As for the proportional parts, which correspond to each other in the human body, a multitude of authors deal with them, from whose proportions those of Architecture are drawn, as can be seen in Vitruvius, Viñola, and others, in which the curious will find an expanded field. Contenting ourselves here with what is most precise to this Science, reserving for the second Book the lines, sections, planes, and angles, in which man is considered divided, for the understanding, and exercise of what the common people call Skill.
Having explained the first Predicate Substance, which produces man (the subject of the Science of the Sword) by a straight line, we properly move on to the second category, the first accidental part, where the division of the real being is admitted, according to the common doctrine of Philosophers and commentators of Aristotle, Ancient and Modern: although there are those among these who, by specifying the term, want it to be considered, not the real being, but the knowable being, because this admits more latitude, intending to apply to the being of reason, with certain circumstances, that are not so common to Logicians, who understood the division of the real being, according to the ten Predicaments, in which the Philosopher distributes it, whom we follow in the most received, and most of our intention.
Quantity, or How much, is placed, as Aristotle taught, in the second place, or Category, where it is admitted as a supreme Genus, for two reasons: the first, because Quantity (whose term we use as more common) is the most proper and similar to the substance, as it does not admit contraries, nor more, nor less in its species: the second, because Quantity is the subject of other accidents, as they cannot subsist without it, because it is first, that there is an interceding or middle subject, so that other accidents subsist in the substance, or in the same accidents: these, as the Metaphysicists and Theologians debate, can be without the same essential substance, but not without Quantity, which is the interceding subject for the substance, with respect to other accidents.
We define, then, Quantity, or How much, saying: It is that which is divided into parts of Succession, Extension, and Description: in parts of Succession, like time, which is divided into parts that are causes of continuous movement, or are causes of discrete movement. Of continuous movement, like the cause in past, present, and future time: of discrete movement, as we will explain in its place, following the Philosopher: As far as the same time is considered successive, although it is divided by the past, present, and future, it does not cease to continue in the succession, because its parts are conjunctive, and what is conjunctive is successive: then, time is successive, and a proper species of continuous Quantity, which is understood to mean, that they are parts of the succession.
In parts of Extension, it is understood, as in the line, which admits extension in conjunctive lines, produced by the first petition of Mathematicians; since, although the line lacks breadth, which is not given in it, because it lacks wide (or width, as the common people say) extension, at least extension is given in the production of length without breadth.
As with the line A.B., which is considered to be 4 and produced to 8, it will be extended by another 4, and its common extreme terms will be points that end at 8, and consequently the conjunction of 4 on the line extended by another 4, if given movement, will be in the extension of 8, as shown in Figure 1, which is in the Diagram of Quantity, or How much.
Thus, a man with a sword in his hand, whose arm is considered as two units in conjunction with the sword, considered as 4 units, will have an extension of 6 units. Consequently, he will be able to perform an extended length motion or movement of 6 units. In this consideration, the Quantity will extend from 2 to 6 units because the arm is considered as 2 units and the sword as 4 units, and the entire movement of the arm and sword has an extension of 6 units. This is demonstrated in Figure 2 of the said Diagram.
En Deſcripcion conſideraſe la Quantidad, como numeral, donde el Todo ſe divide en partes diſcretas, y diſiuntas, como el binario en dos vnidades, el quaternario en dos binarios, &c. eſto baſte quanto à la primera difinicion. Otros, coligiendola de las palabras del Philoſopho, dizen: Quantidad es aquello, que mira à la coſa en quanto; para cuya explicacion, los Modernos ſe valen, ſuponiendo, que qualquiera corporea entidad, padece en sì afecciones, que no miran à la miſma inmediata entidad, ni ſe producen de ſu effencia.
Aristotle explicitly defines it as follows: How much is divisible in that in which it subsists. According to this definition, which is relevant to our intention, it can be said that How much, or Quantity, is that which admits division into parts, of which it is composed, in such a way that any of them can persist on its own and be a something, or How much, whose division is considered in potentiality in terms of continuous Quantity, and in act in terms of the discrete.
From everything, following the Philosopher, it can be deduced that Quantity is the numerosity of parts of the same substance, from which arises (in terms of the physical) diverse extension in the position of the subject, such as in the impenetrable figure, hardness; in the light one, lightness; in the linear, length without breadth; in the surface, width without depth; and in the Body, all dimensions combined, etc. All these attributes are attributed to Quantity.
Everyone admits Quantity as the supreme Genre in its Category, although it can be challenging to understand how essence applies to our intent; thus, we grant the most general Genre, Quantity, in its Predicament. And it will be said that in the distribution into predicates; since these, being Species of the supreme Genre, will be genres of those species that are its predicates. For example: Line is a species of the supreme Genre Quantity, and is the Genre of its predicates, like straight, curved, spiral, etc.
From the aforementioned, it can be understood for this Science, what the predicates of this Category are; to whose demonstration it could be said:
Quantity is the supreme Genre, to which entities compete for the reason of measurability, from whose affections, or passions, the predicates that are species result, according to the knowledge with which we investigate the affections of the same things, which we call accidents.
This description can be a thesis of this Predicament, whose terms in abstract are measurability in potentiality; and in concrete, it is measurable in act; because just as the essence, if perceived quidditatively, in it questions are resolved by the term What: so too in this Predicament, questions made by Quantity, or How Much, are satisfied by measurability, or measurable, divided into continuous, and discrete, as Aristotle taught, and from the affection of these terms results the equality, the inequality, the majority, the minority, etc. and for our intent we consider, That Quantity is nothing else but the number of parts understood by extension, to perceive their measures and ranges by continuous Quantity, or by discrete Quantity, in which following the Philosopher, the Category, or Schema, is formed as it is at the end of the explanation of discrete Quantity. Thus, without a doubt, it will be recognized by the table, or Schema, that the term Quantity, or How Much; is the supreme Genre in this Predicament, and is distributed into continuous and discrete Quantity.
Five main species belong to the continuous, which are, Line, Surface, Body, Place, and Time: and these are considered as Genres of their lowest Species: the Number, and the Speech belong to the discrete Quantity, which are also considered Genera, in relation to their Species, up to the most inferior, as will be explained, and reduced to the most proper of the Science of the Sword. The principles for the demonstration in all Sciences are definitions, distinctions, and immediate propositions, these in Science, as in Mathematics (which it uses as subalterns) are reduced to problems, or theorems: those that demonstrate on a given principle, that can be formed: and these, which investigate and recognize the qualities, and passions of the proposition already formed.
From the immediate propositions, other proper principles result in this Science, as in others, which as Bradwardine noted, can be called Maxims. From these and other principles, the common ones are produced, which the Philosophers call Suppositions, and the Mathematicians Petitions, to whom Bradwardine gave the name of Complex Principles, and we need all of these in this Predicament, because it includes the species of continuous and discrete Quantity.
The geometric terms that serve this Science, are either practical or speculative: if practical, they define the Species by Genus and Difference, which looks at the demonstration, as is seen in Euclid’s Elements, thus it will be said: Point is that which has no parts, line, length without width, etc.
If in the speculative, Bradwardine notes well, that the Definitions are not so much by Genus and Difference, but in another way, that looks primarily at the essence, like I assume Genus; e.g. The Point is the beginning of magnitude, or Quantity; the Line is the trace that limits the clear and obscure, constituting a single dimension. Surface is that which admits two dimensions; Body is that which admits three.
In this Science, in terms of definitions, sometimes they are formed as speculative, other times as practical, always trying to ensure they are essential, as noted in their proper place, where we discussed the precepts, with which we enter more clearly into the explanation of this Predicament, whose first species of continuous Quantity is the Line, considered as the Genus of its Species.
Understood that the Point lacks parts, and that without them, although it is the beginning of Quantity, it does not constitute it (from which it follows that it is the end of that species, which can be terminated in points, as will be any dimension, or Quantity, that does not admit width), we will come to know the first dimension, and this is commonly called Line, which we consider with Mathematicians As length, which does not have width, whose ends are points, leaving to Philosophers the dispute, whether the line is composed of points and parts; and how does it consist, or can it consist of indivisibles? and the other questions in which they are vigilant.
We carry, then, that the Line is caused, and understood by the trace of the movement of some physical or metaphysical point, ending in points, and this dimension is the one that is considered as the first species of continuous quantity, and is Genus of its species, which are relevant to this Science are the Straight, the Curve, the Circular, and the Sinuous, and the others, that will be noted in their proper places. Note that because in the following Book we put all the geometric definitions that pertain to this Science, with the application that is made of them for the practicality of Dexterity, we omit here to demonstrate the figures of this continuous Quantity.
A straight line is the shortest extension between two points at which it ends. A curved line is one that forms an arc, terminating at two points, not following the shortest extension, but rather a more deviated one; thus it can be more or less curved, depending on how much it expands in the deviation of its ends.
A circular line is one that, equidistant to a central point, is considered to be caused by the regular movement of a point, which, if it ends at another point before returning to the same one, is called a portion of a circle; if it ends halfway through its course, it will be half a circle; if it ends at the quarter point, it will be said to be a quarter circle, from which, without cutting off the end, there are wounds that are called quarter circle, half circle, or a larger portion of a circle, according to how they are considered by the movement described with some point, like the tip of the Sword, etc.
The spiral line is that which, in increase or decrease, is composed, forming a coil or spiral, a term used for the movement described by the center of the Sun in the sphere, relative to its course, with which it deviates, declines, and retreats from one Tropic to another, its maximum increase being at the Equinox, and its maximum decrease at the Tropic; and for our purpose, a spiral line will be considered on the central point, and in a spiral it will increase to another external point; and conversely, in decrease, starting from the external point, and ending at the central or interior point: this line is very typical in the Science of the Sword, due to the movements of its tip in the execution of some wounds, or Tricks, as will be demonstrated in their proper places, where it will be noted in what modes, and species it is of continuous Quantity.
A perpendicular or vertical line is considered to be one that, with respect to the opposite, or above another straight line from a point of elevation, falls directly onto the plane of the horizon, or onto a parallel to the same plane, in which it will form a right angle if it ends on the parallel, or the horizontal plane; and similar lines are named Vertical lines after the term Vertex, point of elevation, above the vertex of the opposite, and from this principle the terms Vertical Cut, etc. arise, as will be noted in their proper places.
A diagonal line is that transverse one, relative to the plane of the horizontal, caused by the point that moves along a transverse course, with respect to some Parallel, which is imagined Parallel to the Horizontal, from which the terms Diagonal Cut, etc. are produced. And these Horizontals, and Diagonals, and their close and distant species, are typical of the movement that in this Science is called Reduction, or Retreat, as far as the Horizontals, and in the Transverse, Diagonal; as will be demonstrated in their places, when it comes to action, and passion, and movement, and stillness.
I omit many other types of lines, because they do not belong to this Science, but to Geometry, where all their differences, like the Quadratrix, the Conchulis, etc. are recognized. Those lines that are very relevant to our purpose, we pointed out in the Book of Art, adapting them to the understanding of the Art of Fencing. Concluding, parallel lines are those that are equidistant from each other.
A surface is that, whose boundaries are lines in the Plane, and circles in the Spheres; and so, according to their forms, concave or convex, considering latitude without depth: all surfaces admit the properties of continuous Quantity, insofar as they are potentially divisible; and of the discrete, insofar as they are actually divisible.
