Abstract
One of the main objectives of the present work is to show that Newton in fact gave a different definition of mass. Although Newton was an adherent of the atomistic doctrine of the ancient Greeks and himself often expressed the view (both in his Principia and especially in his Opticks) that all bodies consist of approximately identical solid (hard) particles or atoms of a single matter, he, being a principled opponent of hypotheses not grounded in experiments, refused to define mass as the quantity of this single matter and refused to build his dynamics on the basis of such a “hypothetical” definition.
Full Text
From the History of Physics
On Newton’s Doctrine of Mass
V. G. Fridman
Introduction
It is very widespread—one might say generally accepted—to hold that Newton defined mass as the quantity of matter in a body. This definition is usually contrasted with another, regarded as more modern and better, namely: the definition of mass as the measure of the inertia of a body. One of the principal aims of the present work is to show that in fact Newton gave a different definition of mass. Although Newton was an adherent of the atomistic doctrine of the ancient Greeks and himself often expressed the opinion (both in the Principia and especially in the Optics) that all bodies consist of approximately identical solid (hard) particles or atoms of a single matter, he—being a principled opponent of hypotheses not grounded in experiment—refused to give a definition of mass as the quantity of this single matter; he refused to build his dynamics on the basis of such a “hypothetical” definition.
In this connection, the usual accusation against Newton—that his definition does not accord with the fact of the heterogeneity of matter—falls away, if one restores the genuine definition of mass that Newton actually gave.
Newton’s genuine definition of mass. To decide the question of what Newton’s actual definition of mass was, we proceeded from the original Latin text of Newton’s principal work, Philosophiae naturalis Principia mathematica *), and also from the well-known Russian translation of this work by Academician A. Krylov.
We used three Latin editions of the Principia published during Newton’s lifetime, in 1687, 1714, and 1723 ), as well as the Latin edition of 1871, published under the editorship of Thomson and Tait, which is a reprint of the third (Pemberton) edition of the Principia*, issued by Dr. Pemberton during Newton’s lifetime, in 1726 ). As additional material we used, above all and especially, A. Krylov’s Russian translation (vol. VII of the complete collected works of Academician Krylov, publ. 1936), as well as Motte’s English translation, reissued in 1803 with interesting additions by the English mathematician Emerson, and, finally, Wolfer’s German translation (Berlin, 1872).
Newton’s definition of mass is given in the so-called “Definition 1,” with which Newton’s Principia begins. A. Krylov gives
) “Mathematical Principles of Natural Philosophy”; henceforth we shall call it the Principia*.
**) The latter was published in Amsterdam (not the Pemberton edition).
***) It was from this last edition that Acad. Krylov made (principally) his translation.
this definition 1 in the following form: “The quantity of matter (mass) is the measure of such, established proportionally to its density and volume.” Unfortunately, this fundamentally so important definition was translated by A. Krylov not only inaccurately, but also with a certain, if one may so express it, arbitrariness. The original Latin text is the following: “Quantitas materiae est mensura ejusdem Orta ex illius densitate et magnitudine conjunctim.”
The exact translation here is: “The quantity of matter is its measure, arising*) jointly from its density and volume.”
The word “mass” in parentheses was added by the translator. Yet such an addition helps to entrench the erroneous opinion, very widespread among us, that Newton defined mass as the quantity of substance. And this is all the more so since later A. Krylov, in his translation, several times renders the words “quantity of matter” as “quantity of substance,” again accompanying them in parentheses with the word “mass,” which Newton did not have (for example, pp. 515, 521). Incidentally, such an incorrect understanding of Newton was widespread everywhere in the nineteenth century as well, long before Krylov’s translation appeared (in 1915).**)
Moreover, Newton did not have here the term “proportionally,” which Newton himself very often uses in a number of other places in the Principia, and even in the explanation of definition 1 itself, where it is said that the quantity of substance in a body is proportional to its weight. Newton is not at all speaking in definition 1 itself of the exact mathematical expression of mass (or “quantity of substance,” as Newton says), which is now expressed by the formula \(m = V \cdot d\). Newton indicates only that the measure of matter which he designates by the term “quantity of substance” arises (orta ex) as a measure from the joint or combined (conjunctim) density of the body and its volume; that in judging the magnitude of this measure one must, consequently, take into account both the volume of the body and its density; but Newton, with the caution inherent in him and characteristic of all his scientific work, refuses to speak here, i.e. in the definition itself, of exact proportionality.
