CAPITVLO QVINTO

EN QUE SE INCLUYEN LAS DEFINICIONES Geometricas aplicadas à la Deſtreza del regimen de la Eſpada

CHAPTER FIVE

WHEREIN THE GEOMETRIC DEFINITIONS ARE INCLUDED Applied to the Skill of the regimen of the Sword

En la delcaracion de los Terminons proprios, y neceſſari0s para la inteligencia del regimen de eſte Arte, y ſu enſeñanza, dexamos definido, que la ciencia de la Eſpada, que vulgarmente ſe dize de la Deſtreza de las Armas, es la que enſeña à dar herida, y defender la del contrario, ſea con Eſpada ſola, ò con las demàs Armas que la ſuelen acompañar.

In the clarification of the proper and necessary terms for understanding the regimen of this Art and its teaching, we have defined that the science of the Sword, commonly referred to as the Skill of Arms, teaches how to deliver a blow and defend against one from the opponent, whether with the Sword alone or with the other Weapons that usually accompany it.

Los medios con que ſe pretende conſeguir eſto, ſon movimientos, aſsi del cuerpo, como del brazo, y Eſpada, y todos juntos eſtos movimientos componen la que llaman treta; y aſsi la treta la difinimos, ò deſcrivimos en la forma figuiente.

The means by which this is sought are movements, both of the body and of the arm and Sword. All these movements together comprise what they call a tactic; thus, we define or describe the treta in the following manner.

Treta es vn compueſto de movimientos de qualquiera de los dos combatientes, dirigidos à la defenſa propria, y ofenſa del contrario.

A tactic is a combination of movements from either of the two combatants, directed at one’s own defense and the offense of the opponent.

La treta que ſe formare de movimientos cientificos, y adquados, para conſeguir el fin que ſe pretende, ſerà perfecta, ò verdadera; y la treta à quien faltaren los requiſitos neceſſarios para ſu perfecta formation, fe llamarà falſa.

The tactic that is formed from knowledgeable and suitable movements to achieve the intended goal will be perfect or true; and the tactic that lacks the necessary requirements for its perfect formation will be called false.

El modo que hemos de tener para hazer juizio de la perfecion, ò imperfeccion de la treta, ſerà examinar todas las partes de que ſe compone, que ſon los movimientos referidos.

The way we must judge the perfection or imperfection of the treta is to examine all its components, which are the aforementioned movements.

Tambien dexamos definido, que el movimiento, ſegun Ariſtoteles en el 3. de los Phyſicos, es vna acciopn del motor en la coſa movida.

We also defined that movement, according to Aristotle in the third book of Physics, is an action of the mover upon the moved thing.

Para alcanzar el conocimiento de eſta accion en las Armas, neceſsitamos de conocer primero el agente, ò motor, que es el Dieſtro, ò ſu adverſario, y la coſa movida, que es la Eſpada, y en quantos modos puede moverla, juntamente con las demàs partes del cuerpo, en orden à la formacion de alguna treta; y luego inmediatamente ver la concordancia, y diſcorandcia, que puede aver entre eſtos movimientos, aſsi del cuerpo, como ſus partes.

To grasp the understanding of this action in Weapons, we need to first know the agent or mover, which is the Swordsman or his opponent, and the moved thing, which is the Sword, and in how many ways he can move it, along with the other parts of the body, in order to form some treta. Then, immediately, we must see the harmony and discord that may exist between these movements, both of the body and its parts.

