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This is a hand-edited transcription produced from Scans by Bayerische Staatsbibliothek using OCRmyPDF and translated with GPT4 via ChatGPT

CHAPTER ONE.

SPECIFIC KNOWLEDGE OF THIS ART, and what arguments it uses

The Philosopher defines that Art is the habit of acting with reason on the truth. And conversely, art is also to act on the false with reason (though apparent) concerning that in which a diverse thing is possible. An example in this Art of wielding the Sword, which is the habit of acting on the truth with certain reason; and it is also a habit of acting on the false with apparent reason, as demonstrated in this Book.

From this, it follows that the habit and determination in any Art is due to the collection of its precepts, as defined by Pedro Gregorio. Although the precepts and rules are often considered universal rather than individual, as in Medicine, where the subject named Juan or Pedro is not debated, but the human body, because science does not apply to all particulars individually, as individuality cannot be the principle, according to the Philosopher. Therefore, Art is knowledge based on universals, and experience is of particulars, as taught by the same Philosopher. Adding that all Arts and Sciences, not only in terms of parts but also in terms of genres, exist in something perfect, where what pertains to the genre is sufficient; because in individual Arts and Sciences there are certain principles and general precepts that, according to the subject matter, are precisely distributed (like a calculation) into parts. And subjects are limits in which one transcends from one Art or Science into another Science or Art, like from Arithmetic, whose subject is number, one transcends to the Heavens due to Astrological considerations, and from the passions of Geometry to bodily magnitudes. For this Art, Geometry and Arithmetic are accepted as subordinate, which are the basis for understanding continuous and discrete quantity, deriving precepts to understand and demonstrate the regime of the Sword, in defense and offense, using this Art to transcend into others, and others into it. Because, as the Philosopher said: All Arts in common have among them a certain bond and correspondence, in which they participate from one another.

Cicero noted to the same end that this is a unique cause, turned by nature in all men: for the most rustic, if any difficulty is proposed, even if it is of an Art they do not know, the mind immediately seeks reason through the principles of common knowledge, necessary for reasoning, even if confused; and more so in the rough, lacking knowledge and principles of Art and Science. And yet, through the exercise of the spirit in experience (though without order), it prepares the understanding in its own way to learn a concept in which it finds peace, and until it achieves it, it does not rest; and from such origins, Arts have been produced, not so much by mere reasoning, but by the ordering into rules and precepts.

Pedro Gregorio used the example of gold, which is purified and extracted from mineral impurities, acquiring its purity and splendor through the techniques that refine it to its finest form; even if these techniques do not add to its essence, at least through artistic purification, it becomes smooth and radiant. The human understanding, clouded by ignorance, if it is cleansed and purified through acquired Sciences and Arts, returns to the lights granted to it by its Creator, being reduced to the intellectual purity that, cultivated by precepts and rules (which constitute Art), is perfected for the management of that instrument dedicated to one’s own defense and, when necessary, lawfully to the offense of the opponent.

The reasoning derives from what the Philosopher considered, noting that all Sciences and Arts have a specific kind of Being, principles, and causes of the subject: for example, Medicine is concerned with health, Geometry with the principles of magnitude, Point, Line, and Surface, and Arithmetic with the principles and causes of number, etc. From this, the same Philosopher concluded that all Beings can be reduced to a common one. Thus, Galen reasoned, proving that there are Sciences and Arts that presuppose one another.

The above is confirmed in this Art of governing the Sword, which, as has been said, is a habit of acting the true by certain reason, and the false by apparent reason. This is characterized and achieved by a collection of rules and precepts. Even if for this purpose it leans on the universals of other Arts, they are not applied as if foreign but as intrinsic to this one: acknowledging the stated maxim, that Arts and Sciences have a certain bond and correspondence among them, participating in each other, especially in ways of proving and demonstrating. Through these means, obstacles and errors of the spirit are eliminated, leaving the understanding free in its purity, refined like gold by the precepts and rules that constitute Art: reducing in this Sword Art the Natural, Reasoned, and Real Beings to the Mathematical Being; achieving in propositions the demonstration by mathematical arguments, whose disciplines in the speculative regard consider things abstracted from all sensible matter, treating the Mathematical Being metaphysically. As for the practical aspect, it moves to the Physical, where the entity and the reason for the sensible matter are conjoined, as Proclus taught. For this reason, the Arts that proceed by the precepts and proofs of mathematical disciplines are ennobled above other Arts and Sciences, due to the clarity of the demonstrations, which, excluding all that is dubious, calm the understanding with the infallibility of the conclusion, as achieved in this Art through mathematical arguments, used in it, which are Problem, Theorem, Lemma, and Corollary. These are necessarily (as essential) explained here, because from them arises the certainty of the proposition, both in mathematical disciplines and in this Art, where it is argued, concluded, and demonstrated by such terms, understanding that the main ones are Problem and Theorem, and from them arise the less prominent ones, which are Lemma and Corollary.