But further on, in the explanation of definition 1, Newton explains in what sense this definition is to be understood. He says that “air of double density in double volume is four times as great, and of triple density in triple volume is six times as great.” This indeed means that in this example Newton in fact regarded the quantity of substance (matter) as exactly proportional to its volume and density; but Newton understood very well that one cannot speak of this proportionality in a general definition of the quantity of substance (which, unfortunately, Krylov did in his translation). The whole point is that Newton clearly realized (as we shall explain below) what difficulties arise here with respect to heterogeneous bodies (for example, when comparing the quantity of substance in gold and in wood)). Newton himself says in the explanation that “the same applies also to bodies of every kind which, for whatever reasons, are condensed,” i.e. it applies, after all, to homogeneous*) bodies, but only in varying degrees of condensation or rarefaction. Extending definition 1 to the case of heterogeneous bodies, Newton encountered major difficulties of principle. These were difficulties due to Newton’s extrapolation of the concept, applicable in ordinary practice, of
*) Or “proceeding,” in the sense of the emergence of the very concept of this measure, which Newton called “quantity of substance.”
) It should, however, be noted that Acad. Krylov later, in 1943, in his collection of articles Thoughts and Materials on the Teaching of Mechanics, and also in the collection Isaac Newton, issued by the Academy of Sciences of the USSR for the tercentenary of Newton’s birth, in his article “Newton and His Significance in World Science” (p. 6), gives Newton’s definition 1 without the word “mass” in parentheses.
***) Our Lomonosov also pointed this out.
****) Of one and the same substance.
“quantity of matter” from the case of comparing homogeneous bodies to the case of comparing heterogeneous ones. It was precisely because of this extrapolation that Newton had to concern himself with a special definition of the very term (or concept) “quantity of matter.”) This concept is quite clear in the case of homogeneous bodies and, for this case, requires no special definition, quite regardless of whether such a body is a mixture, or a solution, or an alloy of several chemically different substances or not; it is necessary only that this mixture, solution, or alloy be the same throughout the entire extent of the body, that a body of a definite composition be compared with another body of the same composition, i.e. that the bodies be compared with respect to the quantity of the same matter. In this simple case it is sufficient to take some volume or weight of the given homogeneous substance as the unit of its quantity and to compare with it the volumes or weights of other bodies. Therefore, by the word orta* Newton evidently wished to express that, for the case of heterogeneous bodies, the very concept expressed by the term “quantity of matter” arises from taking into account both the density and the volume of the body, and, moreover, only them. Here the indication of density was an indication that the quantity of matter in a given body cannot be judged solely by its dimensions, that one must also take into account what substance it consists of (subsequently our great scientist Mendeleev spoke of the importance of knowing “how much and what kind of given substance”).
Now we return to the question of the term “mass.” This term, as we have seen, is absent from the original Latin text of definition 1. Newton speaks of it only later, in the explanation to this definition. Namely, at the end of this explanation he indicates that “this same quantity I shall hereafter understand under the names body (corpus) or mass (massa).” From this, apparently, the erroneous conclusion was drawn that Newton defines mass as quantity of matter. In reality, however, this indication by Newton means only that the term “mass” (massa), along with the term “body” (corpus), is used by Newton subsequently as a synonym (another name) for the term “quantity of matter”), defined in definition 1, i.e. taken not in the everyday sense, which was known even before Newton, but as a special new term of mechanics introduced by Newton. But the indication of a synonym for a term never was and is not a definition of that term. For example, in indicating that rodina means the same as otechestvo [homeland/fatherland], or that an aeroplane is the same as an airplane, or that a locomotive is the same as a steam engine, we thereby by no means define what a homeland, or an aeroplane, or a locomotive is; we do not reveal the content of the concept denoted by one term or another. True, sometimes the replacement of a less familiar term by a more familiar one in a way substitutes for the definition of the first term; for example, by saying that an aeroplane is the same as an airplane, we make, so to speak, the term “aeroplane” more accessible to the understanding of the broad masses; but this is only because these broad masses know what an airplane is, i.e. they know its definition.