Ariſtoteles en 3. de los Phyſicos, hablando del movimiento, dize, que parece ſer del generl del continuo; y el continuo propriamente pertenece à la cantidad continua, cuyas eſpecies ſon Lineas Superficies, y Cuerpos: y aſsi parece, que Ariſtoteles con eſtas palabras nos quiſo inſinuar, que la eſpeculacion del movimiento pertenece à la Geometria, llave de las ciencias Mathematicas, cuyo oficio es tratar de la cantidad continua, y eſpecular ſus propriedades: y eſto claramente ſe dexa entender, por ſer tan conjunto el movimiento, y la cantidad, que no hallamos coſa que pertenezca à la cantidad, que pueda ſer movida ſin que conſu movimiento cuaſe cantidad Phyſica, ò imaginaria; porque ſi es el punto el que ſe moviere, con ſu movimiento cauſarà linea recta, circular, ò mixta, ſegun la naturaleza de ſu movimiento.

Aristotle in the third book of Physics, speaking of movement, says that it seems to belong to the category of the continuous; and the continuous properly pertains to continuous quantity, whose species are Lines, Surfaces, and Bodies. Thus, it seems that Aristotle with these words wanted to suggest that the speculation of movement pertains to Geometry, the key to the mathematical sciences, whose role is to discuss continuous quantity and speculate its properties. This is clearly implied, since movement and quantity are so intertwined that we find nothing that belongs to quantity that can be moved without its movement causing a physical or imaginary quantity; for if it is a point that moves, its movement will cause a straight, circular, or mixed line, depending on the nature of its movement.

Si es linea la que ſe mueve lateralmente, cauſarà con ſu movimiento vna ſuperficie, plana, eſpherica, ò mixta, ſegun la naturaleza de la linea, y de ſu movimiento.

If it’s a line that moves laterally, with its movement, it will cause a flat, spherical, or mixed surface, depending on the nature of the line and its movement.

Si fuere ſuperficie la que ſe moviere, cauſarà cuerpo, contenido de ſuperficies, planas, eſpherica, ò mixta, ò parte de vnas, y parte de otras.

If it’s a surface that moves, it will produce a body, containing flat, spherical, or mixed surfaces, or parts of some and parts of others.

Si fuere cuerpo el que ſe moviere de lugar à lugar, cauſarà cuerpo de diferente figura de la que èl tuviere.

If it’s a body that moves from place to place, it will create a body of a different shape than it had.

Las tres primeras conſideraciones, del punto, de la linea, y de la ſuperficie, ſon muy neceſſarias en la Deſtreza, como ſe mueſtra en ſu lugar; la vltima, que toca al cuerpo, no tiene vſo en eſta ſciencia, porque aunque los cuerpos de los dos combatientes ſe han de mover preciſamente para forma ſus tretas, no conſderamos los cuerpos, que cauſan los movimientos, ſino que mueven por algunas lineas rectas, ò curvas, ſegun la forma del movimiento: imitando en eſto à los Aſtronomos, que aunque conſideran moverſe aquellos cuerpos celeſtes dentero de ſus Orbes, para regular ſus movimientos ſe valen de las lineas, y no de los cuerpos; lo miſmo hazen los que tratan de la navegacion, que para regular el movimiento del Navio en la ſuperficie de las Aguas, ſolo ſe valen de lineas, à que llaman rumbos.

The first three considerations, of the point, the line, and the surface, are very necessary in Fencing, as shown in its place; the last, which pertains to the body, is not used in this science because although the bodies of the two combatants must necessarily move to form their tactics, we do not consider the bodies that cause the movements, but that they move along certain straight or curved lines, according to the form of the movement. In this, they imitate the astronomers, who, although they consider those celestial bodies moving within their orbits, to regulate their movements they rely on lines, not on bodies. The same is done by those who deal with navigation; to regulate the movement of the ship on the surface of the waters, they rely solely on lines, which they call bearings (rumbos).

Y pueſto que todos los que han tratado de movimientos, aſſi violentos, como naturales, lo han hecho debaxo de la eſpeculacion de la cantidad continua, ſerà bien que en eſto los imitemos: que ſi ellos por medio de eſtas imaginaciones han podido regular los movimientos de los Aſtros en los Cielos, y el de los Navios en el Agua, no ſerà mucho, que valiendonos de los miſmos medios, nos prometamos poder regular el movimiento de la Eſpada en el Ayre, y el del cuerpo en el Suelo.

