Wednesday, May 6, 2015

Ancient notions of time, space and the Second Law from Newton’s Principia

*Author’s notes from: Smith, George, "Newton's Philosophiae Naturalis Principia Mathematica", The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = <>., and other sources.


Newton’s Principia, (first edition published in 1687), was meant to address a number of important issues of physics. It laid out a generalized foundation of scientific theory, that I believe, went further than even some of the more known accomplishments discussed in more depth elsewhere(such as its “Rules of Philosophy). Key theoretical assertions of Principia were the notion that gravity came from the objects and their smaller parts, and that force “impressed” on an object would be proportional to its change in speed. There were no “laws of motion” at that time, at least ones encompassing a theory of their causes. Before Principia,“motion” was a concept, and science was still emerging from viewing the world almost exclusively through the lens of mechanics and geometry. And in fact the delineation between “physicist “ and geometer” was made very clear when Newton was criticized in his Principia as being the latter. Kepler’s laws, showed geometric perfection of that time, but were not laws of motion of the object themselves, as the ‘bodies’ were treated as abstract points. Though Kepler did consider the nature of motion, as did Galileo, (and gave a statement that resembles a pre-Second Law) he was not critically concerned with motion itself, or of broadly encompassing inverse square attraction of bodies generalized to all “bodies” as Newton was. Newton made the leap that “mass” was a property distinct from other qualities of bodies, like motion.

[This is meant as a brief survey of some of the more salient features of Principia, and is focused mainly on a review of the laws of motion, principally Newton’s Second Law, and Newton’s Rules of Philosophy. Perhaps no other law is as historically important or currently applicable (to modern physics) as Newton’s second law, and I was intrigued by the fact that this is actually not as clearly stated as one might imagine, its geneology is much more complex. It is not formally given as F=ma, nor even as an equality, so how was it stated and why does Newton receive the ’lion’s share’ of the credit?


In Principia, Newton describes in “definitions” various scalars of mass, time, and others.

Of interest is Newton’s definition of mass, the first definition of this substance which lead to his formulation of momentum. From Principia:

The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity. This quantity I designate by the name of body or of mass.”  

Interestingly, we note how Newton’s definition utilizes proportions, not absolute quantities.

Newton also recognized the problem of the lack of an absolute reference from which to measure “motion”.  To paraphrase the problem he stated “there appears to be no object anywhere that is not moving..” Newton dedicated considerable effort, however, to the problem of “relative motion” or velocity, and specifically, to absolute velocity compared to relative velocity. Velocity is actually not a quantity that can be empirically determined, unlike mass. It is a relative term. For Newton this was a significant difficulty, but he advocated that the velocity should be determined by reasonable means available. The “fixed” stars were considered the best absolute reference point.

Key to the definition of motion was Newton’s definition of time, which he defined in two different ways: “[from Principia] Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.”

Newton defined relative velocity according to the conventions of the time, following Copernicus, Huygens, Galileo, Kepler, and others..

There seems to be some debate about what the actual phrasing of Principia was regarding Newton’s second law.


Translated from the original latin: (an original image of the Principia’s Law I and Law II can be seen here:
Laws I, II in their original Latin: 
“Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.”

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
(Motte’s 1729 translation of Law II from the latin):
“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

According to modern ideas of how Newton was using his terminology this is understood, in modern terms, as an equivalent of:

Law II (‘modern’) “The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.”



Also from George’s Stan. Encylopedia:
“The modern F=ma form of Newton's second law nowhere occurs in any edition of the Principia even though he had seen his second law formulated in this way in print during the interval between the second and third editions in Jacob Hermann's Phoronomia of 1716. Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces. (His stating the law in terms of proportions rather than equality bypasses what seems to us an inconsistency of units in treating the law as neutral between these two.)”


From: “The Principia: Mathematical Principles of Natural Philosophy by Sir Isaac Newton, Cal U Press 2014 I. Bernard Cohen
“Newton never actually  made a formal statement of the second law in the algorithm of fluxions or the calculus. The first person to do so seems to have been Jacob Hermann in his Phoronomia 1716 in which he writes G= M dV:dT where he says “G signifies weight or gravity applied to a variable mass M”

In Newton’s Scholium to Principia, in the second edition, “Newton attributed the second law (as well as the first law) to Galileo, even alleging that it was by use of the second law that Galileo had discovered the law of falling bodies.” However, Galileo was not aware about “change in momentum in the Newtonian sense, since this concept depends on the concept of mass, which was invented by Newton and first made public in the Principia.”

The formulation of the second law would of course, not even be possible without Newton’s original definition of the concept of “mass”.

Smith, George, "Newton's Philosophiae Naturalis Principia Mathematica", The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = <>.

What did Newton’s “alteration of motion” mean? It appears that Newton’s Second Law turns on the geometric concepts. In his Principia, Newton was expressing his “alteration of motion” geometrically as a deflection, just as Newton provides a proof of Kepler’s laws based on using geometry, i.e. the areas of triangles.

I’ve seen it reported that Newton’s second law resembled a “point force expression” such as the kind that is associated with impacts. (J=mΔv). However, this next diagram should clarify that his interpretation of the second law was much broader than that, and was obviously meant to describe heavenly bodies, to conform with Newton’s law of gravity. That is to say, Newton was clearly thinking about deflected motions and the problem of how bodies were kept circling larger planets. What force kept the earth in its orbit around the sun? These were unknown problems of that time.