The real (genuine Newtonian) definition of mass, or corpus ***), is, in essence, found in Newton himself. It is not difficult to obtain it by replacing, in Newton’s definition of quantity of matter, the term “quantity of matter” with the term “mass.” Here is this genuine Newtonian definition of mass:
*) The generalization of scientific concepts often leads to the need for a new definition of them, one that embraces the old ones as particular cases.
**) Acad. Krylov rightly points out in note 5 (p. 23) that “the words ‘quantity of matter’ constitute, as it were, one word, one new term.” Such terms consisting of two words are not infrequently encountered in physics, for example: specific heat, coefficient of expansion, current strength, specific weight, etc.
***) In what follows we shall render this latter term by the word “corpus,” in order to distinguish it from the word “body,” which nevertheless has a different meaning.
“Mass is a measure of matter), arising jointly from its density and volume.” We are not speaking here of the term “body,” for it is now not at all used in the sense of mass, although Newton himself used it in the Principia* in this sense much more often than the term “mass.” We must now dwell on this question in somewhat greater detail.
We noted above that Newton himself indicated that “this same quantity I understand hereafter under the names body or mass.” Thus it is translated (and inaccurately) by Academician Krylov. Newton’s Latin text here is as follows: “Hanc autem quantitatem sub nomine Corporis vel Massae in sequentibus passim intelligo.” We have emphasized the word “passim,” for in translation (otherwise quite accurate) it was omitted by Krylov. Yet, by the very nature of the matter, it is of great significance.
First of all let us note that the Latin word “passim” has very varied meanings, namely: usually, scatteredly, alternately, here and there, indiscriminately, disorderly. In other words, the word “passim” represents what is called a homonym**). Homonyms are very frequent in the Latin language.
Which, then, of these numerous meanings should be chosen? In omitting the word “passim” altogether, Academician Krylov evidently proceeded from its meaning “always.” The word “passim” was also omitted by Wolfers in his translation of the Principia. Mott translated “passim” as “every when,” which means “as when” or “usually.”
Newton himself uses the word “passim” in the Principia one more time, when he says (in the concluding General Scholium) that “the word god passim means ruler.” In Krylov this is translated as follows: “The word god usually means ruler.” Academician Krylov correctly translated here the word “passim” by the word “usually,” for the word “god” means for believers not only “ruler.”
But in the case of the explanation to Definition 1, the word “passim,” in our opinion, should be translated “here and there” or “alternately.” This is confirmed by the terminology which Newton in fact uses in the Principia subsequently — in sequentibus, as Newton says.
Finally, let us note that, according to Newton’s direct indication in the explanation to his Definition 1, the term “mass” is a synonym of the term “quantity of matter.” However, scholars of post-Newtonian times, speaking of mass or quantity of matter, imperceptibly changed the word “or,” which according to Newton meant that the terms “mass” and “quantity of matter” are equivalent, i.e. are synonyms, into the word “is,” i.e. they began to say as though, according to Newton, mass is the quantity of matter in a body, in the directly objective sense.
True, in connection with Newton’s atomism he had a hypothetical conception of matter as an aggregate of solid particles of a single primordial matter, the quantity of which in the various bodies of nature varies with the different degree of their condensation; but, although this helped Newton to form for himself a concretely vivid representation of that density of matter of which he speaks in Definition 1, nevertheless this definition of the “quantity of matter” (or mass), in essence, is independent of such a representation.
