Predemonstrable Foundations

¹ Parts are given in the continuous, contrary to what the most acute Thomas the Englishman [Thomas Aquinas] has done.

² These parts are infinite in act.

For the “indefinite” of Descartes is not in the thing itself, but in the thinker.

³ There is no minimum in space or body, or something whose magnitude or part is null.

For such a thing has no position, since whatever is positioned anywhere can be touched simultaneously by many things not touching each other, and consequently would have many faces.

But if a Minimum could be posited, it would follow that the whole is as many as the minimum parts, which is a contradiction.

⁴ Indivisibles or unextended things are given, otherwise neither the beginning nor the end of motion, body, space, or time can be understood.

The demonstration is this: the beginning and end of space, body, motion, and some time are given. Let that thing whose beginning is sought be represented by line ab, whose middle point is c, and the middle between a and c is d, and between a and d is e, and so on. Let the beginning be sought to the left, at side a.

I say that ac is not the beginning, because dc can be taken away from it while the beginning remains safe; nor is ad, because ed can be taken away, and so on. Therefore, nothing is a beginning from which something can be taken away to the right. That from which no extension can be taken away is unextended; therefore, the beginning of a body, space, motion, or time (namely a point, an endeavor/ conatus, or an instant) is either nothing, which is absurd, or it is unextended, which was to be demonstrated.

⁵ A point is that which has no extension. Its parts are non-distant. Its magnitude is:

• unconsiderable, unassignable
• smaller than any ratio (unless an infinite one can be set out in relation to another sensible thing)
• smaller than any that can be given.

A point is not that whose part is null, nor that whose part is not considered.

This is the foundation of the Method of Cavalieri, by which its truth is evidently demonstrated: the beginnings of lines and shapes are to be thought of as smaller than any given.

⁶ The ratio of rest to motion is not that of a point to space, but that of nothing to one.

⁷ Motion is continuous or interrupted by no periods of rest.

⁸ For where a thing has once rested, unless a new cause of motion is added, it will always rest.

⁹ Conversely, what is once moved, as much as is in itself, is always moved with the same speed and in the same direction.

¹⁰ Endeavor (Conatus) is to motion as a point is to space, or as one is to the infinite; for it is the beginning and end of motion.

¹¹ Whence whatever is moved**, however weakly, and even if any obstacle exists, will propagate its endeavor through all obstacles in a plenum to infinity, and consequently will imprint its endeavor on all others.

For it cannot be denied that it attempts to proceed even when it stops, or at least it endeavors (conetur); and therefore it endeavors, or (which is the same thing) begins to move obstacles however great, even if it is overcome by them.

¹² Multiple contrary endeavors can exist simultaneously in the same body.

For if there is a line ab, and c tends from a toward b, and conversely d from b toward a, and they concur; at the moment of concurrence, c will endeavor toward b, even if it is thought to cease.

The following is the English translation of pages 10 and 11 from the provided file m11.jpg. In these pages, the author explores the relationship between endeavor (conatus), the nature of matter, and the distinction between body and mind.

[10]**

…to move, because the end of motion is endeavor (conatus); but it will also endeavor backward if the opposite is thought to prevail, for it will begin to go backward. Yet even if neither prevails, it will be the same, because every endeavor is propagated through obstacles into infinity, and thus that of each into the other. And if nothing is achieved with equal speed, nothing more will be achieved with doubled or greater speed, because twice nothing is nothing.

¹³ In a time less than any that can be given, a point of a body is in many places or points of space**; that is, it will fill a part of space greater than itself, or greater than it fills while resting.

Either its motion is slower, or it is endeavoring in only one direction; nevertheless, it consists in an unassignable [space] or in a point. Although the ratio of a point of a body (or a point of space which it fills while resting) to a point of space which it fills while in motion is the same as the angle of contact to a straight line, or of a point to a line.

¹⁴ Absolutely everything that is moved is never in one place while it is moved**, not even for an instant or a minimum time. Because what is moved in time, endeavors in an instant, or begins and ceases to be moved—that is, to change place. Nor does it matter to say that in any time smaller than can be given it endeavors, but in a minimum [time] it is in a place. For no minimum part of time is given, otherwise a minimum of space would also be given. For what completes a line in an absolute time can complete a line smaller than can be given in a time smaller than can be given…

[11]**

…smaller than can be given, or to a point; and in an absolutely minimum time, it completes an absolutely minimum part of space, of which none exists according to section 3.

¹⁵ Conversely, in the time of impulse, impact, or concurrence, the two extremities of bodies, or points, penetrate each other, or are in the same point of space. For when one of those concurring endeavors into the place of the other, it will begin to be in it—that is, it will begin to penetrate or be united.

For endeavor is the beginning, and penetration is the union. Therefore, they are in the beginning of union, or their boundaries are one.

¹⁶ Therefore, bodies that press or impel each other cohere**. For their boundaries are one, and as they are “two things in one,” they are continuous or a coherency, as even Aristotle defines it.

Because if two are in one place, one cannot be impelled without the other.

¹⁷ No endeavor without motion lasts beyond a moment, except in minds.

Endeavor in a moment is the same as the motion of a body in time.

Here, I give a true discrimination between body and mind, explained by no one until now.

For every body is a momentary mind, or one lacking recollection.

Because it does not retain beyond a moment its own endeavor and a contrary foreign one (for in both there is action and reaction, or comparison, and therefore harmony—for the sense, and without which there is no sense, pleasure, or pain—there is a need for it).