For its dignity, and capacity, the Circular is placed first, which consists of a line equidistant in all its parts from a point, which is called the Center; and Periphery, the container, which includes the Circular Surface, without ending in points, but in itself. And so, from any Point, or Center, and with any interval, the Periphery of the Circle is described, which includes the flat Circular Surface at the same distance, with respect to the center, from which, any straight lines drawn to the termination of the Surface, will all be equal among themselves, by the definition of the circle, and the terminating line will include itself, without giving other points that terminate it: from this consideration, the Surface is caused, which causes the Ellipse, which is that Line, which moving a point, until its beginning ends, is not regular in its equidistance of all its parts to its central point, but rather it describes a quasi oval.
This definition encompasses not only the figure but also the Surface, which is contained by its boundaries, in respect to its latitude, not in respect to depth, which does not pertain to the Surface, hence Clavius pointed out that not because every Quantity has boundaries, it will be a figure, but rather that, whose common boundaries coupled, include the figure or Surface they end; and from here it follows, that that Surface, which is enclosed, and ended by its common boundaries, or coupled among themselves, will be a figure, and consequently a species of continuous Quantity; and if it is not composed of these parts, it will neither comprehend nor form a figure, nor a Surface of the species of continuous Quantity, but of the discrete, whose division will necessarily be in act, more than in potential.
Understood thus, figures, and Surfaces are divided into various species, and from these are proximate Genres their immediate ones, as a Plane figure is a Genre of its species, which all are Planes; Solid of the Solids, Concave of the Concaves, Convex of the Convexes, and in all it does not go beyond latitude without depth, investigating their Superficial forms by their figures.
A Plane figure is ended by lines that include it, and consequently the Surface: so it will be the same to say figure, as Surface, and so we will use one of the two terms. Superficial Concave, or Convex figures are divided, in respect to interior, or exterior: the interior is named Concave, because it does not go on to be a Solid figure: Convex is named the outer part of the supreme Surface. Some say that Concave is the one that has the body outward, and Convex the one that has it inward.
Figures without angles are those that are enclosed by a Surface, or Line, such as the Circle, the Ellipse, the Lenticular, and their like. Angular, or Polygonal figures, are those that ended by lines, consist of angles. Planar Rectilinear figures take their names from the sides, or angles, or by both, as taught by Euclid, and his Commentators.
Trilateral figures are subdivided into different species by the quality, and value of the angles. In the Right, or Rectangular Triangle (whatever the species may be) its three angles do not admit more or less value than two right ones; the cause touches speculative Geometry, because the two Acute angles are equivalent to the Right one, and the Right triangle can have the quality of Isosceles or Scalene, but not of other species, which it does not admit. Obtuse is that triangle, which consists of an angle greater than the right, and can also be Isosceles, and Scalene. Acute is that triangle, which is constituted of all three acute angles, which are less than right angles.
Triangles are subdivided into other species, whose names they take from the quality, and affection of the lines, of which they are composed, and formed. Rectilinear are those that are enclosed by straight lines, Curvilinear those that by Curved lines. Spherical are those that by portions of circles, which admit location in the Spherical Surface. Mixed are those that are not of one species of lines, but different, are composed, and in all qualities and names are considered, which results from them by the value of angles, and sides, and every Triangle is of the continuous Quantity, because it is included by the coupled terms.
Likewise, those with three sides are named Trilaterals, thus those with four are named by the number of their terms, or sides, Quadrilaterals: admitting this Genus different Species, which commonly are reduced to five, which are principal; the four regular, and the fifth subdivides into other smallest Species, which are included in its terms.
Rhombus is the third regular quadrilateral species, consists of all its equal angles, not right ones; although the opposite ones are similar, because two are obtuse, and the other two are acute. Rhomboid is the fourth regular quadrilateral figure, it does not admit right angles, nor are its sides all Equilateral, although the opposite ones are, as they are similar to each other. And so, from these species are the Rhombic and the Similar, that to the others pronounce them Rhombic and Similar Rhombic.
Pentagon is a figure, that is constituted of five sides, that if they couple at five points, where forming angles, if the sides are equal, will be equal and the figure will be a Regular Pentagon. If the sides are unequal, and the angles, it will be irregular, and consequently a Trapezoid. Hexagon consists of six sides, and equal angles, being regular, and the contrary. So for more sides, like the Heptagon, Octagon, etc. which Euclid widely demonstrates.
Compound figures (calls Thomas Bradwardine) are those that are composed of ingressive and egressive angles, as will be demonstrated. Generically simple figures are all those that constitute simple Polygons. Of these, some are regular, and others irregular; of the regular ones, some fill place, and others do not; the irregular ones do not fill place. The regular ones consist of sides, and equal ingressive angles, and for that reason they are named Equilateral and Equiangular.
To fill, or fill a place (as Bradwardine himself points out) is to occupy the space surrounding a given point in a plane: only three regular figures are considered to be capable of this quality, which are, the Triangle, the Square, and the Hexagon. None of the others admit such a circumstance: the speculative reason is, because it is found in the regular Triangle, in the Square, and in the Hexagon, such order, and disposition, that coupled, they fill the place surrounding the given point in a plane.
The affirmative part is proven, because it results from the number and position of the value of its right angles: Bradwardine himself demonstrates this, saying: If four similar squares are coupled at one point, they occupy the surrounding place, because they compose a square, whose central point will be placed in its circumference. In the Triangle, when six are coupled to the given point, they also fill the circumference; and the reason is, because the Triangle is worth two right angles, and the Square four, which is double the Triangle.
The case for the Hexagon is proven because the six angles of it are equal to eight right angles, according to the common demonstration (repeated by Bradwardine). It is known that in any regular figure, given one central interior point, if lines are drawn from it to the angles of the Polygon, doubling them, and subtracting four from the product, the value of right angles remains: therefore, if three Hexagons are coupled at the given point, they fill the surrounding place, because in the Hexagon there are eight right angles, and three are worth four right angles.
And so from the Equilateral Triangle results a similar consequence, because the angle of the Hexagon is double that of the regular Triangle; and it is demonstrated, because three angles of the Hexagon are worth double the Triangle, whose three angles are worth two right angles: then their double four right angles: and by the final demonstrable consequence, in such a double, six Triangles are required for the filling of the place, which the Hexagon fills; and so it turns out, that three Hexagons fill the surrounding place to the given point in a plane, and six Triangles also. It is also confirmed, because the three angles of the Triangle are worth two right angles: then six are worth four, and so they fill the surrounding place to the given point.
The conclusion, or lemma, is therefore rightly said, that fill the surrounding place six Triangles, four Tetragons, and three regular Hexagons. All can be demonstrated by different problems. In a given point in a plane, its circumference is occupied, and filled, drawing a straight line, on whose basis four Tetragons are formed, which coupled, occupy, and fill the circumference of the given point at the distance of the Line. And not making a delineation of this figure, and others, here, is, to not increase the volume, and that for skilled Mathematicians, it is a prolix matter.
Chosen as the center the given point in a plane, coupled six Equilateral Triangles, their bases will form a regular Hexagon, as facilitated in the common problem to any compass opening on the point, or center, forms a whole Periphery, and with the same compass opening it is divided into six arcs, or points, and from one to another straight lines are drawn, the chords of such arcs, and from the angles to the center; the straight lines, or semidiameters, result in six coupled Triangles, that occupy, and fill the surrounding place to the given point.
If three regular hexagons are joined at a point, they fill and occupy the surrounding space around the given point. When these three hexagons are further extended by joining three more, as seen in the problems resulting from this theorem, they form connected hexagons. It is easy for anyone with a non-vulgar understanding of speculative geometry and continuous quantity to acknowledge that these connected hexagons fill the space around the common point of connection. As demonstrated, all these figures and speculations are of continuous quantity in relation to their entirety.
It’s proven that only three shapes, the Triangle, Square, and Hexagon, can fill the space surrounding a point on a plane. However, it remains to be demonstrated the negative part, that no other regular shape is capable of filling space. This is shown (as demonstrated by Bradwardine himself) because each shape has larger angles than its predecessor, as is evident by the common proposition of the knowledge of right angles that each figure holds. Every subsequent figure adds, but doesn’t diminish, because no angle can be worth two right angles, as will be demonstrated in discrete quantity and in regular shapes such as those proposed. Every angle of the subsequent shape is larger than any angle of the preceding shape, a common principle in geometry. Therefore, it’s a clear conclusion that no figure, after the hexagon, is capable of filling the space surrounding a point on a plane. Because if three angles of the polygon, from the regular shape after the hexagon, are superfluous, and no two angles of them will fill space, just like two straight lines don’t fill a surface. No matter how large an angle is given, it’s worth two right angles: therefore, neither two angles can be worth four right angles by the definition of a plane angle. Thus, the heptagon is not capable of filling the surrounding space around a given point on a plane.
The same reasoning applies to the Pentagon, it also doesn’t fill space because its three angles are not worth four right angles, they are not equal to those of the Hexagon, and four angles of the Pentagon have the Square as their preceding figure in the order of regular shapes. Hence, the intent is demonstrated, which is necessary in the Science of the Sword, which always considers a given point on the plane of the combatants. From this point, the consideration passes to find which shapes are capable of filling surrounding space to understand distances and form stratagems based on proportionate means, which often result from the knowledge of geometric regular shapes that, when connected without leaving a void, fill the surrounding space around a given point on a plane. This consideration belongs to the realm of continuous quantity.
This speculation has been explored by few, for only Campanus (among the many commentators on Euclid, who needed the Pentagon) pondered on the extension of its sides. Consequently, he found the Pentagonal figure, composed of outgoing angles, created in regular proportion by extending the sides to their point of convergence, which generates a continuous straight line, dividing into five, forming five outgoing angles, opposite to the inherent sides.
From this principle, Thomas Bradwardine advanced the speculation in the following three conclusions, reducing the composition of such figures to three orders, which we repeat, as it is one of the most significant principles in the Science of the Sword. This is due to the speculation of the outgoing angles, which, with respect to figures, can provide understanding through continuous quantity. By extending the sides of regular figures, not only is the knowledge of the inherent angles that they are composed of in their formal root obtained, but also of the outgoing ones. This is where the proportions of the extensions are provided, a concept that motivated Thomas Bradwardine (whom I follow), as he was the first and only person to expressly speculate on figures that, through the extension of their sides, accept outgoing angles.
The first of these types of figures is the Pentagon. It is demonstrated that the Triangle is not a figure capable of such quality and order with respect to the extension of its sides, even though it is, with respect to the compound combination of Triangle to Triangle. But simply extending its sides, they not only do not converge at a common angular point, but the extended lines deviate, forming external angles over the internal ones. Therefore, the Triangle is not a figure of this type.
Nor is the Tetragon such a figure, because its opposite sides are parallel lines. Parallel lines are always equidistant from each other, as demonstrated by Euclid, so their extension will never form an angle. Therefore, the Tetragon is incapable of producing a figure with outgoing angles. It is thus evident that neither the Triangle nor the Tetragon, the figures preceding the Pentagon, are capable of simply forming, through the extension of their sides, a figure of outgoing angles, which is the proposition at hand.