Newton’s terminology for designating the concept “quantity of substance (matter).” The terms: mass, matter, stock of matter, sphere, magnitude, body. Thus Newton introduced into mechanics (physics) a new scientific term***) “quantity of substance.”
*) Of course, measures of matter are different and very numerous, for example, temperature, specific heat, coefficient of expansion, specific electrical conductivity, etc. According to Newton, mass is a specific general measure of matter connected only with the spatial dimensions of a body and its density.
) Such as in Russian, for example, the words ключ [key/spring], мир [world/peace], источник [source].
***) Precisely a term, for, as we have already indicated, extrapolation to heterogeneous substances deprived the words “quantity of substance” of their directly objective, so to speak everyday, meaning.
the meaning of which he explains in Definition 1, while stipulating that “this same quantity I shall here and there (passim) understand hereafter under the names ‘body’ or ‘mass’.” It is precisely “here and there,” for, as we shall now see, Newton in the subsequent text uses, in the sense of “quantity of matter,” besides the words “mass” and “body” (corpus), a number of other words (terms) as well.
Newton employs the term “mass” (massa) throughout his enormous work only 7 times*), namely: 1) On p. 25 (of Krylov’s translation) the discussion concerns the “inertia of mass”; Newton says that “the innate force of matter ... is distinguished from the inertia of mass ... only in our way of regarding it.” Here the term “mass” denotes, of course, matter or body. But no more than a line later Newton already speaks of inertiae materiae (i.e., of the inertia of matter). Such use of the word “mass” is very widespread at the present time as well. 2) On p. 243, concerning the fact that the absolute forces of attraction of two bodies are related as their masses**). 3) On p. 330 it is said that the quantity of motion is proportional to velocity and mass. 4) On p. 517 it is stated that the forces of attraction of the Moon and the Earth toward the Sun are exactly proportional to their masses (earum massis accurate). 5) On p. 525 it is stated that “whatever the formation of the planets may have been, all matter heavier than water, while the whole mass was still fluid (massa tota fluida erat), tended toward the center.” 6) On p. 529 it is said that, for an exact calculation of the major axes of the orbits of the planets, one must take into account the sums of the masses (massarum) of the Sun and of each of the planets. And, finally, 7) on p. 589 it is stated that “the mass of the Moon is to the mass of the Earth as 1 to 39,788” (which Newton found from the tidal action of the Moon on the Earth).
Of these seven instances of Newton’s use of the term “mass,” in the first and in the fifth instances this term is used in the sense simply of “body” or “matter,” as is very often the case even now, outside any connection with dynamics; in the third instance it is used in the sense, speaking in modern terms, of inertial mass (a measure of inertia), while in the second, fourth, sixth, and seventh instances it is used in the (modern) sense of heavy mass. Thus only 5 times does Newton use the term “mass” in the sense of modern dynamics.
It is interesting that at the end of the Principia there is a detailed index (Index Rerum) on 7 pages in small type, reprinted also in the Latin edition of the Principia of 1871 (edited by Thomson and Tait). And it turns out that in this index there is no word massa at all. Yet here there are, for example, the following reference entries concerning the term “quantity of matter” (quantitas materiae): “Jovis... quantitas materiae, Lunae... quantitas materiae,” i.e., “Jupiter... quantity of matter, Moon... quantity of matter”; “Materiaе quantitas definitur,” i.e., “the quantity of matter is defined” (meaning, a reference to Definition 1); “Saturni... densitas, quantitas materiae, Solis” ... the same (i.e., Saturn... density, quantity of matter, Sun...).
True, in this index there is also no word corpus, which Newton mentions in the explanation to Definition 1 on a par with the word massa. But this is explained, of course, by the fact that (as we establish below) Newton much more often uses the word corpus in the Principia in the simple sense of “body” than in the sense of Definition 1.
All this once again confirms the circumstance that Newton regarded Definition 1 as the definition precisely of the term “quantity of matter”; and his indication that in what follows, by the words “mass” or “corpus,” he understands quantitas materiae, he regarded merely as a mention of other designations for the term “quantity of matter,” i.e., as its synonyms.
