Harmonic Demonstrations

reason stated.

CLAV. We will say, therefore, that every time I say that the Premises of the Demonstration must be more notable than the conclusion, and precede it: that we must always understand it according to the progression of nature.

GIOS. It is so in fact.

ADRI. Therefore, in such a Syllogism we follow the order of nature: and its progression.

GIOS. Without any doubt. But this does not happen many times.

FRAN. For what reason?

GIOS. Because of our imperfection. Whence it arises that, similarly, philosophers rarely put these syllogisms into being.

CLAV. We understand this well: but it remains that you continue to explain to us what is meant by the premises being primary, and without any intermediary.

GIOS. "Without intermediary" is understood when the middle term that is taken in the demonstration is so joined to the major [term], which is taken: and to the major, which is to be concluded: that no other thing can be found in the middle: or it is the middle between them by predicamental degree: and you will be able to understand this (to give an example) as happening between the Definition and the Defined: there being between Man, who is defined: and the Discursive Animal, which is the definition: no intermediary by which one could show that Man is such. For one cannot demonstrate the definition of things.

CLAV. The example you have brought has illuminated this reasoning in such a way that until now what you have said is understood very well. Therefore, continue with the rest.

GIOS. The Premises must, besides this, be primary: that is, they must be such that one does not find any Science (I will say it this way) higher proposed and more notable than them: and it must be necessary that they are taken as known, without any proof.

ADRI. Therefore, for such a reason they will be, or will be called Indemonstrable.

GIOS. That is very good, Sir: because since that which is demonstrated must arise from the preceding and more notable premises: if the Premises all always had to be demonstrated: and also the Premises of the Premises: our progression would have to be to the more notable and more preceding [premises], always ascending into infinity. Wherefore, not being able to ever arrive at those Premises which are known by themselves, nor being able to pass through the infinite: it would be necessary to stop at some of them: which, because they depend on higher premises, would not be known by us by themselves: and consequently the conclusions that would arise from such premises: could not be rendered manifest. From which would follow that which many have imagined: that no proposition could be demonstrated.

FRAN. This discourse has been very useful to me: because it has brought back to my memory many things: which once (because I did not attend to these studies) had slipped from my mind. And I remember that I heard it said many times: that in any Science, before one comes to discuss anything in it: some manifest propositions are supposed, which must not be denied by anyone who wishes to exercise themselves in that Science.

GIOS. You remember very well, upon my faith: and I rejoice that I will not have the fatigue of repeating such a thing anymore: nor many others, that concur in the Demonstration, because of you: of which I know you, together with the others, to be instructed. Therefore, continuing where I left off, I will say that some call such Principles Positions: and these Positions, or Principles, are of many manners. For some are called common Principles: and some are called proper [Principles]. The proper ones are those that serve a particular Science. Nor do you need to think that the proper Principles of one Science are the same principles of another: because you would be in error. But you must know (as I have also said above) that from the diverse genera of knowable things, diverse Sciences arise. Whence, just as continuous Quantity is different in genus from the Discrete: so Arithmetic is different from Geometry. And just as Magnitude is different from Number: so the principles of Geometry, with which its conclusions are demonstrated, are different from those of Arithmetic. Whence the proper principles of Geometry are (to give you an example) these: One can lead a Line from one point to another: The continuous is divisible to infinity: and other similar things. But those of Arithmetic are: Number is a multitude ordered by Units: The parts of number do not join at a common term: Numbers proceed beyond Unity to infinity: and the others. And those of Music are: The Interval is the relationship of the spaces of the low and high sound: and other similar things: as you will soon see. And these are called proper Principles. But the Common ones are so notable: that they can be supposed not only in this or that Science: but in all other Sciences.

2nd [book of] Physics, Chapter 6.