A Problem, according to Clavius and others, in terms of a mathematical argument, is that demonstration by which the establishment of the proposition one wishes to demonstrate is achieved, with a certain quality, such that based on a chosen principle, different figures can be established, like constructing an equilateral triangle on a given straight line, as Euclid showed, or another of a different kind. Due to the ambiguity it allows in constructing figures, triangles, or quadrilaterals, or others that differ, it is named Problem, analogous to questions, which Dialecticians, for the same reason of ambiguity, call Problematic, because they admit probability from one side or the other of the question. Thus, in Mathematics, the Problem is the one that, based on a given principle, achieves the proof of various demonstrations, not ambiguous, as is allowed in Dialectics, but with the clarity that the mathematical argument requires. Therefore, the Dialectical Problem and the Mathematical Problem are of distinct species, as seen in the Philosopher, and in Euclid in the 14 Problems he demonstrated in his first Book. The same observation can be made in others and in their expositions, recognizing that the Mathematical Problem differs from the Dialectical in the evident certainty with which, without ambiguity, it infallibly proves the demonstration of the proposed question, which the Mathematician calls Proposition.

Mathematicians call a Theorem that argument which considers in its demonstration some quality, or property, qualities, or properties of its own constitution in its expressed formality. For this reason, the term Theorem means contemplation or speculation, as deduced by Cicero (Tulio) when discussing Fate, and in mathematical terms, as noted by Clavius, and as seen in Euclid in the 34 Theorems of the first book, which consist of as many theorematic propositions. The most famous of these is the one he placed in proposition 47, which in order of theorems is number 33. The arguments Problem and Theorem differ in that the Problem proves the proposition, according to how it is constituted, showing clearly how it is made; and the Theorem does not teach to form any constitution, but to investigate, through mathematical contemplation, the qualities of the figure in its form, demonstrating in what is evident or deceptive the proposition, according to its formal qualities. As Clavius noted, if it were proposed in the form of a Problem that straight lines were to be drawn from the ends of a semicircle to its circumference meeting at a point on the periphery, forming a right angle or right angles, such a proposition would be ridiculous, because it is not problematic, but theorematic: as all angles that are created in the semicircle, with lines drawn from their ends to the circumference, are necessarily right, as proven and demonstrated by Euclid. The same is recognized in the distinction between Problem and Theorem; otherwise, it would suggest ignorance of Geometry and Arithmetic and the types of their main arguments, Problem and Theorem. Although both, as the highest genus, are included in the term Proposition, just as the term animal includes both man and beast. Therefore, mathematicians form distinct the terms of the conclusion of the Problem and Theorem because in the Problem it concludes by saying: Quod faciendum erat, which corresponds in Spanish to, this is what was proposed to demonstrate. Although signified by different terms, the purpose of the arguments Problem and Theorem is the same, which, by different means, are reduced to achieve infallible and evident demonstration, which is the difference by which Mathematical Syllogisms are distinguished from Dialectical ones.

From these two primary arguments, Problem and Theorem, (as not always, nor in all propositions is there necessarily such formality), arise two other mathematical arguments, less prominent, but also demonstrative and evident. They are chosen so that what has been demonstrated can be more easily grasped and understood, deriving in a mathematical way another infallible final syllogism in more concise terms. Of this kind is the argument called Lemma, which, as a derivative, is used for other demonstrations, not so much as primary, but for some specificity derived from the foundational arguments Problem and Theorem. Because of the Lemma, it is said to be a construction for the demonstration of some Theorem or Problem that was foundational in the demonstration, facilitating its understanding through the mathematical syllogistic argument named Lemma, so that it is clearer, more concise, and easier to grasp. Hence, Cicero, by the word Lemma, interpreted it as Assumption, which is the same as something taken from another.

The fourth mathematical argument, widely used, is what they call Corollary, whose term is more translative than proper. Going back to its etymology, Marcus Varro, Suetonius Tranquillus, and Caelius Rhodiginus provided various interpretations, most of which are unrelated to the role of the Corollary as a mathematical argument. In this context, it is understood as a complement that tightens and completes the main argument and demonstration, whether it be a Problem or a Theorem, drawing infallible consequences from the primary argument with which the proposition is adorned and completed. Thus, Caelius, citing Pliny, said that a Corollary is the same as a supernumerary, alluding to Varro’s notion that a Corollary is what is added as a complement or capstone. Mathematicians use the argument Corollary with various phrases, saying: Ex hac propositione constat, Sequitur etiam, Ex hac propositione pari ratione, Manifestum est, Constat enim, Ex his perspicuum quoque est, Ex hac propositione colligitur, &c. and in other ways, all of which mean that the Corollary is a mathematical consequence that is added, encircles, or caps, or is supernumerary to the foundational proposition Problem or Theorem, following the nature of the main argument and demonstration, depending on the proposition to complete it and surround it with evidence. In any manner, if Corollary derives from Corolis, the curious can refer to the cited authors and the expositors of Euclid and to Apollonius of Perga. It’s worth noting that even though the learned Father Christoval Clavius of Bamberg, in the prolegomena he wrote for the exposition of the Elements of Euclid, frequently used the Corollary, he omitted its explanation, having delved into the arguments Problem, Theorem, and Lemma (a noteworthy omission). Here, it has been necessarily touched upon because in this Art of wielding the Sword, commonly called Destreza, the method of proof by mathematical arguments is pursued, wherein demonstrative evidence is found, making use of mathematical disciplines as arts that are subordinate to this one.

To further facilitate understanding, I will place in the second chapter of this book the definitions of the Method given by ancient authors. Following this, I will continue with the other materials that make up the Destreza, aligning myself with little variation in the names and terms with those that are accepted as good. This is because my intention is not to complicate, but to assist as much as possible, in order to simplify the fundamentals and true understanding of this science, so that its operations are so well-regulated and adjusted that practitioners can enjoy in them the fullness of perfections that are possible within the potential of man.