*) The rarity of Newton’s use of the term “mass” is explicitly pointed out by Krylov on p. 243 of his translation.
**) Ut massa corporis A, ad massam corporis B. Here it is especially clear that the term corpus was not always, for Newton, a synonym of the term “quantity of matter,” but only passim (here and there).
In Krylov’s translation of the Principia the term “mass” occurs about 200 times, but this abundance is explained by the fact that Krylov modernized Newton in this respect (which, however, he also notes several times in footnotes). It must be observed, however, that in his translation Krylov used the term “mass” not only in place of the terms “quantity of matter” and “body,” of which Newton speaks in Definition 1. For example, Krylov about 40 times translated by the word “mass” Newton’s word “corpusculum,” which means “particle.” The word “corpus,” used by Newton most often in the sense of “body,” Krylov, modernizing, also not infrequently translates by the word “mass” (though by no means always).
It turns out, however (as is seen from a comparison of Krylov’s translation with the Latin text of the Principia), that Newton, in the modern dynamical sense of the term “mass,” used a number of other terms as well, and some of them were used by him much more frequently than the term “massa,” which Newton used only seven times. These other Newtonian terms are the following: 1) matter (materia)—36 times, 2) sphere (sphaera)—9 times, 3) quantity of matter (copia materiae)—3 times, 4) magnitude (magnitudo)—1 time, 5) inertia or force of inertia (vis inertiae or vis insita)—2 times.
Here Newton had no uniform terminology.
In all, Newton has 8 different terms corresponding to the modern concept of mass; moreover, the principal term of Definition 1, “quantity of matter,” is used 51 times, while another term, specified in Definition 1 (“body”), is used even more often—68 times. Let us recall, moreover, that the word “corpus” is very often (several hundred times) used by Newton not in the modern sense of “mass,” but simply in the sense of “body,” without any relation to dynamics, when he speaks, for example, of the orbit of a body, or of the motion of a body, or of a body encountering the resistance of a medium in its motion, etc.
It is necessary to give some concrete examples of Newton’s use of these other terms (in the modern sense of the term “mass,” as inertial or heavy mass).
The term “materia.” On p. 392 we read: “The greater the force, the greater the time, and the smaller the mass (in Newton—materia), the greater will be the velocity communicated. This follows from the second law of motion.” Here too we read: “The velocity which a given force can communicate to a given mass (in Newton—in data materia)... is inversely proportional to the mass (materia).” It is clear that here Newton, by the term “materia,” has in mind modern inertial mass (as Krylov translated it). On p. 480 this term is used in the same sense, and in a very responsible formulation of the law of the period of oscillation: Newton says that “the time of one oscillation is proportional to the square root of the mass” (in Newton—materia).
But Newton uses this term also in the modern sense of heavy mass; for example, on p. 518 he says that “the gravitation toward all the planets is proportional to the materia in them” (Krylov here translated it as “quantity of matter”). And immediately there is a curious formulation: “the gravitation of each part is related to the gravitation of the whole as the mass of this part to the mass of the whole” (in Newton—ut materia partis ad materiam totius)*. On pp. 526 and 527 Newton speaks (in connection with the action of gravitation) of the numerical ratio of the matter of the Sun and the planets (materia Solis ad materiam Jovis), etc. Above we saw that Newton, speaking of the ratio of the gravitating masses of the Earth and the Moon, uses the term “massa.” It should be noted that Newton not infrequently on one and the same page uses, in the same sense, several different terms to designate what is now in physics and mechanics called mass. Here Newton indeed does everything passim, intermittently, here and there; here Newton wrote as it “was written” (but definitely avoiding the term “massa”).
* That is, as the matter of the part to the matter of the whole.
