Newton's Principia - the first definition - concerning density

 Def. I.

The Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly.

THUS AIR of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of Body or Mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter. 

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

The first definition pertains to the “quantity of matter”, thus prompting our exploration to start with a historical analysis.

In the 2nd century BCE, Archimedes' work On Floating Bodies (Περὶ τῶν ὀχουμένων) first examined the concepts of weight in relation to floating objects. Ptolemy referenced Archimedes' work in his own writings during the 1st century CE, and the Greek commentator Eutocius of Ascalon (modern-day Israel) produced commentaries on these texts in the 6th century CE. It is likely that these works were translated into Latin by Roman scholars before being collected in the Bayt al-Hikmah, or "House of Wisdom," in locations such as Baghdad. Subsequently, copies of these texts were disseminated to libraries throughout the Umayyad Caliphate, extending as far as modern-day Spain.

In the 12th century, Averroes (Ibn Rushd) discovered these ancient Greek works in the library of Córdoba (Spain) and authored commentaries on them. There is evidence suggesting an interchange of ideas between Islamic scholars in the Umayyad Caliphate and those in Catholic Europe. For instance, Leon Battista Alberti, an Italian polymath, discussed Archimedes' principles in his own writings (Alberti, Circa 1450), reflecting the influence of Archimedes on Renaissance thought. This intellectual exchange occurred approximately forty years before the fall of the Kingdom of Granada in 1492, marking the end of the Spanish Al-Andalus.

In 1533, the lifelong works of Erasmus of Rotterdam were published. Erasmus dedicated his life to traveling across Europe to acquire manuscripts from libraries, among which were the works of Archimedes.

It is well established that Galileo owned a copy of Archimedes (Eutocius of Ascalon, 1544), as evidenced by his direct references to Archimedean concepts in his 1638 publication, Discourses and Mathematical Demonstrations Relating to Two New Sciences (Galilei, 1638). In this work, Galileo not only addresses the treatment of weight but also introduces the relationship between weight and volume for the first time. Notably, the phrase "quantity of matter" is not explicitly mentioned.

A pertinent excerpt from Day One of Two New Sciences states:
"I assume that the same body, if it be of greater bulk, will be heavier, and that this increase of weight will be according to the proportion of the increase in bulk; for example, if one piece of wood is twice as large as another, it will be twice as heavy, and if it be three times as large, it will be three times as heavy, provided that it is made of the same material and that the ratio of bulk to weight remains constant."

In 1644, Descartes published Principia Philosophiae (Descartes, 1644), in which he clearly distinguishes between an object's volume (described as its bulk) and its quantity of matter. A relevant quote from Principia Philosophiae (Part II, Article 18) reads:
"By body, I understand whatever has length, breadth, and depth, and by the term substance we mean nothing other than matter; the quantity of this matter can be measured by its volume, or by the extent of its dimensions and its density."

These foundational ideas provided the intellectual starting point for Isaac Newton, who famously referred to standing on the shoulders of giants later in his life. In 1687, Newton published the Principia (Newton, 1687), in which he formally defines the quantity of matter - commonly known as mass - setting the stage for the entirety of his work.

In beginning with definitions, Newton’s Principia follows the structure of all the major axiomatic studies in geometry. For example, Euclid’s Elements (Euclid, Circa 300 BCE) opens with 23 definitions, which form the foundation of Euclidean geometry. It seems that Newton initially aimed to present an equally rigorous logical framework. In fact, these definitions, along with the introductory lemmas that follow, appear to have been added retrospectively to provide justification for some of his later, less rigorously supported statements.

Turning to the definition itself: the quantity of matter is measured as the product of its density and volume. In modern terms, this corresponds to the concept of mass, which is equal to the product of an object's density and its volume.

                                                                m=ρV                                                                 (1)

What then follows is a clarification by Newton – that explores the word conjunctly – essentially explaining it to mean that conjunctly means to multiple both the density and the volume together. To further demonstrate the point Newton describes how doubling both the density and doubling the volume has the effect of quadrupling the mass.

 

                        4m=(2ρ)×(2V)=2×2×ρV                                                (2)

 Newton subsequently suggests that this principle applies to substances such as fine powders and snow, which can be compressed into a solid form. In these cases, the volume of the solid can be measured, and, combined with its density, it can be demonstrated that the mass increases fourfold.

Newton focuses on the material substance itself and, although he alludes to the possibility of a medium occupying the interstices or spaces between the atoms - or "parts of bodies," as he terms them - he dismisses this consideration for the sake of clarity.

He asserts that mass is determined by weight, which, according to Newton, is directly proportional - a relationship he later verifies through pendulum experiments, as discussed in Book III, Proposition VI, Theorem VI. This forward reference to the final book suggests that Newton revised the text before its publication in 1687, recognising the importance of linking the definition introduced at the beginning to the theorem presented near the end.