(1)
Mathematics in Naples, Naples, Italy
Deceased
Hitherto we discussed the impetus of a projected body in general and on its qualities. Now the impetus resulting from an internal principle must be dealt with. This requires more accurate analysis and discussion of the flow, measures and properties of the impetus. Thus, an impetus either remains of the same magnitude so that it does neither increase nor decrease, or the degree of impetus continuously increases or decreases. Moreover, an increasing impetus can be augmented by similar and uniform increments or by irregular and unequal increments. All this must be discussed accurately and precisely, as well as the coherence and connection of a flowing impetus with a movement in the space of the world.
Firstly, it must be noticed that any degree of velocity which is effective at an instant in time must be considered as a permanent quantity since it can be divided and subdivided in so far as it is slowed down more or less. Consequently, the degree of velocity is not something momentary and indivisible since there is no beginning of velocity but velocity itself is brought to a well-determined degree. Since it originates from the flow of its indivisible moment, velocity itself can appropriately be considered as analogous to a simple linear length.
A continuous flow of the same degree of linear velocity in a well-determined time produces a quantity of another kind analogous to a surface. As a result of a continuous flow of the same impetus occurring in time, the degree of linear velocity is repeated as many times as instants can be counted in, and ascribed to, the elapsed time. As a continuous transverse displacement of a line generates a plane rectangular surface, the flow of the same linear impetus also results in a plane expansion. The actual difference between both surfaces is that the former produces a permanent and existing rectangular quantity whereas the latter gives a successive quantity, since repeated degrees of velocity do not co-exist and the past ones evanesce together with the elapsed time. One particular degree of velocity alone exists together with the instant in time during which it occurs. The flow of the same degree of linear velocity also determines the course or displacement of the body moving at this velocity because at any instant of time the same degree of velocity carries out an effort or some momentary motion corresponding to this degree of velocity. Consequently, this movement produced by this flowing degree of velocity is called uniform and corresponds to a rectangular surface resulting from the flow of one and the same degree of velocity.
If the degree of velocity, while flowing, increases or decreases in intensity, then of course as many unequal degrees of velocity, either increasing or decreasing, must be conceived as there are instants of elapsed time. Transverse displacement of an increasing line results in a plane surface expanding more and more such as a trapezium. An increasing line displaced transversely from an indivisible point gives a triangular figure, a rectilinear triangle if the increments of the displaced line are always equal, otherwise a curvilinear triangle. Similarly, for the flow of an impetus, if it starts from an indivisible moment and increases gradually while flowing, it will generate a triangular surface. If the increments of the impetus are equal in equal times they will give a rectilinear triangle, otherwise a curvilinear one. After these explanations on uniform motions, I will mention the primary and simpler properties in these premises.
Axiom V
The surface generated by the flow of the same degree of velocity during a longer time is bigger than that generated during a shorter time.
Axiom VI
Conversely, the time during which more surface is generated by the flow of the same degree of velocity is longer than that during which less surface is generated.
Definition II
To make short, the surface generated by a flow of impetus is called surface of the impetus or of the velocity.
Proposition LXXV
If two mobiles are displaced in an uniform movement at any velocities but during equal times, the surfaces of the velocities will be proportional to the travelled distances.
Two mobiles A and C are displaced in a uniform movement during equal times AB and CD (Table 16.1, Fig. 58). A travels a distance R at a velocity EA in the time AB whereas C in the time CD equal to the time AB travels a distance S at a velocity FC. The flow of the velocity EA results in the rectangular surface of the impetus AG since, at all the indivisible instants of the time AB, all the degrees of velocity which finally form the rectangle AG, while flowing, add to each other as a result of the same velocity EA which is a linear quantity. Similarly, in the time CD, the flow of the velocity FC generates the rectangular surface of the impetus CH. I claim that the ratio of the surface of the impetus AG to the surface of the impetus CH is equal to the ratio of the travelled distances R/S. The times AB and CD are equal. There are thus as many instants in either of them. But at any instant both velocities EA and FC are exerted. Thus, the ratio of all the velocities which are exerted in the surface of the impetus AG to all the velocities which are exerted in the surface of the impetus CH is equal to the ratio of one velocity EA to the other FC. But the ratio of the distances R/S is equal to the ratio of one particular velocity to the other EA/FC (since the distances are travelled in an uniform movement in equal times). Consequently, the ratio of the surface of the impetus AG to the surface of the impetus CH is equal to the ratio of the travelled distances R/S.
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