or of remaining at rest, and which offers resistance to hindrances to the state of the motion. Truly I have been inclined toward using this definition rather than the KEPLER one, since it has not yet been agreed, how bodies with disturbing forces resist changes in their motion. In addition this particular force of resistance has its own origin from the facility of remaining at rest or of continuing to move; and therefore should be explained from this basis.

PROPOSITION 10. THEOREM

  1. When the space [our inertial frame of reference] from which the relative motion is determined, is either absolutely at rest or moving uniformly in a direction, then the given laws prevail for a state of relative rest or of relative uniform motion.

DEMONSTRATION

If the space is from absolute rest that the relative motion is decided, then the proposition by itself is clear. For in this case the state of rest and relative motions are in agreement with the absolute motions, and thus any body also will be in a state of relative perpetual rest or uniformly moving in a fixed direction (10). For if indeed that space itself [our inertial frame of reference] should be moved uniformly in the same direction as these bodies, which are relatively at rest, then they will have the same absolute motion that the space itself has. Whereby with each thus progressing uniformly in the same direction by their own nature, thus they are able to continue : and in this case the law is observed (66). Truly a body, that is relatively moving uniformly in a direction, it also, if it has uniform rectilinear motion, proceeds absolutely with uniform rectilinear motion, as becomes apparent from the following proposition, where it is made transparent. Hence the relative motion here is in agreement with this law and therefore it can continue without an external force. Q. E. D.

[It appears that Euler has abandoned the idea of an inertial force as a working hypothesis in favour of inertial frames, which he calls ‘spaces’, and the rectilinear motion of a body has the same nature in all such spaces; he still maintains, however, that there is a reference inertial frame, which is the one defining absolute rest or motion, determined by the fixed stars. The model suggested by Newton and this one are of course in agreement with each other.]

Corollary 1.

  1. Therefore a body not affected by some external cause, that is relatively either at rest, or moving uniformly in some direction, from the evidence the motion of this will be as that indicated, either absolutely at rest or absolutely to be moving uniformly in some direction.

[Thus, motion in any inertial frame can be compared to the absolute motion of the reference frame of the fixed stars.] Corollary 2

  1. Likewise such relative motion will be preserved, each in its own state, for ever. For it is not only the body itself that moves, and the body is moved absolutely uniformly in some direction, but also that space itself, relative to which the body is considered, which progresses according to the same law. On account of which each will be continued on its own, and the relative motion will itself continue in this state without any approaching external cause.

Corollary 3.

  1. Since the idea we have concerning all motion, is that it is relative (7), and these laws also are not sufficient to recognise how much of the motion the absolute motion any body shall be. For when we see a body progressing in uniform motion without being affected by an external cause, then we cannot conclude more than to say that this body is either in a state of absolute rest or to be moving with an absolute uniform motion in a given direction. Truly how much of this motion is absolute cannot be defined, and neither the direction that it has.

[Thus, there is no unique reference frame in which the motion of the body can be defined; and this is especially the case if a special background reference frame of the fixed stars has to be included.]

Corollary 4.

  1. Which therefore shall be deduced from the nature of the bodies, that remain in their own state of rest or of uniform motion [p. 30], not only will they pertain to absolute rest and motion, but also to their relative state of rest or motion, and in which space the body, from its own motion is considered to move uniformly in given direction.

Scholium.

  1. And also we will not be very concerned with absolute motion, since the relative motion itself is continued by the laws. And therefore this relative motion we will more often change into other motions of the same kind, as the laws expounded are still observed : if it is allowed, we will contemplate the body in relation to that of another body progressing uniformly in this direction too. From which reason the body will not cease to move uniformly along a straight line, and that with innumerable different ways that this can be done, the most convenient of these can be selected.

PROPOSITION 11. PROBLEM

  1. A body can move uniformly with absolute motion along the line AL (Fig. 8), and another body likewise moves uniformly too along the line AM. The size of the relative motion of the other body progressing along AM is sought in relation to the body progressing along AL.

SOLUTION.