In Newton the term materia is often used also in the sense of “body” or “substance,” i.e., in the literal, objective sense. For example, on p. 25 he speaks of the inertia of matter as inertia materiae, but there too, for some reason, the expression inertia massae is also found (see above). On pp. 454, 455, 458, and 466 Newton likewise speaks of inertia materiae *); on p. 267 there is mention of adding or subtracting attracting matter (materia).
The term copia materiae. It is used by Newton only 3 times, but twice in a very, one may say, responsible place in the Principia, in the famous Proposition VI of Book Three, to which he refers in the explanation of Definition 1. In the explanation of this Proposition VI, the subject is Newton’s experiments with pendulums, in which the role of the oscillating body was played by identical little tubs filled with different substances of equal weight—as Newton indicates—with gold, silver, lead, glass, sand, common salt, wood, water, wheat. These historic experiments were intended, in Newton’s view, to prove experimentally the most important fact (for both classical and relativistic physics) of the proportionality of weight and mass for the case of homogeneous bodies and (what is especially important) for heterogeneous ones.
And here in Krylov’s translation we read (after Newton establishes experimentally the equality of the periods of oscillation of pendulums of the same length, whose little tubs were filled, for example, with gold in one case and with wood in another): “Consequently, the quantity of substance (mass) in the gold... was related to the quantity of substance in the wood as the action of the moving force on all the gold to its action on all the wood, i.e., as the weight of one to the weight of the other.” In Newton’s original there is no term “quantity of substance” here, nor is there the word “mass” in parentheses; he says copia materiae—in both cases, i.e., stock of matter. It is clear that here copia materiae denotes, in the modern sense, gravitational mass.
The third time the term copia materiae is used by Newton on p. 525, where it is said: “the whole quantity of the substance of the Earth (copia materiae) is approximately 5 or 6 times greater than if it all consisted of water.” Incidentally, this is an interesting example of Newton’s practical application of Definition 1, according to which the quantity of substance is its measure arising from density and volume. Here too Krylov, in sense, translated Newton quite correctly, but... modernized him. It is interesting that the term copia figures in the very last phrase of the Principia, where Newton says that there is “not a sufficient stock (copia) of experiments” for studying the properties of the ether. Krylov translated Newton’s term copia here with literal accuracy.
The term vis inertiae or vis insita, in the modern sense of inertial mass, Newton uses only twice: 1) on p. 55, where it is said that “in impact and reflection of bodies (corpora in the Latin text), whose velocities are inversely proportional to the masses (reciproca ut vires inertiae **), equivalent”; 2) on p. 420, where the discussion concerns a resistance proportional to vis insita ***) (in Krylov—mass). This substitution of terms by Krylov, however, is qualified by him right there, in a footnote on p. 420. It is interesting that Euler also quite often used the term “inertia” or “force of inertia” instead of the term “mass.”
The term “sphaera” (sphere or ball) in the modern sense of mass is used by Newton 9 times (more often, therefore, than the term “massa”). On p. 249 Newton says that the forces of attraction of homogeneous balls are “proportional to the volumes of these balls” (in Newton—ut sphaerae). On p. 251 the discussion is that “the forces of attraction by each individual ball... are proportional to the masses of the attracting balls” (in the Latin text—ut sphaerae).
*) Matter or substance, as translated by Krylov.
**) That is, inversely to the forces of inertia.
***) That is, to the internal force, as Newton considered the force of inertia, in contrast to an external force applied to a body.
The term sphaera Newton also uses in the especially important Proposition LXXVI (Book 1), where the famous formulation of the law of universal gravitation is given. In translating the text of this formulation, in several Newtonian special cases and in the general case, Krylov everywhere uses the term “mass”; Newton, however, uses here the term sphaera (p. 251), and everywhere here the discussion concerns the attraction of spheres, which is especially important for astronomy. And indeed the latter was at the center of attention in the Principia.
It must be noted that such a “geometrical” approach to the attraction of spheres in Newton arose because Newton is not interested here in the magnitude of the force of attraction, but only in its dependence on distance; for Newton it is important here that these spheres be homogeneous in density. At the same time, this is also an interesting example of the application of Definition 1 to the case of equal density, when, consequently, only volume plays a role.
