Indivisible points do not seem to exist. They cannot be assigned to a substantial body except in thought and imagination. They are neither parts nor elements composing a material substance. This appears from the fact that a point added to another point twice, ten times, thousand times, etc. does not make more, and several points do not seem to be different from one point alone since a multitude of points, as well as one point alone, do not occupy any space. On the contrary units added together create a numeral magnitude so that units inasmuch as they are parts, and elements are considered as numbers rather than points of the quantifiable substance. Therefore, as a number cannot be created from infinite figures and a being [an entity] cannot be made from an infinity of non beings [non entities], similarly a quantity cannot be made of an infinity of non quantities which are neither parts nor elements of a quantity. Indivisible points are not quantities and are neither parts nor elements composing a quantity. Consequently, material substance which is quantifiable cannot be composed or created from an infinity of indivisible points.

Let the quantities A, B, C, D, E, F, G be in infinite number and equal to each other (Table 5.​1, Fig. 10). I claim that they form an infinite expansion. Let us take a finite quantity of any magnitude RS of its kind with the particles A, B, C, etc. Of course, either RS is a multiple of A, i.e. either A is a measure of RS, or it is not. Firstly, let us suppose that RS is measured by A. Thus the ratio RS/A is the ratio of a finite number to the unit. Therefore, among the infinite number of parts A, B, C, etc. a number of parts can be taken which is greater than the number of parts RS. This greater number of parts forms an expansion X. Unquestionably X is greater than RS and an aggregate of an infinity of particles A, B, C, etc. forms a larger expansion than the finite number X. Therefore, the aggregate of an infinite number of particles forms a much larger expansion than RS. This expansion actually which is larger than any finite quantity, is necessarily infinite. Thus, an aggregate of an infinite number of quantifiable particles equal to each other forms an infinite expansion.

Secondly, let A and RS be incommensurable with each other (Table 5.​1, Fig. 10). A portion SV can be added to RS so that RV becomes a multiple of A. Then the aggregate of an infinite number of equal particles A, B, C, etc. forms a larger expansion than RV, as was shown above. Therefore, it creates a much larger expansion than RS. Therefore, we shall conclude that it is infinite.

Quantifiable particles in infinite number and unequal form an infinite expansion.

Let the parts AB, CD, EF, GH, IK, etc. be in infinite number and unequal (Table 5.​1, Fig. 11). I claim that they form an infinite expansion since they are all quantifiable parts in infinite number. Therefore, there is among them a smallest one. Let it be AB. From all the other larger ones, portions CL, EM, GN, etc. can be taken which all are equal to the smallest part AB. An infinite number of unequal particles AB, CD, EF, etc. form a greater expansion than an infinite number of smaller particles equal to each other AB, CL, EM, etc. On the other hand, according to the previous proposition, an infinite number of quantifiable and equal particles of the same kind AB, CL, EM, etc. form an infinite expansion. Consequently, even much more, the infinite number of unequal parts AB, CD, EF, etc. larger than the equal ones will form an infinite expansion. Q.E.D.

If some parts of an aggregate are displaced while the others are immobile, or if all are agitated by unequal movements which do not coincide, and if they cannot be assimilated to the parts of a hard and consistent body, the parts of this aggregate are necessarily separate.

A sign must some way be sought for from which we can clearly determine whether a body is actually divided into different parts or whether it is one continuous, even if these parts are inconspicuous and invisible as a result of their smallness or transparency. This sign is the movement and the immobility. If one part A of a composite body moves in different ways while other adjacent parts B, H, E, CI, etc. are immobile at the same place, it is sure that the agitated part A is separate and distinct from the others. But, when all the parts of the composite body move, we must find how to decide whether the parts are continuous and united or separate from each other. Of course the inequality of the movements does not always indicate division rather than continuity. Indeed when a solid wheel rotates, its hard particles, although tenaciously bound and tied to the wheel, move at unequal velocities proportional to their distances from the fixed axle of the wheel. Consequently, in a rotating fluid, if the circles described in the same time by its particles are the larger the more the particles are remote from the axis of rotation and are proportional to their distances from this axis, we can of course doubt whether the particles of the fluid are continuous or whether they are actually separate.