 “If the body A should [see Fig. 1], at its place A where a force is impressed upon it, have a motion by which, when uniformly continued, it would describe the straight line Aa, but shall by the impressed force be

‘Figure 1’

deflected from this line into another one Ab and, when it ought to be located at the place a, be found at the place b, then because the body, free of the impressed force, would have occupied the place a and is thrust out from this place by that force and transferred therefrom to the place b, the translation of the body from the place a to the place b will, in the meaning of this Law, be proportional to this force and directed to the same goal towards which this force is impressed. Whence, if the same body deprived of all motion and impressed by the same force with the same direction, could in the same time be transported from the place A to the place B, the two straight lines AB and ab will be parallel and equal. For the same force, by acting with the same direction and in the same time on the same body whether at rest or carried on with any motion whatever, will in the meaning of this Law achieve an identical translation towards the same goal; and in the present case the translation is AB where the body was at rest before the force was impressed, and ab where it was there in a state of motion. [M, 541]”

Regarding the meaning of the second law, we should also note the following passage from Newton’s Scholium, Principia, (1729 translation, p20

“…If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.” So it is clear that motion is clearly differentiated here from “momentum” and we note in the first line, how it is the change in motion (the velocity) that is directly proportional to the force impinged. I believe these details are interesting to flesh out the nuances of what the second law truly was, and to show its clear aspects emerging in various parts of the Principia. The Scholium’s were essentially addendum’s to the Principia, Newton’s.

But I believe in this next excerpt, there can be little doubt about what Newton meant regarding his Law II as we find also from Motte’s 1729 translation of Principia “If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively.” One won’t find a formal equation showing these proportions, they are derived geometrically but still when taken together, capture the modern forms of F=m dv/dt. [this is of course only a cursory treatment on the interpretation of Newton’s Principia available elsewhere].

“I do not feign hypotheses” Isaac Newton

I’ve seen reported on the internet, some very misleading stories about Newton investigating “metaphysical” phenomena and alchemy. This next passage should help to clarify that those types of stories are just that and are obviously attempts by some pseudo “academics” to reinvent history. I know that these sources are in fact on the internet as alternative histories” so the following statement from Newton’s “general Scholium” added to his second edition of the Principia will hopefully clarify Newton’s position on anything speculative. In fact, it should stress that fact that Newton was obsessed, (just like many ancient mathematicians who came before him) with obtaining absolute quantities whenever possible. Anything short of this was a “hypothesis.” It is interesting that at that time, “hypothesis” did not carry the weight (of legitimacy) it does now.

In the General Scholium Newton added this statement:

“I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies, and the laws of motion and law of gravity have been found by this method. And it is enough that gravity should really exist and should act according to the laws that we have set forth and should suffice for all the motions of the heavenly bodies and of our sea. [P, 943“[Newton’s Scholium to Principia, George Smith Standfor Encyclopedia]]”


Regarding the last sentence of the quote above, the almost preemptive nature of Newton's wording here reveals something interesting- there was a great deal of skepticism about Newton’s formalized law of gravity, as it was an action at a distance force which seemed to have no theoretical basis. Huygen’s proposed a theory based on “vortecies” to account for the motion of the planets held in place by this force. But the issue of gravity’s inverse-square, action at a distance relationship cause significant problems for ready acceptance of the idea, despite its mathematical accuracies in mechanics, as there seemed to be no explicable mechanism for gravity to operate. Huygens was not satisified with Newton’s explanation that gravity came from the individual parts (composing an object) but favored the notion instead that objects would tend towards a center. “..were there no Earth, bodies would not cease to tend toward a center because of what we call their gravity.” (Huygens, Discourse on The Course of Gravity 1690). Newton’s Scholium was published a full 26 years after the first edition of Principia, and was intended to respond to critiques such as “vortices” as well as Leibniz’s theories. (Smith, 2008 Stan. Encyc.)


Finally, it should be noted that Newton’s universe, was essentially a virtual one. These laws took place in imaginary time frames (those applied artificially by the observer). It did not take into account what would become “the arrow” of time. Carnot demonstrated with the second law of t hermodynamics, that such a universe could not operate in reverse, and that there were aspects of the physics of these Newtonian quatnities, that were irreverisibly lost in all processes. Something not accounted for in the Newtonian view.

All in all, it is clear that Newton did formulate the second law, with its change in motion being proportional to an impressed force on a body, a body having mass (Newton defined bodies in at least two ways, geometrically and with his ‘mass’ concept) but it didn’t take form until midway between the II and III versions of Principia. After all, it would not have been possible to realize the notion of real forces without at least one of Newton’s key terms he defined for the first time, mass. But one has to examine that evidence (and it has been in more detail elsewhere) through the lens of the form of ancient mathematics of that time, of geometry.




1.      From “Newton’s Rules of Reasoning in Philosophy” (from Principia, editions 2nd (1713) and 3rd (1726) tans. A Motte 1729:

“Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes.

Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

Rule 4: In experimental philosophy we are to look upon propositions inferred [collected] by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.” Sources:

The level of experimental objectivity, over 300 years ago in Principia’s “Rules”, is astounding, and far beyond its time. And these philosophical rules of scientific evidence should still hold and be applicable today (and particularly the section I’ve underlined here. Again these are good reference to refute the contention that Newton engaged in anything but the most objective science: “…For since the qualities of bodies are only known to us by experiments, we are to hold for universal all such as universally agree with experiments; and such as are not liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede from the analogy of Nature, which uses to be simple, and always consonant to itself...” The last quote seems to directly contradict Newton’s later “historical” publications…but is another topic.

2.       (I.Bernard Cohen, 1963”Pemberton’s Translation of Principia..w Motte’s Notes” Isis Vol. 54, No. 3, Sep., 1963

No comments:

Post a Comment