The Pentagon is the first figure which, simply by the extension of its sides, can form a figure of outgoing angles. This is demonstrated and exemplified clearly, as its extended sides will converge at a given point, forming an outgoing angle. Thus, the other sides will converge with each other, forming outgoing angles, and consequently, a single movement to different endpoints creates the entire figure. For, from one endpoint to another, a figure of outgoing angles opposite to the incoming angles is formed by continuous quantity. With this, this conclusion is demonstrated, upon which we could expand with various speculations that serve our purpose, leaving an open field for the discourses typical of this Science; and generally, we refer to Bradwardine. Here, for more clarity, we will recognize different orders in figures of outgoing angles, both those that are capable of being produced from continuous quantity, as well as those that have their composition from discrete quantity. This refers to which figures (when their sides are extended) form outgoing angles, as in the proposed Pentagon; and which figures require two or more regular figures to produce a composite one, which is formed of outgoing angles. An example would be the Hexagon, which is not capable of forming a figure of outgoing angles by extending its sides, because it has opposite sides consisting of parallel lines; and therefore, no matter how much they are extended, they will always be parallel by their very definition. With this, it is demonstrated by the preceding, that the Triangle, the Tetragon, and the Hexagon are not capable figures of outgoing angles, and consequently the Pentagon is the first simple figure of such quality, and for this reason it is referred to as a first order figure in continuous quantity.
In summarizing this topic of figures capable of outgoing angles, by extending their sides to points of convergence, it is demonstrated that the Pentagon is considered in the first order, because it is the first regular figure that admits such qualities in itself, without any preceding or subsequent figure that can have them. Consequently, the regular figures of the second order, capable of outgoing angles by extending their sides, are recognized to be the Heptagon, the Octagon, and the Nonagon, because in the second order they admit the qualities of outgoing angles by the extension of their sides in the formality of continuous quantity, like the Pentagon, as demonstrated in the following theorems.
In figures with more sides, continuity is not admitted in their forms, unlike the three proposed, which by continuous movement of lines ending at equidistant points, without lifting the pen, the figure is formed, as demonstrated in the Heptagon, where from the given point the straight line can run from point to point, and in its convergence, they will be outgoing angles.
In the Nonagon, it is demonstrated in the same order, because by extending its lines until they end at the given point, the outgoing angles result over the interior Nonagon, with its sides extended to the referred points of its angular convergence. And for the purpose of the Science of the Sword, the interior positions are recognized in such figures, from whose sides the reaches and angles that the Sword can form are produced, governed by the movements of the arm, feet, and body of the mover; and consequently, in the operations of the opponent. Although accidents in execution vary the regular positions, this does not prevent the knowledgeable and experienced practitioner from recognizing the positions in which they place themselves, and having perceived in the mind the figures of outgoing angles, they make use of them on occasions, as they can avail themselves.
These considerations and demonstrations on the surfaces speculated by continuous quantity seem sufficient, without going into the extensive offer of Geometry, because this discipline is assumed to be subordinate in the Science of the Sword. Thus, as a warning, we will try to stick to the essential and compulsory, leaving the figures of outgoing angles of the third order for the discrete quantity. These result from some perfect regular figures placed in others of the same species, as will be demonstrated in its place.
The grandeur of Bradwardine’s discourse on these figures of outgoing angles is inescapable, leaving in the interior angles such perfect regular figures, like the Pentagon, the Heptagon, the Octagon, and the Nonagon; and that he had not foreseen the strength of reason, and grandeur, that assist these figures. Indeed, in the internal parts of the ingoing angles, they leave these figures formed. I also point out that the same figures are found in the outgoing and exterior angles of said figures that are found in the interiors, as by drawing chords from the ingoing angle to one of the points, and continuing from one to the other, the same interior Pentagon will be found formed in the exterior, and thus the other referred figures of this order, these exterior chords of great consequence for the Science of the Sword, since through them the fencer will give his compasses, serving him as routes, or paths to occupy good places in the outgoing angles, as will be demonstrated in its place.
Euclid gives a generic definition, saying: A solid figure is one that has length, width, depth, or thickness, these are the three dimensions that every body consists of. And for the formation, he adds: The boundary of every solid is the surface or surfaces, made up of the planes or convex bodies, which include the solid figures, according to their species. There are straight and sloping planes, which are considered according to their forms, oppositions, with which they compose different solid figures.
The inclination of one plane to another plane is considered by the acute angle, constituted by inclining straight lines. Plane to plane, similarly inclined (that is, of equal inclination) are those whose angles are equal: that is, of one value. Parallel planes are those which in their entirety are equidistant, without inclining, nor declining with respect to one another. Similar solid figures are those contained by similar planes, equal in their number.
Peleteario, Tartalla, and Clavio understand by similar solid figures, those that consist of an equal number of angles, and planes, proportional to each other. It results thus, that equal, or similar solid figures, are those that are contained by equal, or similar planes in their size, and number. For understanding, it is necessary to know, what solid angles are, noting, that in general they are caused by the concurrence of planar angles, or curved, or mixed. It is thus defined: A solid angle is that, which besides two angular planes, not consisting in a plane, but converging at a point, contain and constitute a solid angle.
If it is a simple Conic solid figure, it is caused, in Euclid’s consideration, by the movement of a right triangle, the perpendicular being the resting line, or in another inclination, placed in very short extension of the Cone to the base, and the Diagonal, or Hypotenuse is considered the movable of the triangle, whose revolution with its trace, leaves formed a solid figure circumfluent conoidal, caused by entire movement, that consists of two planes; one circular by the line of the base of the triangle; and another from the base to the pyramidal Cone, caused by the Hypotenuse.
It should be noted, with the same Euclid, Campano, Peleteario, Clavio, and other Expositors, that if the resting line, and the base were equal, the Cone will be Orthogon, because the perpendicular on the base, the Hypotenuse will form angles of 45 degrees, and in the revolution the angle, terminated at the point of the vertex, is double to that of the base; and so right if the line of the base (which is the one in the whole revolution results semidiameter of the circle, which describes the revolution) were greater than the resting perpendicular; the angle of the pyramid at its vertex will be Amblygon.
And conversely, if the resting line were larger than the revolving one of the base, the angle of the vertex in the pyramid will be Oxigon, the more acute, the more the resting line of the figure exceeds the mobile one of the base. According to the proposal, it can be defined that pyramid is a generic name, which includes all solid figures, which are composed of planes, continuing from the one that is the base to the Vertex point, or Cone, where they all converge.
The species is distinguished and is defined by the plane (figure of its base) as demonstrated by Euclid’s expositors, and noted by Bradwardine, because the base of the pyramid gives the name to the whole pyramidal body, hence, if it is a Triangle, it will be called triangular; if Quadrangular, quadrangular; if Pentagon, pentagonal; if Hexagon of the pyramid, they are necessarily all triangles, because they end in the common angle, which they call solid.
Prism is a solid figure, which is composed of planes, and of them, two are different from the others; although similar, equal, and parallel to each other, as the Pyramid is explained, and named for the terminating base, so the Prism by the number of its sides, which constitute the surface of its body, hence if that solid figure, whose two opposing planes, Parallel, equal, and similar, that constitute its plan, were triangles, they are named triangular Prism; if Quadrangular, quadrangular, etc. as in the pyramid.
Tetrahedron is a solid figure, composed of four triangular surfaces, one is the base, and the other three end in a solid angle, and can be the triangles, that end on each surface, all equilateral, or the base diverse, or all different; and in the same way the following solid figures are considered. Octahedron is a solid figure, contained under eight triangles, or equal and equilateral triangular surfaces. Dodecahedron is a solid figure, contained under twelve equal equilateral and equiangular Pentagons. Icosahedron is a solid figure, contained under twenty equal, and equilateral triangles.
These five bodies are called regular, because all the planes, by which they are contained, can be Equiangular, and Equilateral, as is recognized in their definitions. Philosophers call these bodies Platonic, because the consideration of them, according to their doctrine, is for those five elements that constitute the Universe, Heaven, Fire, Air, Water, and Earth, whose speculations I omit here, as they are not of our intention. The curious should refer to the expositors of Aristotle. Ptolemy in his Great Construction, and those who deal with the division of the Sphere according to substance.
Geometers will find the construction of the five regular Bodies, defined in Euclid, and his expositors, who in various ways have sought to demonstrate the composition of the five regular Bodies, not easy on paper, and easy in wood, or in other material suitable for the formation of their structure, which as it is not of our concern; we could skip them, to exchange demonstrations; and for the same reason we do not engage in the solid Parallelepiped figures (which are those quadrilateral) that are contained by six, or by coupling the parallel surfaces, and forming a solid body, as demonstrated by Euclid. And even less are of this intent the solid figures, which are considered inscribed, and circumscribed one in another, although all the repeated (perfectly solid) ones are of the continuous Quantity.
Complex principles are those that embrace, or comprehend the concept, with such demonstration, that the understanding accepts them without repugnance, as felt by Thomas Bradwardine, dividing them into two genres; one to which is given the name of Petitions; another more common to all sciences, which are understood by Axioms, Pronounced, or common notions: and of both used Euclid, and we use in the Science of the Sword, and we place them in this place, not because they are of this Category, but because we need such principles for the understanding, and forced arguments of it, and the others.
From any given point to any other point, a straight line can be produced; and without an intermediate obstacle, it can be physical: and having one, or not being visible, or actual, but imaginary, it will be metaphysical: and in both ways straight, for being the shortest extension from point to point.
Any given straight line can be produced in continuous augmentation, because in every line its ends are points, and from an ending point, a line can be produced and continued to an ending point, which is the ultimate end of the produced line: and we deny the production into infinity, because it is not human to comprehend, nor of this Science of the Sword.
Given any point as a center, a circle can be described at any interval, according to the chosen radius, or according to that point that ends the interval, to which it is described. This is proven because an end is that which is extreme of something, as Euclid defined, from where it follows, that on any point chosen as center, a circle can be described: in respect, or proportion of any quantity, or size, greater or smaller quantity, or magnitude can be given, whether considered in Line, Surface, or Body, or in number.
From such principles it results, that equal circles are those, whose diameters, and radii among themselves are equal, and consequently, those circles are unequal, which differ in their diameters, or radii. This is proven by common sense, because the circle is a figure contained by a regular circumference around a point, which is the center, from which all straight lines drawn to the Periphery, are among themselves equal; and consequently, the other figures, that do not consist of such circumstances, are not circular, but of other shapes. Thus all the curved lines, which are caused by the movements of the Sword, not being uniform to a given central point, will not be portions of circle, nor whole circle, but of other species, that can be caused by curved lines.
If a straight line falls upon two straight lines, intersecting them, if the two converge at a point, forming an angle, the line falling upon the intersections will constitute figures and angles, which will be complementary to two right angles, upon the angle formed by the convergence of the two given lines.
Two straight lines do not include a surface, nor do they constitute a figure; therefore, the first figure is a triangle. These principles are specific to Geometry and the Science of the Sword: they are commonly named Petitions, or Assumptions, because they are accepted or admitted as complex principles of the first kind, without needing any other proof or demonstration, as they are self-evident, providing sufficient evidence.