And given that all who have dealt with movements, both violent and natural, have done so under the speculation of continuous quantity, it will be good for us to imitate them in this. For if, through these imaginations, they have been able to regulate the movements of the stars in the heavens and the movement of ships in the water, it won’t be much to hope that, using the same means, we can regulate the movement of the sword in the air and the movement of the body on the ground.

Siendo eſto tan claro, que no admite controverſia, pondremos todos nueſtro esfuerzo en buſcar el camino de facilitar la Deſtreza de las Armas, por medio de las Mathematicas, moſtrando el vſo de algunas lineas, ſuperficies, y cuerpos, las quals nos ſirvan de guia, y norte para ſu perfecta inteligencia: mas antes de poderlo hazer, declaràmos algunas Definiciones tocantes à la Geometria, aunque en los Terminos proprios dexamos eſpecificado gran parte de ellas; pero ſin figuras, pro el embarazo que ſe ſeguia: y en eſtas, que aora declaramos, es neceſſario ponerlas; para que luego le moſtremos al Dieſtro la aplicacion que tienen en la Deſtreza de las Armas.

This being so clear that it does not admit controversy, we will put all our effort into finding the way to facilitate the Art of Fencing through Mathematics, showing the use of some lines, surfaces, and bodies, which will serve us as a guide and north star for its perfect understanding. But before we can do so, we define some definitions related to Geometry, even though in the proper terms we have already specified a great part of them; but without figures, because of the complications that ensued. And in these definitions we now declare, it’s necessary to include them, so that later we can show the fencer their application in the Art of Fencing.

Definicion del Punto.

1. Punto (ſegun Euclides Defin. 1. lib. 1. de ſus Elementos) es cuya parte es ninguna.

1. A Point (according to Euclid, Defin. 1. Book 1. of his Elements) is that which has no part.

Definition of the Point.

Dos diferencias de Punts ſe hallan entre los Mathematicos, el vno Mathematico, ò imaginario, como eſſencia ſin cantidad, abſtrahido de materia, y ſolo objetado por el entendimiento; y el otro Phyſico Real, ò material, que por pequeño que ſea ſe puede dividir en partes ſiempre diviſibles: ſeñalale le letra A. en la figura del numero 1.

Among mathematicians, two types of Points are recognized: one is Mathematical or imaginary, an essence without quantity, abstracted from matter, and solely objectified by the understanding; the other is Physical, Real, or material, which, however small, can always be divided into divisible parts. This is marked with the letter A in figure number 1.

En la Deſtreza de las Armas tambien conſideramos dos diferencias de Puntos, vnos Phyſicos, que ſe conſideran en las puntas de las Eſpads; y otros imaginarios, ò Mathematicos, que ſe conſideran en la interſeccion de ciertas lineas imaginarias, aſsi en la plano inferior (que ſe entiende por el Suelo) como en los cuerpos de los dos combatientes, ò en el Ayre, con las lineas que conſideramos paſſar por los brazos, y Eſpadas de los dos combatientes.

In the Art of Fencing, we also consider two types of Points: Physical ones, which are considered at the tips of Swords; and imaginary or Mathematical ones, considered at the intersection of certain imaginary lines, both on the ground plane (understood as the Floor) and on the bodies of the two combatants, or in the Air, with the lines we consider passing through the arms and Swords of the two combatants.

Definicion de la Linea.

2. Linea es (ſegun Euclides) vna longitud ſin latitud. Aſsi como el Punto ſe conſidera de dos maneras, el vno Phyſico, y el otro Mathematico; tambien la Linea ſerà Phyſica, ò Mathematica: por que ſi la Linea, ſegun Platon, es el fluxo, ò movimiento de el Punto, ſi el Punto fuere Phyſico, la Linea que cauſare ſerà Phyſica; y ſi el Punto fuere Mathematico, abſtracto de materia, tambien la Linea ſerà Mathematica.

