Reasoning

in that faculty in which it is placed as a principle, one could not find any way to demonstrate it; for, wishing to demonstrate it, other principles would be needed, in such a way that such Suppositions and Petitions would end up not being principles. But being taken as known and understood, from them, as from sufficient premises, the conclusions which belong to the said faculty are then demonstrated. Whence if in any particular science, such as Music and Astrology, or in any other, anyone wishes to deny any principle whatsoever, in such a Science it will not be granted to him to dispute, nor will he be fit in any way to learn it. 1. Physics, Chapter 2 ADRI. Up to here I have learned many beautiful things; however, if there is anything else you have to say, I pray you to continue. GIOS. As for this part, I do not want to tell you anything else; but believe me, sir, the good part is just beginning. ADRI. God be praised then; continue, and say what you will, for we are prepared to listen to you. GIOS. I want you to know that every Proposal which is proposed for demonstration can be of two kinds: for either it leads us to Speculation, or it makes us operate. That which leads us to speculation is called a Theorem theoretical proposition; but the other is called a Problem practical task, and this is asked for by such a name because from it we learn the way to divide, compose, describe, draw, and form every quality of surface figure, with all those accidents that can occur in many arts, as in painting, perspective, chorography, cosmography, geography, sculpture, architecture, and other similar arts. Besides this, I want to tell you that every Theorem or Problem, which is completed by its parts, must have within itself six things: the first is the Proposal, which by the Greeks is called Prothesis proposal, in which is contained the Given and the Question, of which two things every perfect Proposal is composed. And the office of this part is to teach what is sought from the Given. The second is called Exposition or Explication of the Given, called Ekthesis exposition, whose office is to receive the Given and prepare it for the Question. CLAV. Tell me, if you please, what each of these two things is. GIOS. I will make you understand with an example. If I were to say: One can place the Tone upon a given string according to its proportion, the given string is truly called the Given, and the placing of the Tone is the Question. CLAV. I understand very well; continue your speech, and pardon me if I disturb you at times. GIOS. Rather, you give me pleasure. But to return to our purpose, I say that the third part is named Diorismos determination, that is, Determination of the Question, whose office is to set aside what the Question is. The fourth is called Construction, called by the Greeks Kataskeue construction, which is that which, to find the Question, adds those things that are missing from the Given. Added to these is the fifth, called Apodeixis demonstration, that is, Demonstration, which scientifically gives us the proposition by means of granted and presupposed things. Lastly, there is the sixth, called Symperasma conclusion, which we can call an Epilogue or Conclusion, which turns again to the proposal, confirming what has been demonstrated. ADRI. Are all these things found in every Theorem or Problem? GIOS. No, sir; but in each one are necessarily found the Proposal, the Demonstration, and the Conclusion; because it is necessary to know first the Question, that is, what is proposed in the question; and then to demonstrate it with the due means; and after it is demonstrated, to conclude it. So that none of these three things can ever be missing. In some places, the others are often used, and in many they are left on the side, as one can see in the 10th of the 4th book of Euclid, which says: We can constitute a Triangle of two equal sides that has at one and the other of the angles at the base double the other angles: where the Determination and the Exposition are missing, especially when the Explication of the Given is sufficient, in such a way that no other addition is needed to demonstrate what is proposed. But the Construction is often not found in many Theorems. And when in the Proposal there will be no Given, then the Exposition will be missing. But the proposal will most of the time have the Given and the Question, though not always; for at times it will have only the Question, which needs to be known, or to be done, or reduced to effect, just as in the said Problem or Proposal it is seen; for it is not said: From what Given does one need to constitute the Triangle of two equal sides-