Consequently, in the rotation of a wheel, if one part recedes from the centre more or less than previously or if it moves more quickly or more slowly than what corresponds to its distance from the axis, this particle is necessarily distinct and separate from the wheel. In the wheel AEH rotating about its centre D (Table 5.​2, Fig. 5), if the particles A, B, C describe in the same time the circles AEH, BFI, CGL proportional to their distances from the centre AD, BD and CD, one cannot determine whether these particles are separate like sand or whether they are welded to the solid wheel since they behave like the particles of the hard wheel. But if, during the rotation of the wheel, the particle A leaves the circle AHE and moves over the tangent AM or the spiral AN, this is an obvious indication that the particle A is neither annexed to nor united with the solid wheel but is separate from it, because it moves continuously more and more away from the centre D, as to N or M.

Moreover, if particles keep their distance from the centre and, in the time that the wheel carries out a whole revolution BFB, another particle A describes more or less distance than the circle AEA, for example the arc AEH or the arc AEO, it is obvious that the particle A is not part of but is separate from the solid wheel.

Similarly, during the straight displacement of an aggregate AEH (Table 5.​2, Fig. 6), if its particles move at unequal velocities, for example if, while A describes the straight line AB, another part E travels the straight line FC shorter than AB, and another part H travels the distance HD shorter than EC, this is an obvious indication that the particles A, E, H are distinct and separate from each other.

From this a very general rule can be deduced: whenever an aggregate of bodies moves straight ahead and its parts are displaced at unequal velocities or their paths are not parallel, or if, while rotating about a centre D, all its particles describe spirals such as AN, or if they describe circles at velocities which are not proportional to their distances from the centre, or if they are the slower the more they are remote from the centre (for example if the particle C moves more slowly than D, B more slowly than C, and so on), unquestionably these inequalities indicate that the particles composing the aggregate are completely distinct and separate from each other. Therefore, these movements do not coincide and cannot fit the parts of a continuous, consistent and hard body.

Whenever a body is supposed to be fluid its parts must necessarily be separate. A fluid must have a natural conformation and structure, and all the requisites to be able to carry out the movement which we call flowing, i.e. one of its parts can move while the others are immobile or, if all move, they travel unequal straight distances in the same time or, if they move in a circle, their revolutions are not proportional to their distances from the centre of rotation. These motions cannot be carried out unless the parts of the fluid are separate as was just shown. Consequently, whenever a body is supposed to be fluid, its parts are necessarily separate.

A fluid cannot have adjacent parts forming one continuous whole.

Moreover, if the parts of a fluid were not separate, i.e. if water had all its parts connected and agglutinated forming one continuous whole, and these parts moved straight or circularly in a fluid of its own kind, i.e. stagnant water, its minute particles would have to retain the same invariable arrangement, situation and structure as if they were parts of a solid body or a solid wheel. Unquestionably, they would then describe in the same time equal straight lines or unequal circles increasing proportionally to their distances from the centre or firm axis, and could not behave otherwise. But all the water of a lake does not move equally straight nor does it rotate together with its internal portion displaced or rotated. We see that the most remote parts remain absolutely immobile while the intermediate ones run or rotate very quickly. There is no immediate passage from the greatest velocity of the internal parts of the fluid to its most remote and quiescent parts which act as a vessel. But, as appears in murky water and in smoky air, one passes by gradual increment from the parts of water moving straight or rotating very quickly, through slower and slower parts to the extreme and immobile parts. Consequently, water cannot have adjacent parts forming one continuous whole. After these premises, we arrive at the main proposition.