Let the speed of the body progressing along AL be a; and the speed of the other motion along AM be b : and these bodies set out at the same time from the point A. It is evident that, if the two distances AL and AM are taken in the ratio of the speeds a and b, [p. 31] both bodies arrive at the same time at L and M . Therefore draw the lines ML with the line AM making the angle AML, the sine of which is to the sine of the angle ALM, that AL makes, as AL to AM, i. e. as a to b, L will designate the place at which the body progressing along AL at the same instant, at which the other will be present at M. Since truly the relative motion of any other body with respect to this body can be sought, this body actually moving along AM can be considered as being at rest at A. Therefore with the point M known from the translation of A to arrive at L, through N draw AN parallel and equal to ML, from A. In a similar manner when the body moving on AM arrives at the nearby place m, the other is found at l, and ml is parallel to ML, since Mm : Ll = b : a = AM : AL. Indeed with the point m in a similar manner translated from A by taking An = ml there comes about l in n, and n is on the same line AN. From these it follows that the body in absolute motion along AL is to be moving in relative motion along the line AN. Moreover the relative speed will be to the absolute speed as Nn to Ll, or as ML to AL. Which ratio when constant on account of the kind of the given triangle ALM, a body in absolute motion along AL is in uniform relative motion on progressing along the line AN. For with the position of the line AN found by taking the angle LAN so large, in order that the sine of this to the sine of the angle NAM is as b to a. Hence the absolute speed along AL will be as the relative speed along AN as the sine of the angle MAN to the sine of the angle LAM. Q. E. I.

Corollary 1

  1. Therefore a body progressing in absolute uniform motion along a line will also be progressing uniformly along [another] line in relative motion, if the manner of the body, from which the relative motion is indicated, should be progressing uniformly along some line. And that is what we have assumed in the preceding demonstration (77). Corollary 2.

  2. Moreover the construction of the line AN and of the relative speed has been most easily found by being set up in this manner. By assuming, as we have done, that AL and AM are in the ratio a to b and draw ML, then AN is drawn parallel to this line ML from A, and in this manner the relative motion will be described. Indeed the relative speed will be to the absolute speed as ML to AL

Corollary 3

  1. Likewise it prevails from the same reasoning , that if with AL not the absolute motion, but travels relative and has the same relation to AM . Then truly another relative speed of the body will be produced by the body traveling along AL with respect to the size of the speed of the body traveling along AM.

Corollary 4

  1. It is therefore apparent, how the absolute motion can be changed into relative motions in an endless number of ways, which will always be uniform and made in along straight lines, but only if the absolute motion and the movements of the bodies from which the relative motions arose, were of this kind.

Scholium

  1. In the solution we have taken both bodies setting out from the same point A : but the solution is just as successful, if both bodies in the beginning start out from different points that were put at A and B (Fig. 9). For the body A progresses uniformly with an absolute motion along the right line AL, and indeed the other B similarly along the right line BM, thus so that the speeds are in the ratio a to b. AL and BM are taken in the same ratio a to b, and both bodies arrive at the same time at L and M.

But since the relative motion of the body A with respect to that of B is required, body B must be considered to be at rest at B. Therefore, for this reason, body B is moved from M to B, as body A arrives from L at N, by drawing BN parallel and equal to ML: I say that the point N is on the line passing through A, thus as the body A is moved relatively along the line AN, in a uniform motion. For draw NL and it will be equal and parallel to BM. With the appearance of triangle ANL from this construction : whereby NL to AL will have the given ratio ; hence, since NL = BM, the ratio AL to BM is the given ratio, which hence, if both were taken once in the ratio a to b, will always be in the same ratio. From which it is apparent that the point N lies on the line AN and the relative speed along AN is to the absolute speed along AL as AN to AL, i. e. in the given ratio. Therefore the relative motion is made along the right line AN and it is uniform. Corollary 5.

  1. If therefore the absolute motion of the body A (Fig. 9) is given along the right line AL, and the uniform relative motion of this is given along AN [p. 34] with whatever speed. then the motion of the body B can be found that arises with respect to the relative motion of the body A. Indeed with the two distances AL and AN assumed that are traversed in the same given time, a line BM is drawn through any point B parallel to the line NL: and this will determine the path traversed by a given body B, and the speed of this will be to the absolute speed of the body A along AL, as NL is to AL. Truly the body B will be at the point B at the same time that body A is at A.

Corollary 6.

  1. Therefore there are innumerable motions of the body B, since the point B could be assumed as we pleased, from which the same relative motion of the body A came about. But the speed of the body B was always the same and the direction of this followed the parallel line NL.

Corollary 7

  1. Also the uniform absolute motion stretching out along a line is able to be changed into any relative and likewise uniform motion made along a straight line. Indeed the line AN to be drawn can be chosen arbitrarily, and any speed assigned to that line. Indeed this same uniform motion is given to B, and the line of progression of the body B, from which here the relative motion arises.

Corollary 8.

  1. In which case that relative motion alone without any external force will be able to continue. Indeed the absolute motions along AL and BM, since they have been made uniform along right lines, [p. 35] by themselves are to be continued. Indeed for as long as this motion lasts, so long also should the relative motion along AN also continue.