Certain common principles are found in philosophers and mathematicians, which embrace the concept, and they name them with different synonymous terms, such as Axioms, Pronouncements, common notions, etc. They consider them as evidences, to which the understanding agrees, without more proof than the intelligible explanation, enough to give to understand the sentence that is pronounced. Here I could compile many of these principles, but I exclude those that are not proper to the intention, reducing the most essential to a small number.
- The whole is equal to its parts, because all parts together do not make up more than one whole.
- Any whole is greater than its parts; this is understood categorematically, but not without syncategorematically.
- Any of the mathematical or physical species, which consist of a perfect dimension equal among themselves, are equal.
- If, according to their constitution and formal terms, they are unequal among themselves, they will be unequal.
- If equal quantities are added to equal amounts in their entirety, they will be unequal.
- If equal amounts are added to unequal quantities, the totals result in unequal amounts.
- If equal quantities are subtracted from equal amounts, the residuals result in equal amounts, and vice versa.
- If portions or equal quantities are subtracted from unequal quantities, the residuals are unequal.
- If equal forces or powers or equal quantities are assumed, the results are equal, and vice versa.
If in forces, powers, or unequal quantities, supplements are applied by accident, science, or art in operation, which exceed the inequality, they will not only reach or terminate unequal terms, but the lesser force, power, or quantity, as much as it exceeds by accident, science, or art in operation, will result larger by the excessive supplement.
This question arises from those who believe, or hold the opinion, that there is a fourth type of continuous Mathematical Quantity, whether considered in abstract or concrete, while the correct understanding of Philosophers, Physicists, and Metaphysicists is that there are no more Mathematical species than Line, Surface, and Body, which properly belong to the Category, or Predicate of continuous Quantity.
Those who claim otherwise, deviate from the Philosophical doctrine, which truthfully only admits these species of Mathematical continuous Quantity, considering them as parts that comprise a whole, which is the Body, adding (regarding the physical consideration) two other species of permanent continuous Quantity, which are, Place, and Time, as taught by Aristotle.
Some Moderns, without considering the entity of the species, add Angle as the fourth, and deducing from their arguments (those that weigh with greater acuity) argue thus: Every Mathematical continuous Quantity is constituted of terms, the Angle is constituted of terms: therefore, the Angle is the fourth species of Mathematical, or Physical continuous Quantity.
Against this is Aristotle, and the whole current of Logicians, Physicists, and Metaphysicists: and the wise Mathematicians feel, and must feel, the same. This is proven because in two ways the species of continuous Quantity are understood in terms of Line, Surface, and Body. One is, considering the species, like pure Philosophers; another, like pure Mathematicians. Neither in one way nor in the other is the Angle the fourth species of continuous Quantity, and there are no more primary species in it than Line, Surface, and Body, without admitting the Angle as the fourth species, as Aristotle has, and his Expositors carry, without disagreement, admitting immediate fourth and fifth Species, Place, and Time, and everything else, that can concur as a species of continuous Quantity, the Philosopher wants it to be by accident.
If Discrete, its generic species are Number, and Speech in terms of distinctions, derived from reason, because in the continuous Quantity the terms are common comprehensive; if in the discrete, they are distinct: from where by maximum received, Scotus said, speaking of Quantity: That being is a union of the extremes.
For this reason, Ancient and Modern Logicians, as Philosophers, admit as a conclusion that, speaking properly, the Line, Surface, and Body are not three species of permanent continuous Quantity, but rather, in a certain way of understanding, they are parts for the Whole, and can be called incomplete parts, since Length, Surface, and Depth (which are all dimensions of permanent continuous Quantity) all are found in the Whole, which is the Body, from where the concept can understand them as parts composing the Whole.
Mathematicians, who do not pay so much attention to the entity of things in themselves, as to the demonstration by Problem, or Theorem (without getting bogged down in the way of proving by syllogism, and reasoning, like Logicians, Physicists, and Metaphysicists) although they do not add more to the continuous Quantity than the three species, Line, Surface, and Body, they consider, at least, in each one, distinct passions or affections, as if in a perfect dimension others did not concur, distinguishing Lines from Surfaces and Bodies, although the Line, and the Surface precisely signify, and demonstrate the Length, and Latitude, which are found, as parts of the Whole with the Depth, constituting a finished, complex Body of its common terms, achieving the perfect being of continuous, permanent Quantity by the union of its extremes.
From the previous two conclusions, to which the discourses of true Philosophers and Mathematicians are reduced, a third conclusion is drawn, with which it is reconciled, and it is understood, what is continuous Quantity, and how it is reduced to the three main species, Line, Surface, and Body, without admitting the Angle as the fourth species of continuous Quantity, as it proceeds from reason, and is demonstrable, because although when they concur, forming the Body as perfect, like common united extremes, this does not prevent the Line, and Surface from being distinct species, when they terminate without the union of the Whole, ending from their common united extremes, thus the Line is a pure species of continuous Quantity, as Length, without admitting Latitude, including in its united extremes, which are points: and thus, the Surface is also a proper species of the continuous Quantity, when it is contained by its common extremes, which are Lines; and consequently, the Body is the third perfect species of the continuous Quantity, when its Depth, or thickness is contained by its common united extremes, which are Lines, and Surfaces. For example: The cube consists of the three dimensions, Length, Width, and Depth, constituted in its whole six copulated Surfaces, and each Surface of four lines, with which it receives its being from the union of its extremes, constituted of its common united terms.
More perfectly, the spherical Body, which without admitting an angle, receives a perfect being of a solid Body, and continuous Quantity, by the union of its common extremes; in whose composition Euclid wanted the three Species, Line, Surface, and Body, to concur, from which the Whole, or Spherical solid results; and so he defined it by the circular line in the Quantity of a semicircle, saying, the Sphere is the transit of the revolution of a half circle, &c. whose movement described, and formed the extreme Surface, which includes the depth. For these true fundamental demonstrations, the Angle is clearly excluded from the continuous Quantity; because in truth it is not, but passion, or affection of concurrent lines at a point, which produced by the first petition, must necessarily be cut, causing proportional Angles, equal to each other, to four right, being the lines two enough to form an Angle; and consequently, the Angle is not a species of the continuous Quantity, but of the discrete; for this reason it receives its essence, and quality from the numbering, which demonstrates its value by degrees, and minutes, or in other terms (which are more from other Predicaments, than from the continuous Quantity) because of the passions, and concurrences of some lines in others, inclining, and declining, the Angle is caused and as well as from the inclining, and declining planes, with respect to the concurrence in another plane, in which the reason, the discourse, and the mathematical demonstration widely finds an open field, as it will be noted in the proper place: and for more convention, the contrary arguments are demonstrably responded to.
To the first it is conceded the greater in as much as it admits that the continuous Quantity is constituted by united terms. The Angle, then, is not constituted by united terms, but by separate ones in the constitution of its being. Thus, the Angle is not the fourth species of continuous Quantity. This is proved, because this: can be in its parts, and in its whole, without constituting an Angle, and when it constitutes it by accident, the Angle will not be an essential species, but a passion of termination, and concurrence of lines, as they concur on the Surface.
The greater of this argument is proved by the sentence of Scotus, saying: That being is in the union of common extremes. The greater is notable, as proved by Thomas Bradwardine, and it is common that two lines do not constitute a Surface, nor is such admitted, being straight; and they can constitute an Angle, which is not of the continuous Quantity, like the Surface, or the Line alone; and if two curved portions can form two Angles, then copulated in the unity of common extremes, they are not considered by the Angle, but by the Surface, which they include, as complex terms, as noted in the other figures, that uniting in common terms, their extremes compose a Surface, which is the second species of the continuous Quantity; and consequently, the Angles are passions, or affections in the figure, being caused by the lines, which by uniting at the extremes, give being to the figure, as seen in all.
From here it results, that two produced lines (being straight) cause interior and exterior Angles, which are worth among all four right, proportioning themselves among themselves, according to the point of concurrence, by the section, and production of the two straight lines, causing the Angles, without union of extremes: then the Angle is not the fourth, nor another species of the continuous Quantity, but a passion of concurrent lines at a point.
To the second argument is conceded the greater, but explaining, that the continuous Quantity, and the discrete are distinguished as to their dimensions, and divisions, and differ; because the continuous Quantity in itself is apt for dimension, and division as to power; but the discrete Quantity is in act (as says the Philosopher) because the number 5. includes the 3. and the 3. is not 5. because its division in the discrete Quantity is in act, and in the continuous is in power.
The Angle, then, is numerated, and divided by the Number, which is a species of the discrete Quantity, just as the Angle of 5. degrees, although it includes the one of 3. the one of 3. is not the one of 5. but different, etc. then the Angle is not of the continuous Quantity, but of the discrete, because its being receives it from the numeration, caused by the concurrence of lines at a point, and not from the union of all its common extremes.
To the third argument it is answered, distinguishing the greater, and denying its absolute; since as it has been demonstrated, the Line by itself is nothing more than length without width, whose common terms are points; and in this part the greater of the contrary argument does not apply to the Angle, which is not caused by the length alone, but by the division of two lines, without termination of all its united extremes, with which they do not constitute Surface, nor Body, which are second, and third species of the continuous Quantity, that perfects the whole by the union of its common extremes: from where the Angle, although the lines that include it, end in points, they do not unite in all their extremes, nor can they constitute a species of continuous Quantity, but discrete numeration of its interval, caused by its greater or lesser deviation of the lines, which, being concurrent, form an Angle, and thus we achieve the intention.
With what has been referred to, we follow the doctrine of the Philosopher, placing as the fourth species of the continuous Quantity the Place, in which we will briefly gird ourselves, excusing the multitude of questions, and arguments, in which the Logicians, and the Physicists get entangled, from where the dispute is more proper, finding out what place is, and in what way it is considered, as to be given void in nature, it will occupy, or will not occupy place? The other discourses, and difficulties, that are disputed, I omit, for not being of the commitment of the Science of the Sword, in which I will try to choose the most forceful, and convenient.
We consider with the Philosopher that every body occupies a place (according to its capacity and form), hence Plato called the receptacle what is commonly named Place: and from this principle it is defined by description, saying: Place is that concave surface that surrounds and contains the body, in such a way that, without leaving a vacuum, it encircles it and unites with it, in such a way that Place and Body continue by the contiguity with which they join; e.g. the vessel, or the urn full of water, or another liquid, encircles and surrounds the concave surface of the vessel, or the convex of the liquid body, uniting without a vacuum the placing body with the placed body, not by pure continuation, but by unitive contiguity, which is reciprocal in the external surface of the body, and in the internal of the vessel that circumscribes it, excluding the vacuum, as the Philosopher amply proves, discussing our intention the causes that delay or facilitate the movement from place to place, because the lighter the natural placing body is, the more speed it admits in movement; and the denser it is, the more it impedes and slows it down. Take the example in water and air: both elements are apt to surround any moving body, either by itself, or by the impulse of another, touching the mobile body (as proposed) without giving a vacuum, as demonstrated by the Philosopher. Thus, due to the greater density of water, and the greater rarity of air, which makes it light, the mobile body has more aptitude to speed up in the air than in the water.