The term magnitudo (magnitude) is used by Newton (in the modern sense of mass) once (on p. 236). Krylov here speaks of a change in “the masses of bodies” and of the influence of this change on deviations in the motion of other bodies; but Newton in the Latin text speaks of a change in corporum magnitudine, for here he is interested only in changes of the sizes of bodies at any given density.
It must be supposed that Newton himself did not regard the words copia materiae, sphaera, and magnitudo as equivalent to the term quantitas materiae, and therefore he does not mention them, as synonyms of the term “quantity of matter,” in the explanation to Definition 1 (in contrast to the words “mass” and “body” (corpus)).
The term quantitas materiae is used by Newton in the modern sense of mass as a measure of inertia, for example, on pp. 24 or 29, when Newton says that the quantity of motion is produced from quantitas materiae and velocity, or that the motive force is proportional to the accelerating force and quantitas materiae (which is now expressed by the equation \(f = ma\)). In the same sense Newton says on p. 466 that “the inertia of matter is essential to bodies and is always proportional to the quantity of matter” (in this case Krylov translated the term quantitas materiae quite accurately).
But Newton quite often speaks of quantitas materiae also in the sense of modern gravitational mass. We take the opportunity to note that Newton himself never stated directly that he distinguishes gravitational and inertial mass (or quantitas materiae, or corpus, etc.). Here we are speaking of what in fact took place in Newton, from the modern point of view, which distinguishes gravitational and inertial mass and then equates them. In Newton this distinction was not explicitly expressed by him; he simply spoke of quantitas materiae, or corpus, etc., in accordance with Definition 1, and used these terms in the corresponding different cases, in fact distinguishing what are now called gravitational and inertial masses, although Newton himself did not have these terms.
In the case of comparison of homogeneous bodies, the term “quantity of matter” means the quantity of the given matter in a directly objective sense. This is also true for the case of the example that Newton gives in the explanation to Definition 1 to illustrate that the quantity of matter is determined from volume and density, namely, that “air of double density in double volume is four times as much, in triple—six times as much.” Here the discussion concerns the quantity of air, i.e. the quantity of the given matter in a directly objective sense, without mention of any specific units for measuring this quantity, for which in the present case there is no need. And nowadays, too, one often speaks of the quantity of a particular substance in just this way.
Of course, Newton did not invent this term arbitrarily. In human society, long before Newton, in trade and technology people often dealt (as they do now) with estimating the quantity of a given substance.
and they spoke of this quantity of matter, took it into account in commodity exchange, in industry, in agriculture, and in technical work.
Newton was the first to introduce an estimate of the quantity of matter, or mass, of various bodies simultaneously by their volume and by their density, as he says in Definition 1. In practice, however, as Newton himself indicates at the end of the explanation to Definition 1, he retained the judgment of the quantity of matter in a body by its weight, but with the fundamentally important distinction that, in his view, the quantity of matter (or mass) was only proportional to weight, but not identical with it*).
Newton’s use of the term “corpus” is explained by the fact that any quantity of particles of matter forms one body or another (corpus). Therefore Newton uses the term “corpus” both in the sense of inertial mass and in the sense of heavy mass, just as he does the term “quantitas materiae.”
Thus, on p. 223 it is said that “the motive forces ..., proportional to the masses of the bodies, ... cause both bodies ... to move equally.” Newton does not use the term “mass” here and speaks here (in the Latin text) of motive forces which are sunt ut corpora, i.e., stand as the bodies, and which cause corpora, and so on. It is noteworthy that here, for the first time, the word “corpus” denotes inertial mass, and the second time simply a body. On p. 516 there is a remarkable statement by Newton that “the forces by which unequal masses (corpora) are accelerated equally are proportional to the masses (sunt ut corpora), i.e., the attraction is proportional to the masses of the planets” (attracted by the Sun.—V. F.). In Newton’s Latin text it says: quantitas materiae of the planets. Adhering to Newton’s exact text, the following translation should have been given here: “the forces by which unequal bodies are accelerated equally are proportional to the bodies, i.e., attraction is proportional to the quantities of matter of the planets.”