Therefore, it lacks memory, lacks the sense of its own actions and passions, and lacks thought.

[12]**

¹⁸ A point can be greater than a point. An endeavor greater than an endeavor, but an instant is equal to an instant; whence time is represented by uniform motion in the same line, although an instant is not without its own parts, but they are non-distant (like angles in a point) which the Scholastics—I know not whether following the example of Euclid—call signs.

As appears in those things which are simultaneous in time, but not simultaneous in nature, because one is the cause of the other; likewise in accelerated motion, which increases with every instant, and thus immediately from the beginning. To increase, however, presupposes a “prior” and a “posterior”; it is necessary in that case, in a given instant, for one sign to be prior to another, even without distance or extension.

Add to this problems 24 and 25: No one will easily deny the inequality of endeavors, but from that follows the inequality of points. That an endeavor is greater, or a body that is moved faster than another, is evident because from the very beginning it completes more space.

For if in the beginning it completed only the same amount, it would always complete the same amount, because as motion begins, so it continues, unless an extrinsic cause changes it (by foundation 9).

But if the beginnings are equal, the ends are also equal; therefore, at the moment of concurrence, the fast would achieve only as much against the slow as the slow against the fast, which is absurd.

Assume therefore they are unequal.

Therefore, in a given instant, the stronger will complete more space than the slower; but any endeavor cannot traverse in one instant more than a point, or a part of space smaller than can be set out; otherwise in [finite] time it would traverse an infinite line: therefore, a point is greater than a point. Whence the unassignable arc of a larger circle is greater than that of a smaller one; and any line drawn from the center to the circumference, commensurable to the circle, or by its own circumduction the generator of the circle, is a minimum sector perpetually increasing, but within unextension.

Hence the difficulties of two concentric wheels rotated over a straight plane, of incommensurables, of the angle of contact, and so many other things are solved, to explain which the most eloquent Borelli challenged all the Philosophers of the whole world, and in which the Sceptics especially triumph.

An Angle is the quantity of a point of concurrence, or a portion of a circle smaller than can be assigned, that is, a Center; the whole doctrine of Angles is about the quantities of unextended things. An arc smaller than can be given is surely greater than its own chord, although this too is smaller than can be set out, or consists in a point.

But so, you will say, an infinite-angled Polygon will not be equal to a circle: I respond, they are not of equal magnitude, even if they are of equal extension; for the difference is smaller than can be expressed by any number.

Whence, from Euclid’s definition—that a point is that of which there is no part—no error could creep into demonstrations of Extension (as indeed seemed to the otherwise most profound Hobbes, who erred from that head 47). Truly, the Canon of sines, and whatever [pertains] to his quadrature…

The following is the English translation of pages 14 and 15 from the provided file m15.jpg. In these pages, the author discusses the composition of motion, the creation of complex curves like the cycloid, and the principles of rational choice in physical dynamics.

[14]**

…his [Hobbes’] quadrature opposes, he calls into doubt, which I never read from such a Great Man without unexpected admiration: provided a part of extension is understood as a part distant from another part.

If the unassignable arc and chord coincide, the endeavor (conatus) in a straight line will be the same as in an arc: for the endeavor is in the unassignable arc or straight line. Now if the endeavor is the same, then the motion in a straight line and an arc is also the same, that is, circular and rectilineal motion (because whatever motion has begun is continued, or as the endeavor is, so is the motion), which is absurd.

¹⁹ If 2 endeavors are at once compatible, they are composed into one, with both motions preserved.

As is evident in a sphere rotated over the straight line of a plane, where the motion of some point, designated on the surface, composed from the rectilineal and circular, through minimums or through mixed endeavors, is composed into a Cycloid; add concerning the spiral th. 7 & 12. This argument deserves to be treated more diligently by Geometers, so that it may appear which endeavors of lines, when mixed, produce which new lines, whence many perhaps new Geometric Theorems can be demonstrated.

²⁰ A body which is moved, without diminution of its own motion, imprints on another that which the other can receive, while the prior motion remains safe**; hence theorems 5 and 6.

²¹ If something cannot do all things at once, and there is an equal cause for all, and no third thing, it does nothing. Hence the cause of rest th. 11, 12. (22.) If incompatible endeavors are unequal, they are taken away from each other…

[15]**

…the direction of the stronger being preserved, theor. 1: 2, 3. Because two endeavors can be taken from themselves, for the smaller is equal to a part of the larger. So long, therefore, as the thing finds an exit through a part of either, there is no reason why a third should be chosen.

²³ If incompatible endeavors are equal, they are deceived by mutual direction, and a third intermediate direction, if any can be given, is chosen, with the speed of the endeavor preserved**; theor. 7. 8. 9. 10.

This is the apex of rationality in motion, when not only by the brute subtraction of equals, but also by the election of a more proper third, the matter is finished by a certain wonderful but necessary prudence of species; which does not easily occur elsewhere in all Geometry or Phoronomics.

Since therefore all other things depend on those principles—that the whole is greater than its parts, and whatever else is to be resolved by addition and subtraction alone—Euclid prefixed these elements; this one thing, with foundation 20, depends on that most noble [principle]:

²⁴ Nothing is without reason**, of which it is a corollary that the minimum must be changed, and between contraries a medium must be chosen, and anything must be added to one lest anything be taken from either; and many other things which also dominate in civil science.