From here it follows how one Sword hinders another, moved by the impulses of the combatants, and because the arm and the Sword are located in the air, they are more apt to move more quickly than the feet, which move on a dense surface: that being located mostly in the air, in the end, is a dense part on which they rest, which is enough not to speed up as much as the hand and the Sword: from where it is noted, that the farther the feet are from the earth, the more ready is the aptitude with which they acquire speed: and the more united by contiguity, the more impeded and slow. From the same places in which the body, arm, and Sword move (which are in themselves heavy parts) and to where they move, also results in slowing down or speeding up in the aptitude and operation of movements, as will be more distinctly demonstrated in its place: this is proper to consider the rectitudes, in relation to the location of the bodies; for as Aristotle proves, every body occupies a place, and is occupied by its placer by contiguity, whether it is at rest, or in movement, with such consideration, that there are always its differences noted, in relation to the same body, and these are commonly called Rectitudes; but the Philosopher notes them as parts, and differences of location, reducing them to six, which are, Up, Down, to one side, and to another, forward, and backward.
Here, for more distinction and necessity in the Science of the Sword, we use the most common terms, and we name these parts and differences rectitudes, noting them in their six differences, or parts, which are straight to each body, Up, Down, to the right side, to the left, forward, backward; and this consideration is so forceful and necessary in this speculative and practical Science, that it is recognized as one of the greatest foundations of all Skill, because from such principles of place and rectitudes, comes the knowledge of the intelligence, and exercise of the Sword, as is seen demonstrated in this Category and Species, and in that of Time, and Movement, and Stillness, &c.
Noting here, that in the occupation of place, or places, the combatants have certain respects against each other, and each one in itself, according to the positions in which they find themselves place, or places, that they occupy, and they are occupying postures and modes, in which they are placed, and they are placed, and how they acquire, and lose the aptitude to movements, which are slow, which are fast, which are remiss, which impede, and which are impeded, in relation to places, bodies, and rectitudes, and the other circumstances, and considerations, that arise from knowing and understanding, what kind of continuous Quantity is the Place? What is Body placed by itself, and in relation to another? How by the distinction of places are advantages won, or lost? How are we to pass from one place to another? And in what way what in one gives disposition to execute outgoing angles, causing extension in the subject, in relation to his Sword, and local figure, where it is planted; in another it is the opposite due to the ingoing angles, that cause tension, and reduction in the reaches from place to place: because the Sword, is not only ruled by the arm, but by the position, and place, that the body occupies in its placement, in relation to the postures of the combatant subjects, which all this depends on the understanding of this species of continuous Quantity, as considered by Philosophers and Mathematicians: of how.
Whether there is a Vacuum, or Void in the nature of things, is the matter in which a multitude of Ancient and even Modern Philosophers have been lost; although, the most learned denied and deny the vacuum in the nature of things, both in the composition of the Universe, and in the placement and movement of bodies in which it has been touched, and could have been discussed with Lucretius, Pythagoreans, Platonics, Xenocratics, and better with Aristotle, who proved demonstrably, there is no vacuum in nature, neither in relation to the filling of place, nor in relation to the movement of bodies, nor to the consistency and transmutation of Elements into each other, nor in the conjunction of one body, in relation to another body, nor in the same porous bodies, like wool, sponge, ash, &c. that admit tension, and extension by their separate rarity, or by the comprehension, because in the occupation of the air it fills, and without admitting a vacuum. The curious reader can see, among others, Plutarch, who compiles various opinions of Philosophers.
Pedro Gregorio wisely observed from the place in Genesis, that the earth, at the beginning of its creation, was void and without form, he said, that not absolutely, but in a certain way, the term Vacuum is probable, not looking at it by the rigor of the signification, but by the lack of something that completes, as the earth is natural, to express its operations, it is used by common terms, Left place empty: Did not occupy place entirely: Found empty, or occupied empty place, &c. and in such ways the signification is to understand the lack to the fullness, that is acquired, or the complement in what is worked.
In this way, and similar ones in Skill, the use of the terms Vacuum, Void is not absurd, because they do not look at the nature of things philosophically, but at the explanation of operations, with distinct consideration, to explain more briefly, and more intelligibly the proper meaning of actions, as also found in many classical authors, Cicero against Catilina: By the death of your first wife you left the house empty. In this way Passerasio noted that the common people call air-filled what is empty. The same was used by Virgil and Lucretius, and others, for the wide and spacious, as also used by Virgil, and the same Cicero called the idle mind empty, and from there came the proverb: Vacuos habere dies, to signify idleness, and Vacuus equus, the horse, which is not occupied by a person, as Livy said: Vacuus equus errans per urbem. And from vacuum come the common terms, Empty, Emptying, Emptied, &c.
With that, it is sufficiently excused in the Science of the Sword to use the terms Vacuum, and Void, in which erudition can be expanded, which is deliberately omitted here; with that we move on to the fifth species of continuous Quantity, which is Time, leaving for its place the understanding of Movement, and Stillness, which is so much of this Science.
The Philosopher placed in the second Predicament, or Category as the fifth species of continuous Quantity, Time: we leave for the Physicists the debatable, whether Time is contained, or caused by movement, and how they should be understood in their essential being, Movement, and Time, both by comprehension, as by substance, or accident in Quantity or Quantum.
Theologians investigate by other means the principles of Time in the creation of the Universe, and what was the first light that formed day and night? and when created? whether the first darkness was time? and other questions proper to the exposition of Genesis, where the curious can occur, for not being of our concern, in which excusing the long debatable, of the essential, which is found in Aristotle, we define time, saying: Being apart from a thing, a formal transitory duration, which the understanding perceives by permanent extension, divisible, and measurable, according to power, by the regulation of the natural movement of the first movable, with which it admits distinction of past, and future coupled in the point, which is given as present, and from where its measurable continuity in potential is known, teaching the Philosopher, That continuous movement is continuous time, and continuous quantity, concluding the similarity between Time, and Movement, and Movement, and Time.
These doctrines make it difficult to determine whether Movement is also another species of continuous Quantity, separate from Time, which Scotus touched on with his acuity, and after him other Moderns, saying: Movement is seen to be in the genus of Quantity, as is inferred from Aristotle, who said: Time is Quantity by movement: therefore Movement is Quantity. It is confirmed: Time, then, is a species of Quantity, because it is the fifth of continuous Quantity, according to Aristotle: then Movement is more a species of the continuous Quantity.
In response to our intent, Movement is not a species of continuous Quantity, because it does not have extension and measurability in potential embedded in its own concept, as in the opposite persistence of the agent it can be weakened, because against Movement a greater efficacy of the agent can occur. That which by itself is not consistent in all its parts is not by itself extensive and measurable in potential: therefore Movement is not a species of the genus of continuous Quantity.
The discussion about what types of Movement exist, and which are due to alteration, corruption, or transmutation, etc., is omitted because this is not very characteristic of the Science of the Sword, nor of this Category. But the essential will be treated when we talk about Movement and Stillness, following the Philosopher, who although included the matter, De Motu, etc. Quiete made a special discourse: not touching here more than to understand the five species of continuous Quantity, in which Logicians and Metaphysicians agree. One of them is Time, for being successive, measurable, and extensive, concluding with the Philosopher, that all other Species, which are intended to be applied to this Genus, are not truly Species of the continuous Quantity, but by accident: from which it follows, that in the Science of the Sword, the term Time is considered and used as a permanent, divisible, and potentially measurable extension; as by the regular movement of the first mobile, the other movements of Orbs and celestial Astros, we consider time as divided and measurable in instants, minutes, and hours, days, months, years, lustrums, ages, etc.
So, in this Science, through the movement and movements of the body and actions, Time is divided and measured in instants and operations, from where there are wounds that are named instantaneous, and others in other terms, using those of Time, and Times, taking the whole for the part, or the part for the whole, as is more appropriate for the explanation in the understanding, and practice of the Sword, demonstrating the Tricks, the Wounds, the Movements, and the rest, that in this Science constitutes the true Skill, remaining, as they remain, in this Category explained the five Species of the continuous Quantity, which are, Line, Surface, Body, Place, and Time. Agreeing with Euclid, Principle of Mathematicians, that Line is extended Quantity, Verſus longitudinem, coupled with indivisible points. The Surface, extended Quantity, Verſus latitudidem, coupled with indivisible lines. The Body, extended Quantity, Verſus profunditatem, coupled with indivisible surfaces. From which it is clear, that the Line only has one extension, which is the Length: the Surface has two extensions; that is, Width, and Length: saying, the first De formali, and the second De materiali, as the Philosophers note. The Body consists of three extensions, of which two, which are, Length, and Width, expresses them materially; and the other, which is the Depth, explains it formally. Which Aristotle confirms, saying: That the continuous magnitude Ad vnum, is Line, or Length; if at two, it is Surface, or Width; if at three, it is Body, or Depth, that all are Propriè, & per se, Species of the continuous Quantity, as can be seen broadly in the Metaphysicians.
Finally, it is established that Place is the measure of the located, per external: and Time is the number of Movement, according to what is before and after, adding in passing the definition of Movement for the understanding of Time, which is, Act of the entity in potency, insofar as it is in potency as defined by the Prince of Philosophers. We leave, then, to them, different disputes, and so we move on to the Genus, and Species of discrete Quantity, as our effort allows, placing its Schema in the middle of the two Species immediate to the Genus Quanto, so that the Reader’s contemplation can easily register it.
The second Genus, dependent on the Generalissimo, Quantity, or how much, is the part that includes discrete Quantity (which is the same as disjoint) because just as the continuous (according to the Philosopher) is defined by that which is complex of its common terms; so the discrete is that whose parts do not couple in common terms, and of these all those that are reduced to Number, and Speech, are species, as admitted by the Logicians according to the doctrine of Aristotle.
It is proven. The parts of the Number are distinguished in such a way that between one and another unity, which are extremes, there is no common thing that is similarly part of the first and second unity, as explained by Torrejon, and in the same likeness it is noted in the parts of Speech, in syllables, and words, which are constituted in such a way that between the first and second syllable, and between the first and second word, there is nothing that mediates, that is similarly something of the first and something of the second extreme, as is demonstrable; because if it were something similarly something of the first extreme, it would not differ from it, and the same in the second. Therefore, the Number, the syllable, and the word differ from their extremes: ergo, &c.
As for the principle of Number, the difficult part is, and has been, the distinction of units, which is natural, which is entitative, which is predicable, which is mathematical, in which Logicians and Metaphysicians elaborate, taking as their object philosophical reason, in which we get little entangled here, because the Science of the Sword only makes that unity, which of discrete Quantity is species in this Predicament.
In the light of this, it is noted that the philosophical unity, either is the one that in its entity competes with the spiritual, or with the material, or the transcendental: and the one that we consider in this Science, is that unity suitable to the corporeal, which is proper Quantity in the Predicament, because it consists in the integration of its material parts, and this is properly the one that in this Science we admit as Predicamental unity. Thus Euclid defined it (as a pure Mathematician) saying: Unity is that which is understood as one, where the number is the composition of units.
From this it follows that, since the Number is composed of units, it is all of parts, which are constituent ones: then the spiritual and transcendental unity in this Science are not proper to this Predicament, but those corporeal mathematical ones that form the Number, which is called composition of units.