PROPOSITION 12. PROBLEM

  1. The body A (Fig. 10) moves in some manner along the line AL with absolute motion, and the body B along the line BM. The relative motion of body A is required with respect to body B.

SOLUTION

The arcs AL and BM are cut from the curves, which are traversed in equal times. Therefore the body A will be found at L, when B reaches M . But since the relative motion of the body A with respect to B is desired, the body B is taken as at rest. Whereby that known motion is transferred at B from M along the line MB, and the body L reaches N, drawn parallel to LN and equal to MB.

Therefore the curve, on which the point N is found in this way, will be the path for the body A with the relative motion relative described. And by this relative motion the arc AN is traversed in the same time, by which the arcs AL and BM are completed. From which the relative speed at N is also to become known. Q. E. I. Corollary 1.

  1. Therefore in this manner the relative motion of any body can be determined, with respect to the extent of the motion of some other moving body. [p. 36] Corollary 2.

  2. It is also understood from the solution, how from the given curves AN and AL together with the motions along these, it is possible to find a curve BM and the motion along that curve. And the curve AL is defined from the curves BM and AN.

Corollary 3

  1. It is also evident that the curve BM , on account of the arbitrary choice of the point B, can be taken in an endless number of other positions. However since the arc BM, which is described in the same time as AL and AN, always subtends BM equal and parallel to the line LN, which will itself always be similar and equal and parallel to itself, and the motion along that line will always be the same as along BM. Scholium.

  2. And these are the motions, which are concerned with the comparison of absolute and relative motion that I have judged worthy of reporting on. Moreover there is another way in which it is customary to described relative motion, as the motion along AN is called the motion of the body A, which actually is moving along the line AL, and with such for the motion observed from the body B moving along BM. Truly this motion will be observed with the body B placed in relative rest and the point B as considered at rest. Thus the relative motion of the stars with respect to the earth agrees with this motion, that we living on the earth consider as at rest. Indeed the earth from B has moved forwards to M and the star from A to L, we see that star from M following into the region ML and in the distance ML. Since indeed we are not able to apprehend motions from the place B any more, but still now we consider ourselves to be at B, then we will see the star [p. 37] from B and not at L, but at N, clearly in the same place and the same distance. Therefore the right line BN will be equal and parallel to the line ML, as we have found by considering our method.

GENERAL SCHOLIUM

  1. These laws of motion, which a body observes that is left to itself, either at rest or in continued motion, are seen particularly for these indefinitely small bodies, which can be consider as points. For in bodies of finite magnitudes, of which the individual parts have their own motions, the body will exert itself to observe these laws, but which will not always be possible to happen on account of the state of the body.

The body therefore will follow a motion which is composed from the individual exertions of the parts of the body, and this hitherto on account of the insufficiency of the principles has not been possible to be defined, but this discussion is to be differed to the following.

The different kinds of bodies will therefore supply the needs for the primary division of our work. For in the first place, we will consider very small bodies or which can be considered as points. In the next case we will approach these bodies of finite magnitudes which are rigid and are not allowed to change their shape. In the third case we will consider flexible bodies.

Fourthly, we are concerned with these which allow extension and contraction. Fifthly, we put under our examination the solution of the motion of many bodies, that are impeded by others, that their own motion may be completed as they exert themselves. Truly in the sixth case the motion of fluids will be the agenda.

For these bodies we will not only see, how the remainder of the motion is to be continued [p. 38]; but in addition we will inquire, how these are affected by the external causes or forces. Finally from all these inquiries the large scale variation of the whole body can be inferred, whether it is free or not.

For a non-free state, I understand this : when bodies are impeded, by which they are unable to progress in that direction, and which they try to overcome ; the motion of pendular bodies is of this kind which, since the are unable to descent directly, as they try, and so make oscillations. For the free state is to be understood : when bodies are progressing and which come upon no impediments to their motion anywhere, not only on account of their own force, or from disturbing forces pulling on them. Therefore it appears, from the things Mechanics will have as its agenda , and that there are many which have not even been touched upon.

For besides the motion of points, which have been dealt with hitherto, nevertheless there are so few that it will be necessary to derive nearly all from first principles.

I begin therefore with the motion of free points with any kinds of disturbing forces, because these left to themselves will follow the motion shown in this chapter. Hence on account of this I have resolved for the First Volume to be concerned with the motion of free points, and the following Volume truly set up to explore the motion of points which are not in free motion ; in both of which, and which will occur, as with these already dealt with, so likewise for these that follow, I shall derive the motions from first principles using the analytical method.

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

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