2. A Line (according to Euclid) is a length without breadth. Just as the Point is considered in two ways, one Physical and the other Mathematical, the Line will also be either Physical or Mathematical. For if the Line, according to Plato, is the flow or movement of the Point, if the Point is Physical, the resulting Line will be Physical; and if the Point is Mathematical, abstract from matter, the Line will also be Mathematical.

Definition of the Line.

De la Linea recta

Linea recta es la mas breve extenſion de vn Punto à otro, la menor de todas las Lineas, que tienen vnos miſmos terminos, ò lo que eſtà igualmente entre ſus Puntos: como por exemplo, la Linea B.C. que causò del movimiento recto del Punto B. haſta el Punto C. Donde los limites, ò extremidades de la Linea, ſon Puntos, como B. y C. y ſe vè en la figura del num 2.

A straight Line is the shortest extension from one Point to another, the smallest of all lines that have the same endpoints, or that which lies evenly between its Points. For example, the Line B.C., which resulted from the straight movement from Point B to Point C. The limits or ends of the Line are Points, like B and C, and are seen in figure number 2.

Of the Straight Line.

En la Destreza de las Armas conſideramos la Eſpada, Daga, y Montante, ò ſus filos, como linea Phyſica; y ſiempre que ſe moviere la punta de la Eſpada en parte adonde pueda dexar ſeñal, ò veſtigio de ſu movimiento: como por exemplo, en el ſuelo, ò contra alguna pared, ò en el cuerpo de ſu adverſario, la linea que cauſare con ſu movimiento ſerà Phyſica: y ſiempre que ſe moviere en el Ayre, aunque en rigor parece avida de ſer Phyſica, la llamaremos imaginaria, por no caer debaxo del ſentido de la viſta, ſino de la imaginacion; y la tal linea ſerà recta, circular, ò mixta, ſegun la naturaleza del movimiento.

In the Art of Fencing, we consider the Sword, Dagger, and Montante, or their edges, as a Physical line. Whenever the tip of the Sword moves to a place where it can leave a mark or trace of its movement, such as on the ground, or against a wall, or on the body of its opponent, the line it causes with its movement will be Physical. And whenever it moves in the Air, even though strictly speaking it seems to be Physical, we will call it imaginary because it falls not under the sense of sight but under imagination; and such a line will be straight, circular, or mixed, according to the nature of the movement.

De la Curva

3. Linea curva es la que forma vn Arco, y no eſtà en derecho, como lo demueſtran las letras D. y E. num 3.

3. A curved Line is one that forms an Arch, and is not straight, as demonstrated by the letters D. and E. in figure number 3.

Of the Curved Line.

En la Deſtreza la conſideramos quando ſe dobla el brazo ſin hazer Angulo en la ſangria, y quando el cuerpo ſe pone à manera de Arco, inclinandole àzia delante, ò àzia los lados, haziendo el Arco en la profundidad; y tambien ſe conſidera en el plano inferior, y particularmente en los compaſes, que de ella toman ſu denominacion.

In the Art of Fencing, we consider it when the arm bends without making an Angle at the cut, and when the body is positioned in an arch-like manner, leaning forward or to the sides, making the Arch in depth; it is also considered on the ground plane, and particularly in the compasses, which derive their name from it.

De la Mixta

4. Linea mixta es la que ni eſtàn recta, ni circular: como por exemplo, la linea F.N. es recta, y la N.G. es curva, como ſe vè en la figura del numero 4.

4. A mixed Line is one that is neither straight nor circular. For example, the line F.N. is straight, and N.G. is curved, as seen in figure number 4.

Of the Mixed Line

En la Deſtreza ſe forma con la Eſpada, y brazo, quando eſte ſe dobla ſin hazer Angulo en la ſangria; y tambien en el plano inferior para los compaſes mixtos, y para el movimiento de concluſion, dando por ella el compàs mixto de tranſverſal, y curvo.