From this principle it follows that any Number is a species of discrete Quantity, because this is nothing other than a precise concept, which includes measurability, resulting from the division of parts, as we consider in this Science, because any Number in its precise concept includes the division of its parts by measurability, both by unity and by another Number, from which true quantity proceeds, as we consider in the gradual numbering, both of the Sword, and of the value of the Angles, in which we find with measurable equality, and inequality between numbers, and extensions. Consequently, it is inferred that equality and inequality essentially presuppose measurability in the number, or the extension in this Science: any number, and any extension is measurable; e.g., the arm, and Sword in extension of 6. by numeration of 2. and 4. units, arm, and Sword is a species of Quantity in this Predicament, because we find a composition of parts in discrete Quantity, not as in transitory Time, but by the positioning of the parts, which have an order of themselves among themselves, in order to Place, and location, that their discrete terms are recognized, one anterior, and one posterior, with respect to their number, not in the permanence of continuous Quantity, which has different measurability than the species of discrete Quantity.
In this (as Celio Rodiginio noted) there are numbers, which being proper to the discrete Quantity, respect the continuous Quantity by their terms, and composition of units: thus linear are called those numbers of continuous progression 1.2.3.4.5.6. &c. or as 2.4.6.8.10. &c. or as 3.6.9.12. &c. others are called flat, or areal, that proceed from two numbers, which mutually multiply: one by the other; and the one they produce is named flat, or area, whose sides are those numbers, which cause from their multiplication the flat; e.g., 4. in 6. produce the flat 24. whose sides are 4. and the flat, or area 24. that refer to the right parallelogram: and for the same reason, when equal numbers mutually multiply one by another, they produce a perfect square, whose root is one of the causing numbers, as 5. in 5. square power 25. root 5. as it is demonstrated in the square, whose sides are 5. root of power 25. and not only in such powers of flats, but in many others different flats are produced, by the different positioning of the Numbers, which multiply one by another, as the 24. that is caused by 4. and 6. can be caused by 2. and 12. and by 3. and 8. and so for many others.
Solid numbers, and cubes are called those, which are produced from the multiplication of 3. numbers, as from 2. 3. 4. the solid is generated, which all composes 24. because 2. in 3. makes 6. and 6. in 4. form 24. in such solid: and consequently, in the cube, when a number, carried in three multiplications, generates a perfect solid, which on all its faces consists of equality, like 3. in 3. generates the square plane 9. and the 9. in the same 3. the solid 27.
Such numbers corresponding to the 3 species of continuous Quantity with measurability of discrete Quantity, lead us to the understanding of powers, and roots, and the other affections of Numbers, which the discipline of Arithmetic teaches, of which this Science makes use, as of a subordinate: thus we presuppose the understanding of the Elements of Euclid, considering for our purpose the Number in three ways: the first, as a numeral number: the second, as a numbered number: the third, as a counting number. Numeral Number is that which the understanding perceives by pure concept, which admits number; e.g., of the Stars the understanding perceives, that they are in themselves a numeral number, capable of number: the grains of wheat of any measure, also the understanding perceives, that it is a numeral number, as Archimedes perceived of the sand, &c. but not because the understanding perceives the numeral Number, it grants, and distinguishes it in its own distinct numeration, because that is of higher knowledge.
Numbered Number is that in which the number is distributed, and comprehended, according to its composition of units, like 4. which is composed of 2. and 2. or 1. and 3. or like 6. which is composed of its aliquot parts 1.2.3. that form 6. or like any aggregation of units, or individuals, or spirits, &c. of which our understanding perceives distinctly the numbered number, like 100. 1000. &c. 10. men, 10. lions, 10. coins, &c. 20. Angels, 20. demons, 20. Stars, 7. Planets, &c. from which results, not only the knowledge of the numeral number, but also the understanding perceives it in its distinct Predicamental units, a counter in certain numbers, composed of its units, with a term terminated in distinct, and certain numeration. In this Science we consider such the value of the angles, and everything else, that in it the understanding perceives by distinct, and certain numeration.
Counting Number is the instrument of the mind, that in order to perceive things (to our understanding) we consider them by numeration, like the line, the Sword, the division of the angles, according to their composition, degrees, and values, in which we number things, so that the understanding makes pure concept by numeration, according to discrete Quantity, besides this in the Numbers, some are considered certain, others uncertain.
The certain ones are those that the understanding perceives with clear distinction, and with it their numeric terms are reached; e.g. all those Numbers, which the Arithmeticians call rational, because reason, and understanding investigates them, and perceives them, are called certain Numbers, like root of 36. which the understanding comprehends that they are 6. or like root of 27. which is 3. because 27. is a cubic number, and not flat, and so its root is 3. and its power 27. &c. and so all the numbers, which generate rational roots, and rational powers, propositions, equalities, &c.
Uncertain Numbers are called those, that even though in their being, and essence by fractions, or other unfound means, are without being perceived by the understanding for certainty of numbered numbers, although in themselves they have numbered terms they are not perceived: they are called deaf, or irrational, or uncertain, respect, not of themselves, but of our fragile understanding: thus root of 28. (it is said that) is root of 28. but not precise rational, &c.
The types of specific numbers include all those commonly called principal, perceived by their composition of units through their collection, subtraction, multiplication, division, powers, roots, propositions, equalities, etc. Euclid and his expositors, Boethius, Campano, Peletarius, etc., extensively deal with this. For uncertain values, Algebra and Logarithms provide a path, and for the value of angles, tables and the subject of sines (or ‘senos’, as pronounced in our vernacular) are used. Since this Science greatly involves the understanding of angles, it’s necessary for us to delve into their explanation, which is inherent to this category.
An angle is that space, caused and encompassed by the intersection of two lines, which converge at a point from two terminal points of their deviation. This point is the limit, whether the lines are straight or curved, or a combination of both in the various oppositions perceived by the mind and executed in practice. The angles derive their names from the types of lines and the manner in which the existing and insistent lines are found.
A rectilinear angle is one whose uncomplicated terms are straight lines. A curvilinear angle is composed of curved lines, converging at a point on the concave or convex part: and thus they can converge, forming the angle on the curvilinear concave part; and on the convex to convex part, just as on the concave to the convex part. Mixed angles are those caused by different types of lines, such as curved and straight; and for the same consideration, there are mixed curvilinear angles when one leg or line consists of a perfect circular portion and the other is not so curved. As for their species, angles are distinguished by the greater or lesser deviation of the lines, according to how they insist or consist on each other, according to the positions of their disjointed ends and the point of their convergence. A right angle is called that whose lines that cause it, converge perpendicularly at a point, without deviation or inclination, one with respect to the other, like the T.V.X. angle that the figure 1, Stamp 3 manifests.
All right angles are those that are caused by straight lines that, when extended, cut each other in a Square; that is, one falls perpendicularly onto the other, as Euclid demonstrates, and it is seen in the standing T.Y., existing Z.X., that they will cut each other at the point of the convergence V, and all its angles are right, and therefore equal, because there is no inclination of one line with respect to another.
From this, it follows that in any deviation or inclination of one line over another, a difference in angles is created. Such that the larger angle, known as the external, complements the smaller one, known as the internal; and vice versa, because between the two they total two right angles. The larger is called obtuse, and the smaller acute or sharp. This can be seen in the second figure, third stamp, as the straight line A.B., resting on the consistent straight line C.D., forms unequal angles at point B. The A.B.C. angle is obtuse or external, and the A.B.D. angle is acute or internal, inclining towards point D. The larger and smaller angles complement each other to two right angles, with the endpoints A.C.D. being uncompounded terms (that is, separate). The lines drawn to the point of convergence B, as in the ends of the deviation, do not admit coupling, do not include a figure, and so the angles they cause are not of continuous quantity, but discrete, as recognized by philosophers and Clavius, stating that there are three terms: Points that terminate the line, coupling in it, Lines of the Surface, Surfaces of the Body, and this last is not the end of another continuous quantity because there are no more than three dimensions, and every term surpasses its term by one dimension. It becomes evident that the angle is not of continuous quantity, but of the discrete, because it does not admit coupling of its terms, nor does it exceed in one dimension to its predecessor. From this follows the consideration that some angles are complementary to others, because if the existing line is perpendicular to another, it forms right angles, which are all equal, as demonstrated by Euclid and his expositors; and if the existing line is declining or inclining, the obtuse and acute angles that it causes (without one complementing the other) will equate to two right angles.
From these principles, and the others that have been touched upon, it also follows that two straight lines do not enclose space; although by their convergence at one point, drawn from different extremes, they form an angle, whose value is known, and qualified by the gradual number, and not by the figure, because (as has been said) two straight lines are not capable of enclosing space between their ends. Right angles are problematically formed by setting a straight line perpendicular to another.
From this it follows that two lines, extended and intersecting at one point, drawn straight to their ends, constitute equal angles at the vertex, as can be seen by figure 3, Plate 3. For example, the two straight lines A.B-C.D intersect at point E. It is said that the angles at the vertex are equal to each other because the angle A.E.D, and B.E.C (which are at the vertex) are equal to each other, because the straight line D.E. extended to C. insists on the straight line A.B. at E. And the angles at the vertex A.E.D and C.E.B are equal, and A.E.D and D.E.B are worth two rights, as well as A.F.C and C.E.B and all equal to four rights, being each other’s complements, and the angles A.E.D and C.E.B of the vertex are equal acute angles, and consequently F.F are obtuse, as Euclid demonstrates, and Proclus and others prove, drawing various corollaries.
From these principles, not only the formation of angles is investigated, but also their differences, their equalities, and inequalities, knowledge of which is essential in the Science of the Sword, and for its operations, in which it is necessary to know the value in which they are graduated; otherwise, one will proceed with a confused concept, making the understanding by numeral number, and not by numbered number, nor numerating number.
For this reason, the great Mathematicians, and Philosophers found that in the gradual and numeral value of the angles the Periphery is a competent measure, which shows the distance of the deviation of the lines from which they are caused, converging on each other. For this they chose a certain gradual numeration in which the circle is distributed, whether it is larger or smaller, with the understanding that the numeration should be one, and the parts proportional to their whole, as in the Sphere larger circles are placed, which are those that divide it into two equal parts: and in smaller circles, which are the more so, the closer they are to their Poles, being Parallel to the circle; for example, the circle, which is named Equinoctial, is a larger circle because it divides the Sphere into equal parts: those which are called Tropics, are smaller, and the closer they are to the Parallel Poles, the smaller they are; so also in a plane the circle, which includes the others that are Parallel to it, is the one that approaches the center, they are smaller; and therefore, dividing, as all are divided into a gradual numeration of parts, those degrees, or parts of the larger circles, will be larger, and in the smaller circles smaller, as is common; and this division of numbered number, which we number, for more convenient was done in 360. degrees, or parts, considering that the sixties are the most capable for operations, because they are composed of the two numbers, called perfect, both Mathematicians and Philosophers, the one of the Mathematicians is 6. because its aliquot parts 1.2.3 joined together form 6. without excess or lack. The one of the Philosophers is 10, considering it by the fingers, which are 10, and so these are named numbers digits. Having found these principles, they took the number 6. in itself, and it gave its power 36. these taken by 10. produce the numbered number, which we number, for the graduation, and division in parts of every circle, which are 360. From here it went on to make fractions of integral numbers, and each degree was divided into 60. minutes, each minute into 60. seconds, and so on to tenths, to which rarely is arrived, and these are called Astronomical fractions, unlike the Mathematical ones, which divide the Predicamental unit into smaller ones, as is suitable to the ultimate fraction (as the Arithmetic part teaches) which they call squares.