If this is true, the smallest particles forming a fluid are always fluid, if at all possible. Thus, by dividing a fluid body indefinitely and ad infinitum, we shall never arrive at its smallest particle which is not fluid. It will always be fluid. Something is fluid as long as it can carry out the movement which we call flowing, i.e. it is fluid as long as some parts can move while others are immobile or carry out movements different and unequal from those which fit hard and continuous bodies. Thus, so that no particle of the fluid body lacks this quality of fluidity, it is necessary that the definition of fluidity applies to it always, namely any of its parts must be able to move while others are immobile or carry out different movements from those which fit hard and continuous bodies. But the adjacent parts of this mass cannot move partly and be immobile partly or be agitated in varied way by movements different from those which fit continuous bodies unless they are separate and distinct. Consequently, there is no particle of a fluid body, however small it can be imagined, which is not divided and subdivided into several other particles. Therefore, one can never reach the end of the enumeration of its many particles. Their multitude will be greater than any number, i.e. greater than any finite quantity, thus infinite. An infinite number of separate parts, if they were quantifiable, equal or not, would make an infinite expansion. Thus, a fluid ball of one palm would have an infinite magnitude. But is not what occurs. Therefore, they will not be quantifiable but indivisible points. But this is also impossible since an infinite number of points cannot make a quantifiable expansion. Therefore, it is untrue that the smallest particles of which a fluid is made and in which it can be divided are always fluid. Q.E.D.

From this it is concluded that a fluid body is composed of non fluid minute particles.

If this is not true, i.e. if the particles composing fluid water are always fluid, the water can be divided always successively and ad infinitum into particles which are always fluid. This is against the opinion of Aristotle who, contrary to Anaxagoras, denies that any natural body can retain the same nature if reduced more and more through continual division into small and minute particles. If the flesh of animal is thus divided it will be reduced to particles which are no longer flesh. Also, as his exponents say, after successive similar division is carried out in elements, finally the most minute particles no longer retain the nature of the element. Hence it can be inferred that, when fluid water is subdivided successively ad infinitum, one finally arrives at particles which are not fluid, i.e. one particle cannot move while the others are immobile. Therefore, all can be agitated together in one movement, i.e. they must have a solid consistency.

But leaving the authority of Aristotle and the peripatetics, let us ponder the force and energy of the argument. If it is true that water divided in any way and subdivided always retains its fluidity, the definition of the fluidity mentioned above always applies to these very last fluid particles, namely any small particle of a particle can move whereas the adjacent parts are immobile. All sublunary bodies are perforated by countless pores and small holes. Consequently, water would penetrate all hard bodies. Let us indeed imagine a very narrow pore in a wooden, glass or metal vessel. A portion of water superimposed on this small hole which it matches has a size and magnitude equal to the amplitude of the hole. According to the always divisible nature of a quantity, central particles can be imagined narrower than the size of the pore. These central particles of water can move while the adjacent ones are immobile, as required by the nature of a fluid mentioned above. Consequently, they will necessarily be able to run freely through the size of the pore. Therefore, there will be no vessel through which water cannot penetrate. It must be noticed that water could be impelled and introduced through these pores by a motive virtue however small, namely by the force of its own gravity or by any other force blowing it in or impelling it, as we will show later. This is obviously untrue since fresh water or the thinnest spirit of wine cannot penetrate the pores of glass even if violently impelled. Consequently, it is untrue that a fluid can be divided ad infinitum into parts always fluid. Therefore, by dividing, we must finally arrive at particles of water which are no longer fluid, i.e. such that one very small of these particles cannot move while the adjacent ones are immobile. Consequently, these extreme particles of the fluid are consistent. Q.E.D.

Water can be impelled through the pores of glass by a very small force. We indeed see that a long beam stagnant on water can be pulled transversely by the force of a small hair. Consequently, this very small motive force can overcome the resistance of as many parts of water as can be assigned to the size of the beam. It is thus conjectured that the force necessary to impel a small and pointed floating straw must be almost insensible as a result of its extreme smallness. However, such a small force moves one particle of water while the adjacent ones do not move. Therefore, a force equal to this is sufficient to overcome the pertinacity which binds and unites the particles of water. Consequently, the energy of the pertinacity of the parts of water has a very small vigour and, therefore, can be overwhelmed by a small impulsive force.