In the Art of Fencing, it is formed with the Sword and arm when the arm bends without making an Angle at the cut; it is also seen on the ground for mixed compasses and for the concluding movement, representing it by the mixed compass of transversal and curved.

De la Flexuoſa

5. Linea flexuoſa, tortuoſa, ò Eliaca, es aquella que và ſerpeando como el Rio, ò culebra: divideſe en dos eſpecies, vna caſi entendida, y otra rebuelta; la entendida H.I. la rebuelta K.L. numero 5.

5. A wavy, tortuous, or Eliac Line is one that snakes like a River or serpent. It is divided into two types: one almost straight, and another that is winding; the almost straight one is H.I. and the winding one is K.L. in figure number 5.

Of the Wavy Line.

En la Deſtreza ſe conſidera la H.I. quando la punta de la Eſpada con movimiento accidental, ò extraño, que vayan mixtos con los remiſos, deſcribe pro debaxo de la otra Eſpada porciones de lineas curvas à vn lado, y à otro; y la K.L. quando tambien con movimiento accidental, ò extraño, que ſe mixto con otros, và formando lineas curvas, ſin ſecacion en el curſo, ò yà que con la Eſpada vaya incluyendo à la de ſu adverſario, para no dexarla libre.

In the Art of Fencing, H.I. is considered when the tip of the Sword, with an accidental or strange movement mixed with slackening movements, describes portions of curved lines underneath the other Sword, to one side and the other; and K.L. is considered when also with an accidental or strange movement mixed with others, it forms curved lines without interruption in its course, or even when using the Sword to encompass the opponent’s Sword to restrict its freedom.

De la Eſpiral

6. Linea eſpiral, ò voluta, es la que à manera de caracol ſe và rodeando à algun cuerpo colunar, ſin bolver à la parte do començò: ſemejante à las bueltas que dà el Sol entre la Equinocial, y los dos Tropicos; ſeñalaſe con letra M. num. 6.

6. A spiral or volute Line is one that, in the manner of a snail, wraps around some columnar body without returning to where it began, similar to the turns the Sun makes between the Equator and the two Tropics. It is marked with the letter M in figure number 6.

Of the Spiral Line

In the Art of Fencing, it is considered in many aspects and propositions, such as when forming the two general movements, weakness above and below strength; and when the fencer’s Sword follows his opponent’s from the inside continuously, until making a concluding movement, ending up striking, forming the mixed angle. It is also seen when having blocked on the outside at the nearest end of the opponent’s Sword, if the opponent lifts his arm and guard, the fencer will envelop and wrap the Swords on that side, until leaving the opposing Sword below his right arm and his own striking above the jurisdiction of the arm, simultaneously making a concluding movement. Thus, this spiral line can be formed both from the inside and the outside of the Sword.

En la Deſtreza ſe conſidera en muchas partes, y propoſiciones, como quando ſe forman las dos generales, flaqueza encima, y debaxo de la fuerza; y quando la Eſpada del Dieſtro figue à la de ſu contrario por la parte de adentro vnitivamente, haſta haer movimiento de concluſion, quedando hiriendo, formando el Angulo mixto: y tambien quando eſtando pueſto el atajo por de fuera en el extremo propinquo en la Eſpada del contrario, ſi eſte levantare el brazo, y guarnicion, incluirà, y rodearà el Dieſtro las Eſpadas por aquella parte, haſta dexar la contraria debaxo de ſu brazo derecho, y la fuya hiriendo por encima de la juriſdiccion del brazo, haziendo al miſmo tiempo movimiento de concluſion; y aſsi eſta linea eſpiral ſe puede formar tomando la Eſpada por la parte de adentro, y la de afuera.

De las rectas

7. Lineas rectas paralelas ſon las que tiradas en infinito, y en vn miſmo plano, jamàs concurren: ſeñalanſe con las letras N.O. y P.Q. numero 7.