Through these and other principles, we proceeded to investigate the proportion of parts of a curved, circular line to the straight line, considering how the circle and its diameter relate to each other. This has challenged many ingenious minds, and until now (as everyone admits), a demonstrable proportion has not been found, leaving us with the laborious task of the Quadrature of the circle, on which many have written, to which we refer, acknowledging that the most widely accepted and closest to certainty is what Archimedes called the Spiral, which is the circle with its diameter, as 7 is to 22, which is a triple sexquiseptima proportion, sufficient for what is intended in the Science of the Sword, as well as for the investigation of the value of angles, and how they relate to the circular portions that number them, and these with the right sines, and the complement tangents, and secants, arcs, chords, and arrows: this Science uses tables of sines, or sines, for whose light figure 4 is proposed, of stamp 3, assuming that the total sine (which is the semidiameter) some divide it into 100000, and others into 10000000, and some into more, and into less, of which the most practicable is chosen, following Ptolemy from the Ancients, and Pitiscus and others from the Moderns, who all admit the distribution of every circle into 360 degrees, or parts, and each one into minutes and seconds, giving a numbered value of 90 degrees for the right angle, which is a quadrant, and to the proportion the oblique, obtuse, and acute angles.
The regular proportion between the curved and straight line (as mentioned before) is up until now a numeral number, but not a numbered number; thus, the common opinion received and put into use is followed, through which the tables of right sines, and complementary tangents, and secants, have been created, in which the following definitions are accepted: 1. Curved lines for straight lines are reduced by definition of the Quantity that straight lines have when applied to the circle, with respect to the radius: 2. Straight lines, applied to the circle, are subtense, sines, tangents, and secants: 3. Subtense is the straight line inscribed in the circle, which divides it into two parts, and when they are equal, it is called the Subtense diameter: 4. Line subtense is maximum, and not maximum: 5. Maximum Subtense is the diametric, like the straight line G.C. 6. Non-maximum Subtense is the straight line, which divides the circle into two unequal parts, one subtending for the larger portion; and the other, for the lesser, like I.B. that the lesser part of the Periphery subtends the arc, less than the semicircle, like I.F.B. and the other greater, than the semicircle G.C. because it is the arc I.H.B. 7. Sine, or sine, is either right or versed. 8. Right sine is that which is included in a smaller arc than the quadrant, like B.C., known by the straight line B.E which is the right sine of the arc B.C., and perpendicular falls on the semi-diameter A.C. at point E. 9. Versed sine is the one of the arc greater than the quadrant on the subtense diameter, or maximum, like B.G., and is considered by the same straight line B.E. produced in D., and similarly for subtending to the arcs B.C. and B.G. this is by half the straight line B.E.D. that similarly subtends to the arc B.C.D. or to the arc B.G.D.
10. Consequently, the right sine to the quadrant arc of the larger and smaller, up to the semicircle, is the same, as the right sine of the arc B.C. and the arc B.G. is the same line B.E. because being half of the straight line B.E.D., it is as much subtended to the arc B.C.D. as to the arc B.G.D. 11. Therefore, the right sine of the complement, in any way it is understood as the sine, complement to the arc of the smaller quadrant, like the sine of the complement of the arc. B.C. by the arc B.F. is the line B.K. and conversely reciprocal sine of the complement of the arc B.F. is the line B.E. 12. Tangent is the line, which ends in the secant, and at the end of the diameter perpendicularly, as of the arc B.C. is tangent the L.C. 13. Secant is the line, which produced from the center to the convergence of the tangent, cuts the arc of the quadrant, like the line A.L. that cuts the arc at B. and converges with the tangent at L. and so the similar ones.
14. Definition of the Quantity, that the lines have, applied for the circle, is the construction of the tables of sines, tangents, and secants, which are formed by the computation of right sines, and not of the versed ones. The reason is, because to the lesser versed sine, as to the right sine of complement, is equal the radius; e.g. lesser versed sine E.C. as to the right sine of complement A.E. is equal to the radius, or whole sine A.C. then if subtract the right sine of complement A.E. from the radius A.C. remains the versed sine, or sagitta E.C. and also in the greater versed sine is equal the radius, as is considered together the excess to the right sine, as the greater versed sine G.E. is equal the radius G.A. together with the excess: then if for the radius G.A. the excess A.E. is added, the versed sine for the arc G.F.B. will be obtained. And so, in the tables, work is not done on the versed sines, because the right sines are half of the sub-tenses. From where it follows, that if the maximum sine is found by the maximum subtense, also the non-maximum sine can be found by the non-maximum subtense, because the reason that there is from the whole for the whole, can be from the half for the half, just as the reason that there is from 10. for 6. is the same as there is from 5. for 3.
The entire larger or smaller circle (as has been said) is divided, and numbered into 360 parts, or degrees, which are proportioned, according to their diameters, in which we accept the received ratio of 7 to 22. From this it follows that the degrees, or parts of one circle, in relation to another, are similar, but not equal, not being of the same diameters, and radii, because the parts, or degrees, are larger as the circles are larger; and smaller, as the circles are; e.g. the circle E.D.F.G. its parts are larger than those of the circle L.P.J. even though the arc 45. degrees, number B.D. is similar to the arc O.P. 45. degrees, also as the quadrant D.E. is 90. degrees, as also as the quadrant D.E. is 90. degrees, as also the quadrant P.L. and as D.E.G. 180 degrees, so P.L.K 180. degrees, proportionally similar, in relation to each other, larger in the larger circle, smaller in the smaller: the angles, for the same reason are measured, and their value is qualified by discrete Quantity, according to the arc, which is cut by the lines; from whose convergence to a point the angles are created; e.g. the lines E.A.D. converging at the point A. because their interval is the arc E.D. 90. degrees; the angle they form is a right angle, and therefore, extending the lines E.A. in F. and D.A. in G. all the four angles they form are right angles, because the lines do not deviate or incline with respect to others, because all right angles are equal, and in the same regulation they are found, with respect to the smaller interior circle, as demonstrated by Pitiscus, by the doctrine of Euclid: the same reason of value, and measure, is recognized by the arcs in the formation of all the other Angles (which as has been noted) either are Right, or are Oblique: if Right, all are equal, because they are regulated by the quadrant, which is worth 90. degrees: if Oblique, some are larger than Right, others smaller, explained by the terms, Obtuse the larger than Right, Acute, the smaller than Right; e.g. the Angle G.A.B. is Oblique Obtuse, its value 135. degrees, the excess to 90. 45. and so the same the Angle K.A.O. and therefore Acute the Angle B.A.D. 45. degrees, like the O.A.P.
Complements of the Angles are considered by the arcs, because the Angles caused by the affection, and convergence of any lines one in another, Simul sumpta, are equal to two right ones, like the Angles E.A.D.-E.A.B.-E.A.G. where the lines converge at point A. on the straight line G.D are equal to two right angles G.A.E and E.A.D. by their own structure. Consequently, in the triangle A.B.C. right in C. the angle A.B.C. of the acute ones, the other one is equal to the angle B.A.E. because the angle E.A.B. is the complement of the angle B.A.C. by its structure: then the angle A.B.C is the complement of the angle B.A.C and so it can be reasoned in the other complements. With such information, one comes to understand the reason, because the angles caused on the circumference are half of those caused at the center, and it is manifested by the figure 6. of the Stamp 3.
The angle A.B.C proposed for its demonstration in the circle, whose center E. is the diameter, the straight line B.E.D excited from the angular point B. caused at the vertex of the circumference by the center E. dividing the circle A.B.C.D diametrically draw the radii E.A and E.C. it will be said, that the partial angles A.B.D-D.B.C are subduplicates for the partial angles A.E.D and D.E.C taken together: then it follows, that the angle A.E.D. for the angle A.B.D. is double, because the parts of the angle A.E.C for all A.B.C. is double: and consequently, the angle A.B.C is the measure, and graduation of the A.E.C caused at the center E. and although it is also the value of the angle A.B.C caused on the circumference at the point B. the value is diverse, because the angles A. and C. are equal to the partial A.B.D and D.B.C and the entire angle A.B.C. is half of the A.E.C as has been demonstrated, and proved by Pitiscus.
Given a straight angle, cut it into two equal parts, as seen in figure 7, of Stamp 3. This bifurcated section is achieved, given the angle: and let it be B.A.C. on the straight line A.B. at the chosen distance, take the point D. and on the straight line A.C. the corresponding E. couple the points D.E. with the straight line, of which they are extremes on D.E. constitute an equilateral triangle, which is D.F.E draw the straight line A.F. which divides the given angle B.A.F into the partial B.A.F and F.A.C follows that they are equal, because the sides D.A-A.F. of the triangle D.A.F are equal to the sides E.A.-A.F. of the triangle E.A.F. corresponding to each other, because D.A. is as E.A. which by its construction are equal, and therefore the triangle D.F.E. is equilateral by its construction: then the angle D.A.F. and the angle E.A.F. are equal partials, from which it follows, that the given angle is bifurcated into two equals.
From this demonstration common Practitioners derive a difficulty, which Euclid did not teach, nor is found; that is to know, the division of the angle into odd angles, because the bifurcated division only generates pairs, as Clavius noted; and so the angle is divided into 2, into 4, into 8, into 16, &c. but not into 3, into 5, into 7, &c.
Clavius himself responded to this deceptive doubt and formulated a demonstration by dividing the curve, or the Periphery, understanding the 27th Proposition of the 3rd of Euclid: and here by the same, and its consequences, recognizing by the doctrines of sines, their arcs, and subtenses, that the value of the angles is found, with which the veils are run to the difficulty, which is easily overcome, speculating the propositions of Euclid, where he taught, that in the circles in equal segments, the angles are equal to each other (as has been demonstrated) in which the Expositors of the Geometric Elements expand: to further demonstration, we admit here as thesis two theorems, repeated by Euclid, on whose principles we will base the hypothesis of our argument, and demonstration, as essential for the Science of the Sword, and proper to this Category in the Genre of discrete Quantity.
In equal circles, as in figures 8 and 9, Stamp 3, A.B.C-D.E.F. are the centers G.H. If equal angles A.G.C-D.H.E are constituted for the center, it will be said that the Peripheries A.B.C-D.E.F. two points at the vertex, which will be B.E. where the straight lines A.B C.B-D.E-F.E are derived. Draw the straight lines A.C-D.F. and because the angles B. and E. are medians for A.B-C.B-D.E-F.E will be the bases the straight lines A.C-D.F. because the angles B. and E. are subduplicates of the equal angles G. and H. and so they are equal to each other, because the segments A.B.C-D.E.F. are equal, and similar; and because the sides A.G-C.G. of the triangle A.G.C are equal to the sides D.H-F.H. of the triangle D.H.F because of the equality of the circles, and the equality of the angles, which contain G.H. by equal hypothesis, will be equal the bases A.C-D.F. and as the similar segments A.B.C-D.E.F are on equal lines A.C-D.F. they will be equal to each other, because if equal segments are removed from equal circles, the segments A.C-D.F. remaining are equal to each other, which is what is proposed.