At the same time, on p. 29 Newton writes that “the motive force of gravity,” or weight, is proportional to the mass of the body” (in the Latin text it says: ut corpus). On p. 519 it is said: “the attraction toward these bodies, being as many times less than the attraction toward the Earth as the mass of the body is less than the mass of the whole Earth (in Newton’s Latin text: ut sunt hae corpora ad Terram totam, i.e., ‘as these bodies stand to the whole earth’), will be far less than what could be sensed”**). Here Newton gives a remarkable explanation of why, despite the universality of gravitation, we do not feel the attraction of our body to various individual terrestrial bodies, the existence of which was subsequently demonstrated in the famous experiments of Cavendish, Maskelyne, and others. Most often, however, Newton uses the term “corpus” in the sense of inertial mass.
Let us cite one more, especially striking example of Newton’s mixed, so to speak, terminology, when over the course of literally one page and even a few lines Newton uses a number of different terms instead of the term “quantity of matter.” On p. 517, where the explanation to Proposition VI in the third book and the corollaries from it are set forth, Newton uses the term quantitas materiae 8 times, the term massa once, and the term materia twice. In addition, in other places of the same explanation to Proposition VI Newton also uses the term corpus—3 times—and the term copia materiae—2 times. Krylov here, in translation, almost everywhere used the word mass, which, in meaning, is of course correct, but once again conceals from the contemporary Russian reader Newton’s style and terminology.
The fact that the term massa, so rarely used by Newton himself, ultimately displaced all the other terms used by Newton
) Newton himself says this on p. 63 of the Principia*.
**) Krylov here translated Newton’s text while adhering exactly to Newton’s explanation of the term “corpus” in Definition 1.
in the same sense as the term massa, which is explicable by the following reasons. First, the word massa is, so to speak, international. In many modern languages, in everyday speech people speak of a “mass” when they have in mind a large multitude. Meanwhile, such terms as quantitas materiae, corpus1, are specifically Latin. When original national cultures began to develop, scientists of different countries began to write their works in their native, national languages (instead of Latin); but at the same time an international scientific connection was necessary, as follows from the very essence of science. This led to the gradual establishment of the use of the term massa as an international scientific term denoting the measure of what, in a spontaneously materialist way, is fundamental for natural science—the measure of matter, as Newton called it. Second, the word “massa” is simpler than quantitas materiae, both in pronunciation and in printing, and this simplicity plays its role (which cannot be ignored) in the development of scientific terminology and scientific symbolism2. Third, Newton’s younger contemporary J. Bernoulli (1667–1748), who enjoyed great influence in the scientific world, and later Euler as well, most often used the term massa. Fourth, the term “quantity of matter” was displaced by the term “mass,” as it seems to us, also because physicists instinctively felt the inadmissibility (in the directly objective sense) of the former term for the case of heterogeneous bodies.
The attitude of contemporary physicists toward the definition of mass as the quantity of substance (matter) attributed by them to Newton. Those who, for various reasons, consider it inadmissible to define mass as the quantity of substance are occupied with “correcting” Newton, i.e., with proving that Newton made a serious mistake on this question. Those contemporary scholars, on the other hand, who sympathize with such a definition of mass and defend it, rely on Newton’s high authority to confirm the correctness of their position on this question.
But both the former and the latter, without realizing it, are, so to speak, unjustly blaming Newton: they attribute to him a definition of mass that he never gave and, in principle, could not have given, for Newton understood very clearly that it is impossible to define mass as the quantity of substance in the case of heterogeneous bodies. Thus contemporary physicists, by attributing to Newton the definition of mass as the quantity of substance, distort the genuine Newtonian doctrine of mass and display in this respect an underestimation of Newton’s great creation. In this sense one would like to exclaim: “back to Newton,” to the genuine Newton, to his genuine definition of mass.