7. Parallel straight lines are those that, extended infinitely and on the same plane, never converge. They are represented by the letters N.O. and P.Q. number 7.

Of the straight lines

En la Deſtreza ſe forman quando ſe elige el medio de proporcion, conforme à el Arte, para ir inmediatamente à poner atajo por la parte de adentro; y tambien ſe conſideran en otras muchas propoſiciones, y caſos, aſsi en el plano inferior, como en el ſuperior: y en los cuerpos de los combatientes ſon paralelas las lineas Verticales, Colaterales, y Diametrales.

In fencing, they are formed when choosing the middle proportion, according to the Art, to immediately set a block on the inside; they are also considered in many other propositions and cases, both on the lower plane and on the upper one. In the bodies of the combatants, the Vertical, Collateral, and Diametrical lines are parallel.

De las concurrentes

8. Lineas concurrentes ſon las que tiradas por vn plano, concurren haziendo ſeccion, ſegamento, ò cortadura, como ſe ſeñala con las letras R.S. y T.V. que concurren en X. num 8.

8. Converging lines are those that, drawn through a plane, converge, making a section, segment, or cut, as represented by the letters R.S. and T.V. that converge at X. number 8.

Of converging lines

En la Deſtreza ſe conſideran quando dos lineas, ò Eſpadas ſe agregan, ò ſujeta la vna à la otra, tambien en otras propoſiciones; y à eſtas dos lineas ſe dizen no paralelas.

In fencing, they are considered when two lines or swords join or one is subjected to the other; also in other propositions. These two lines are said to be non-parallel.

De la Elipſis

9. Linea Elipſis es ſemejante à vna figura oval, diviſa en dos partes, con vna linea Diametral, como demueſtra la letra Y. numero 9.

9. Elliptical line is similar to an oval figure, divided into two parts, with a Diametrical line, as shown by the letter Y. number 9.

Of the ellipse

En la Deſtreza la conſideramos en el cuerpo del hombre, en el Arco que haze con el grueſſo del cuerpo, y pecho, porque la eſpalda es caſi llana.

In fencing, we consider it in the human body, in the Arch made with the bulk of the body and chest, because the back is almost flat.

De la hyperbolica

10. Linea hyperbolica, ò parabolica, es la que tiene ſemejanza, ò ſimilitud con algun monte, ò mas propriamente con el monton de trigo: demueſtraſe con la letra Z. del numero 10.

10. The hyperbolic or parabolic line is the one that resembles or is similar to a mountain, or more precisely to a pile of wheat. It is represented by the letter Z. number 10.

Of the hyperbolic line

En la Deſtreza ſirve para la deſcripcion de los compaſes, y ſe conſidera en el viage, aſsi del compàs curvo, mixto de trepidacion y extraño, como del tranſverſal, mixto de trepidacion, y extraño; y tambein en algunas revoluciones, que haze la Eſpada en el Ayre.

In fencing, it is used to describe the steps and is considered in the journey, both of the curved step mixed with trepidation and strange, and of the transversal one, mixed with trepidation and strange; and also in some revolutions made by the sword in the air.

De la circular

11. Linea circular es la que ſe cauſa del movimiento de vn punto, quando ſe mueve ſiempre en igual diſtancia de otro punto de eſtà inmobil: como por exemplo, ſi eſtando firme el punto A. ſe moviere el punto B. ſiempre en igual diſtancia del punto A. la linea que ſe cauſare con ſu movimiento, ſe llamarà circular, numero 11.

11. A circular line is one that results from the movement of a point when it always moves at an equal distance from another point that is immobile: for example, if point A is fixed and point B moves always at an equal distance from point A, the line that results from its movement will be called circular, number 11.

Of the circular line

En la Deſtreza fe forman eſtas lineas circulares con el brazo, y Eſpada, y tambien las conſideramos en los planos ſuperior, è inferior, como ſe explicarà en la aplicacion que adelante harèmos de ellas.