Therefore, let the two equal angles B. and E. be constituted for the Peripheries, it will be said that the Peripheries A.C-D.F. on which they ascended, are equal. Also, as before, let the segments A.B.C, and D.E.F. be similar: it follows that they will be on the equal lines A.C-D.E. and as the angles G.H. are equal, because they are doubles of the equal angles B.E. they will be as before the straight lines A.C-D.F. equal: then the equal and unequal circles, similar in the segments, make equal angles, which is what is proposed.
The theorem implies that the angles in the center and on the circumference are gradually numbered by the propositions of the Periphery, which include between them the lines that cause the angle, with such notice that in equal or similar segments, the angle of the circumference is subduplicated from the one caused in the center, and the one in the center is double the one caused in the circumference, from which comes the knowledge of the value of the angles, by the portions of Periphery, that is included in the interval of the lines, causes of the angle.
To understand this proposition, as demonstrated in the antecedent in the equal circles A.B.C-D.E.F. whose centers G.H. insist angles in the center A.G.C-D.H.F of equal Peripheries A.C-D.F then the angles A.G.C. and D.H.F. will be equal and not being equal, angle G. will be smaller, because it is caused by A.G.I. that should be equal to angle H. by the Periphery D.F. for the Peripheries A.I.D.F to be equal to each other (which is absurd) because they would be part, and all equal to each other: then for certain are equal A.G.C-D.H.F. and if they also insist on the same equal Peripheries A.C.D.F. the angles B. and E. it turns out to say that the angles in the Peripheries are equal, because otherwise, as A.B.C is larger, angle E. will be made equal to angle A.B.I. and they would give the Peripheries A.I-D.F. equal, because as first they were given in the Peripheries A.I-A.C. equal parts, and all: then the angles A.B.C and D.E.F. are equal, because in equal circles, equal angles insist on equal Peripheries, which is the intention of this demonstration.
Let the angle of figure 12, Stamp 3, B.A.C., be 45 degrees, either assumed or measured: if measured, it will be by the portion of arcs B.C-D.E, which are similar, because the common center is A. Thus, from any of the arcs B.C-D.E., 15 degrees will be B.H-I.C., and the intermediate arc H.I. will also be 15 degrees and consequently, similar proportions in arc D.E., 45 degrees will be 15 D.F. and 30 D.G. in equal divisions: then, angles of 15 degrees each D.A.F-F.A.G-G.A.E. result, as are also B.A.H-H.A.I-I.A.C of arc B.C measured, and thus the angles M.N.O. are also equal, because the angle B.A.C is the same as the angle D.A.E., since arcs D.E. and B.C. are similar by Bartolomè Pitiſco’s first (as we have demonstrated), and by proposition 9 of Euclid, the line A.F.H. extended to L. is perpendicular to the very short extension B.I., dividing the angle B.A.I., which is the same as D.A.G., into two equal angles; and similarly, line A.F.H. extended to L. is perpendicular to the very short extension B.I., dividing the angle B.A.I., which is the same as D.A.G., into two equal angles; and in the same way, A.I is perpendicular to the very short extension H.C., and F.E. divides the angle H.A.C., which is the same as F.A.E., bifurcating. Then, the three angles B.A.H-H.A.I-I.A.C are equal to D.A.F-F.A.G-G.A.E. because they insist on equal Peripheries; thus, the given angle B.A.C. is divided into three equal parts, which is what was proposed, and demonstrated by Euclid’s 26.27.p.1, Pitiſco’s first 53, and Clavio’s second Scholio in 9.p.1., and consequently, the angle is divided into odd parts 5.7.9.11.&c by the same principles.
As Father Clavio rightly warns, no one should be amazed that the division of the angle into 3.5. &c can be exhibited in practice, because the demonstrations depend on previous propositions; as he said in definition 10 of the 1st, what is relevant to the purpose is deduced from its own places, where the division is sought: straight lines, whatever they may be, are divided into equal parts, as Euclid taught, giving precepts for dividing the angle and the line into equal parts, demonstrating how to divide the line and angle bifurcating, because precise demonstrations are not always required in practice, but rather the effect of what is intended is correctly proven; and whoever is not content with bare practice, giving the demonstration that is required, can resort to investigating the demonstration by the necessary principles, speculating, because not all propositions that are relevant to the purpose are always cited, and in this Clavio himself is content with Euclid’s 27. Prop 1., which (well understood with the adjacent ones) removes the veils to what has been ignored: a cause that has reasonably moved and required to demonstrate that angles are not of continuous quantity, but of discrete one; and consequently, divisible into pairs and odd numbers, like 2. 4. 8. &c and like 3. 5. &c., because the understanding perceives it this way and can perceive it in the Science of the Sword, which does not submit to the absurdity that angular divisions are by bifurcation, but by gradual numbering, as it is convenient for this Science for its understanding and exercise, and for more teaching, and b
The angle, whose interval includes an arc in the quadrant, or semicircle, is divisible into even or odd angles, according to the gradual number that admits division: and the right and complementary sines will be proportional to each other, according to the degrees and minutes, which the secants consist of on the Periphery, because all sines are constituted from perpendicular lines that fall on the semi-diameter, being Parallel among themselves, like the subtendents, or subtenses, proportioning the tangent sines, and secants, according to the graduation of the arcs; and consequently, the cords, or subtenses Parallel to the diameter, the closer they are to it, the larger they are; and the farther apart they are, the smaller they are; and drawing, and constituting in a number of equal degrees, as distances cause, the closer they are to the diameter, the more separated they are; and the farther they are from it, the closer they are to each other: and extended the diameter, and drawn Parallel Diagonals from end to end of the cords, they will end in the extended diameter; in such a way, that the terminations will be equal to the cords, from whose ends they are produced, passing through the ends of their predecessors, as all is demonstrated by figure 13 of Stamp 3.
In the circle A.B.C.D. of this figure, whose center E. the quadrant D.A. 90. degrees from 9 to 9. its half the arc N.D. 45. degrees: the angle N.E.D. wants to be divided into 5, and it is achieved with the secants E.O-E.P-E.Q-E.R-E.S because each interval D.S-S.R-R.Q-Q.P-P.O. are terms in the tangent D.O. of the secants, which runs the arc D.N. the S. in 9. degrees, the R. in 18. Q. in 27. P. in 36. O. in 45. which are 5. equal divisions of the angle N.E.D each one includes 9. degrees of the arc V.N. 45. degrees: then it follows from the 26th and 27th of the 3rd of Euclid, that the arcs are equal, in which they are cut by the secants; and consequently, the S. is perpendicular to the very short extension between D. and the last term of degree 18. at the point, which is cut by the straight line E.S. and so it results by the 9th of the 1st and in the same way E.R. secant is perpendicular, and bifarious, it cuts the arc by the degree 9. to the 27. in which it is perpendicular by the 9th and 10th Prop. of the 1st of Euclid; and so, it can be demonstrated in the divisions, being the ones demonstrable proofs of the others: then the intention of dividing any given angle, in equal parts, even, or odd, is achieved, being demonstrated by those cited in the previous Theorems; and consequently, the perpendiculars on the semi-diameter E.D. from the points of arc, at which the grades 9. 18. 27. 36. 45. are terminated are right sines, which are found in the tables, according to the computation of the radii, or total sine with the tangents, and secants, which correspond to them (which is not from here its theory, nor its practice) with which we refer to the common tables, the more adjusted, the more number is made the computation, or following that of Pitis (which is one of the most modern) the right sine of 9. degrees is 17364. 82. and the tangent 17632. 70. and the secant 1010542. 661. and to the proportion the complements, equaling the right sine, and of complement in degree 45. to which corresponds 70710. 68. and the tangent, as the radius, or total sine, or semi-diameter, computed in 100000000. whose secant 141421. 36. with which we have demonstrated, that the angles are divisible, not only in even parts, but in equal odds, like 4. 8. 16. &c. pairs, and 3. 5. 7. 9. &c. odds.
Therefore, in the proposed circle A.B.C.D with the diameters A.C-B.D intersecting at right angles at the center E. draw parallels H.F, which is the subtense of the arc H.B.F, 30 degrees away from the diameter A.C., and likewise the parallel I.G., 30 degrees of arc away from the H.F. It follows that the diameter A.C. is greater than H.F. and H.F. is greater than I.G. by Prop. 8.3, explained by Tartalla, and Campano. It also follows that the parallels to the diameter, the farther away they are from it, the closer they become to each other, as Ptolemy broadly proves in the Almagest, demonstrating the diversity of artificial days, and to our purpose for this science of the sword, extend the diameter A.C. to K, which is enough distance for the parallels to terminate, which are drawn from extreme points to the extremes of the parallels to the diameter, which are subtenses of 30 by 30 degrees, like H.C.I.F-B.G. When produced to terminate on the diameter, they will be at the points C.L.K., whose distances are equal to the lines, through whose extremes they pass to terminate on the diameter A.C. produced by the first petition of the first of Euclid. It follows that H.C., which terminates at C., includes the same diameter A.C., because the cause of its termination is the extreme C., and therefore, being parallel to H.C., the straight line I.F. does not terminate on the diameter, produced at the point F. passes to terminate at the point L., and because the extreme of its causation is F., a point of H.F. from the term C. to the term L. is equal C.L. to H.F., therefore, Parallelogram H.F.L.C. has equal opposite sides H.F. to C.L. and H.C. to F.L. and by the same structure the extreme G. of the I.G. drawn parallel B.G. to I.F. passes to terminate at the point K. of the diameter A.C. produced at L. and from L. to K., point of the termination of B.G.K., and therefore, as C.L. is equal to H.F., L.K. is equal to I.G., whose extreme G. was the point of its causation, passing to terminate at point K. Likewise, Parallelogram I.G.K.L. has equal opposite sides, recognizing to which terminating points the tip of the sword can be directed, and at what distances it will reach more or less, depending on the locations and movements, and what angles are formed in the different positions, and the other consequences that this theorem produces in this science.
It is concluded, therefore, how numbers are understood, what are numerals, what are numbered, and what are numerating, what we number, what are certain, what are uncertain, and for what reasons; how the angle is not a fourth kind of continuous quantity, but is included in the discrete, how the value of angles is recognized and graded, and their composition of right and oblique angles, how they are divisible, and how in equal parts, not only even (as practical geometers concede) but also odd, as the speculative ones have found, reaching the mysteriousness of different propositions of Euclid, and especially the repeated ones in this predicament, and what is the unit predicament in this science of the sword, etc. with which it only remains in this predicament to hint at what is the other kind of discrete quantity, which the philosopher called Speech, which, as it is not proper to this science, but to Rhetoric, Poetry, and Music, we will only say that Speech (according to philosophers) is considered in the composition of syllables, periods, sentences, verse, and song, which is all of discrete quantity, because the means are not similar parts of the extremes, but differences between them, with which harmony is composed, whether by voices or by numbers, finding the reason and cause of consonance, and dissonance, which, as it is not of our concern, is omitted, referring the curious to the professors of Oratory, Poetry, and Music.
