First.

and universally: because they are such that Man, by nature, once he has understood the meaning of the words they contain, is immediately aided by the intellect and clearly knows their truth. As for example: when it is said that the Whole is greater than the Part, once the meaning of these two words, Whole and Part, is known, then one immediately knows, without any other aid, that such a position is true: so that anyone who would wish to make one believe otherwise would not be given credit for his words, and would be reputed a fool. Clav. In vain, truly, would he labor who wished to persuade me of the contrary. Gios. This principle is called Common, because it serves in different Sciences. Whence in Geometry the Geometer applies it to measurable quantities; in Arithmetic the Arithmetician accommodates it to numerable quantities; and in Music the Musician makes use of it and adapts it to Intervals, or to sonic Quantities. These common principles, or propositions, are called Dignities original: "Dignità" — in this context, axioms, and because of the excellent knowledge they hold, they are supposed as the most well-known principles and the principals of every Science. But the other principles and other proposals are of a different nature, because some of them are found which, even if they are not naturally known by the one who wishes to learn a Science, nonetheless it is necessary that he concede them as true and not seek any other demonstration of them in such a Science, as there is nothing more known therein that stands above them. And these such propositions are of two kinds. The one is that which, in affirming or denying some thing, is called a Definition: which declares many terms necessary to that Science. And such Definitions are accepted as true without any other proof; just as, when dealing in Astrology with Spheres, circles, and other such terms, before all things it is supposed that the nature of the celestial Circle consists in the circular figure, comprised by a single line that has the point in the middle, from which lines drawn to its circumference are all equal. Desl. The same can also be said of the Sphere, and of every other term necessary in such a Science. Gios. It is so. Whence the Astrologers, by means of such definitions, prove the properties of celestial bodies, which are truly their Subject. Fran. This same thing could also be said in Music of the sonorous Bodies which contain the interval, such as Strings: for by dividing or measuring a straight line, placed in place of a string stretched over a space, it is a very accommodated means for the Musician to prove the conclusions of his Subject. Gios. You understand it very well. Therefore, we will move on to speak of the other kinds of propositions which are Dignities; and they will be when in a Science some things are supposed which contain within themselves affirmation or negation, and they ought to be called and esteemed as propositions. And even if they are not manifest by their nature, they must, however, be supposed as known in the Sciences. And these are of two manners. Clav. Do not fail, I pray you, to provide the examples. Gios. I shall do so: listen to me. The first manner is when the one who has to learn that Science, hearing such proposals, assents to them easily, not being previously inclined by himself more to accept them as true than to deny them as false. And to give you an example: If I were to say that the one who wishes to learn and understand Music has to suppose that all Intervals of the Diapason octave are equal in proportion, and you, hearing this, believed it, for not having beforehand an opinion yourselves that they are more equal than unequal, these Positions are called Suppositions. Adri. I understand perfectly, so you may continue. Gios. The second manner of these Positions are those where, on the contrary, the one who has to learn the Science, hearing the Positions that are proposed to him to believe, assents to them because he is told that it must be done so, but not because he knows it or it seems to him that it is so, having indeed previously held the contrary by himself. And to come to an example, I will say: If to you, who desire to learn the things of Music, it were proposed that one has to suppose that the Unison single note is that which does not have any interval, at which Position you might perhaps marvel, as it would seem strange to you, if you did not have knowledge of this Science, that one could find something that is not dissonant and that does not have an interval. Adri. Therefore, having understood everything well, we can say that all Positions, Dignities, Definitions, Suppositions, and Petitions or Demands as well, must be esteemed as principles of that Science in which they are placed. Gios. That is truly so: but note also that even if any of the named Petitions and Suppositions could be demonstrated in another Science, nevertheless