In fencing, these circular lines are formed with the arm and sword, and we also consider them in the upper and lower planes, as will be explained in the application we will make of them later.

Definicion del Angulo

12. Angulos es la inclinaciond e dos lineas que ſe tocan en vn punto, y no eſtan en derecho

12. Angles are the inclination of two lines that meet at a point and are not straight.

Definition of Angle

Definicion del Angulo

13. Angulo Plano rectilineo es el que eſtà contenido de dos lineas rectas, concurriendo como ſeñalan las dos Eſpadas C.E. en punto D. numero 12.

13. A Planar rectilinear angle is one that is contained by two straight lines, converging as marked by the two Swords C.E. at point D. number 12.

Definition of Angle

En la Deſtreza nos valemos del genero generaliſsimo Angulo, como ſe explicarà en cada eſpecie: y el rectlineo le conſideramos en qualquiera parte que las dos Eſpadas ſe cortan, ò tocan el plano del Orizonte; y tambien en el inferior ſe cauſa, aſsi en las lineas que en èl conſideramos, que nos ſirven de rumbos, como quando ſe dan los compaſes tranſverſales, mixtos de trepidacion, y extraño; y en las lineas imaginadas de vn ombro à otro de los combatientes. Para algunas propoſiciones tambien ſe forma Angulo plano rectilineo.

In fencing, we use the most generalized kind of Angle, as will be explained in each species: and we consider the rectilinear one anywhere the two Swords intersect or touch the Horizon plane; and also in the lower one it is caused, both in the lines we consider in it that serve us as directions, and when the transversal steps are given, mixed with trepidation, and strange; and in the imagined lines from one shoulder to the other of the fighters. A planar rectilinear angle is also formed for some propositions.

Curvilineo

14. Angulos planos curvilineos ſon los que ſe componen de lineas curvas, como ſeñalan las letras D. en el vno, E. en el otro, y F. en el tercero, en la figura del numero 13.

14. Planar curvilinear angles are those composed of curved lines, as marked by the letters D. in one, E. in the other, and F. in the third, in the figure number 13.

Curvilinear

En la Deſreza los conſideramos quando ſe batalla con Armas dobles, y tambien en las lineas que en el plano inferior deſcriben los pies para diferentes compaſes.

In fencing, we consider them when fighting with double weapons, and also in the lines that in the lower plane describe the feet for different steps.

Mixtilineo

15. Angulo mixtilineo es el que ſe cauſa del tocamento de linea recta, y curva, como ſeñalan las letras L. y N. que concurren en M. numero 14.

15. A mixtilinear angle is one that results from the touch of a straight line and a curve, as marked by the letters L. and N. that converge in M. number 14.

Mixtilinear

En la Deſtreza, quando ſe encurva el brazo, y con la linea recta, que ſe conſidera en el pecho (à quien llamamos de la contingencia) conſtituyen Angulo en la colateral derecha, ò con el brazo, y Eſpada, conſtituyendole en las empuñaduras: tambien la conſideramos en el plano inferior para los compaſes mixtos.

In fencing, when the arm is curved, and with the straight line, which is considered in the chest (whom we call of contingency) they constitute an Angle on the right side, or with the arm, and sword, constituting it in the grips: we also consider it in the lower plane for mixed steps.

Recto

16. Angulo recto ſe dize quando vna linea recta, cayendo ſobre otra linea recta, hiziere los Angulos de la vna y otra parte iguales entre sì: cada vno de los Angulos ſe llama recto, y la linea ſe dirà eſtàr perpendicular ſobre la otra: como por exemplo, la linea C.D. ſe dize eſtàr perpendicular ſobre la baſis A.B. y haze los Angulos rectos, numero 15.