Although fresh water is not absolutely pure and without some mixture of earthy and solid parts, pure parts of water cannot be denied. They intervene between the most minute earthy fragments. According to their nature, since they are fluid, they could penetrate the interstices between the small grains of sand mixed with the water and would be able to pass through the pores of the grains of sand.

The pores of glass or of metal may not be straight but twisted in amazing ways, and circuitous. Nevertheless their size could not prevent and impede completely the passage of flowing water. At the worst this passage could have to be slowed down and to occur in a longer time than if it occurred through straight and equally wide pores. But this is untrue, i.e. water contained in a glass vessel never exudes. It must thus be admitted that the smallest particles are not fluid but are consistent.

The pores of any vessel are absolutely pervious and their paths are neither impeded nor occluded by plugs. This is proved by the fact that some fluids penetrate through them, such as quicksilver through the pores of gold; water, oil and quicksilver also through the pores of wood and vessels of terracotta. Therefore, all the other fluids should necessarily pass and flow through these, at least slowly, if it is true that there is no fluid part of a fluid which is not fluid also. Consequently, air should flow out of a vessel of terracotta or wood whenever condensed there after being sent in violently.

If we wish to argue, not for the pleasure of arguing, but according to the laws of nature, four prerequisites appear to be necessary for the constitution of a fluid. The first is that the body be divided and subdivided into small, minute particles. The second is that their shapes become as spherical as possible. The third is that their surface either be most perfectly smooth like a mirror or at least can very easily run and flow one over the other. Finally, all must have an equal motive force by which they strive downwards, i.e. they must be equally heavy. All these requisites could be confirmed even using a rough example. Several crystalline globules are taken and put in a vessel. Firstly, these globules adapt to the shape of the vessel. If a hand is introduced down to the bottom of the vessel, the globules yield and move to the sides. Moreover, one or another crystalline ball can be moved while the collateral ones are immobile or hardly moved. Besides, after being shaken, the globules would be flattened down, not in a heap like grains of wheat or of sand. As a result of their lightness they would run easily and descend towards the lowest positions. In this way the upper surface would be flattened and become about horizontal. If these small crystalline spheres were smaller and more minute, these operations would be carried out much more easily. If finally they were reduced to an unutterable smallness they could not be seen nor palpated but would present an appearance of continuity, as occurs with the most minute powder. They could then have all the effects of fluidity and, however, this mass would be an aggregate of countless hard and consistent crystalline globules.

But an audacious opinion which is thriving by now deserves that we lay some stress on this examination. People indeed admit that a fluid is composed of separate, small and smooth particles but they say that its decisive condition is that the particles by which it is formed be agitated in different ways. Either this movement is innate or it is carried out by its passage through some thinner substance rolling the particles towards any direction.

Two main arguments are brought forwards to confirm this opinion. Firstly, in melting metals we see that the minute particles of metal are actually agitated by the violence of the fire. The same is observed in wax and in the other bodies which are liquefied by the action of the fire. It is obvious in boiling water that the igneous exhalations penetrating through the pores of the basin produce countless bubbles moving very quickly through the water itself. It thus can be suspected that the fluidity results from this most impetuous ebullition. Even if sometimes these bubbles in the fluid are not visible and the melted body appears absolutely quiet and placid, as occurs in molten lead, nevertheless the mass of lead is considerably increased by the melting. Moreover, smokes getting out of it carries quite a few parts of lead. It is thus obvious that molten lead is agitated continuously and its parts are twisted and moved in different ways.

The second argument is taken from fermentation. If some grains of salt, or some other soluble and fermentable substance, are immersed at the bottom of the water, we see that the taste and colour of this ferment quickly impregnate and alter all the water. This could not happen unless the saline particles were carried all over the water. This could in no way occur without agitation of the parts of this water.