16. A right angle occurs when a straight line, falling upon another straight line, makes the angles on either side equal to each other: each of the angles is called right, and the line is said to be perpendicular to the other: for example, the line C.D. is said to be perpendicular to the base A.B. and makes the right angles, number 15.

Right

En la Deſtreza ſe conſidera quando no ſe afirma el brazo derecho, como nace en el cuerpo, ſn que participe de alguno de los extremos de arriba abaxo, à vno, ni otro lado, ſiendo el cuerpo la linea A.B. y el brazo, y Eſpada C.D. y tambien ſe conſideran, y forman eſtos Angulos rectos, aſsi en el plano inferior, como en el tocamento de las Eſpadas.

In fencing, it is considered when the right arm is not affirmed, as it arises in the body, without participating in any of the extremes from top to bottom, to one side or the other, the body being the line A.B. and the arm, and sword C.D. and these right angles are also considered and formed, both in the lower plane and in the touching of the swords.

Obtuſo

17. Angulo obtuſo es mayor que recto, pues ſube la Eſpada deſde la letra C. que ſe conſideraba recto, haſta la letra B. num. 16.

17. An obtuse angle is greater than a right angle, as the sword rises from the letter C, which was considered a right angle, to the letter B, num. 16.

Obtuse

En la Deſtreza ſe conſidera quando el brazo ſube à la rectitud alta derechamente, ò inclinandoſe juntamente à qualquiera de los lados; demanera, que el brazo, y cuerpo no eſtèn paralelos al plano del Orizonte, como ſenala la C.D. antes ſe hallen como ſe demueſtra con la D.B.

In Fencing, it is considered when the arm raises to the high straightness directly, or leaning together to either side; in a way, that the arm and body are not parallel to the Horizon plane, as C.D. indicates, but are found as demonstrated with D.B.

Agudo

18. Angulo agudo es meor que recto, pues baxa la linea del brazo, y eſpada deſde la letra C.D. donde ſe conſideraba ſer recto, haſta la letra B. donde es agudo, num. 17.

18. An acute angle is less than a right angle, as the line of the arm and sword drops from the letter C.D. where it was considered to be right, to the letter B, where it is acute, num. 17.

Acute

En la Deſtreza ſe hazen con èl las poſturas baxas; deſuerte, que el brazo no eſtà del todo vnido al cuerpo, ni la mano, y Eſpada al muslo, como ſeñala la letra B.

In Fencing, low postures are made with it; so, the arm is not fully united to the body, nor the hand and Sword to the thigh, as indicated by the letter B.

Definicion de la Superficie

19. Superficie es la que tiene longitud, y latitud tan ſolamente, y del miſmo modo que conſideramos la linea cauſada del movimiento del punto; aſsi tambien conſideramos la Superficie cauſada del movimiento de la linea: y como la linea ſe divide en tres diferencias, recta, circular, y mixta, ſegun la naturaleza del movimiento del punto; tambien la Superficie ſe dividirà en tres, ſegun la diferencia del movimiento de la linea, que ſe llama plana, eſpherica, ò mixta, y los extremos de la Superficie ſon lineas.

19. Surface is that which has only length and breadth, and in the same way that we consider the line caused by the movement of the point; we also consider the Surface caused by the movement of the line: and as the line is divided into three differences, straight, circular, and mixed, according to the nature of the movement of the point; the Surface will also be divided into three, according to the difference in the movement of the line, which is called flat, spherical, or mixed, and the ends of the Surface are lines.

Definition of the Surface

De la Superficie plana

20. Superficie plana es la que eſtà igualmente entre ſus linea, y ſe cauſa del movimiento lateral de la linea recta, quando ſe mueve igualmente: como por exemplo, ſupongamos que la linea A.B. ſe movieſſe toda deſde el lugar en que ſe halla, haſta D.C. igualmente, y rectamente por el camino mas breve, ella avrà cauſado vna ſuperficie plana, terminada de quatro lineas recta A.B.-A.C.-C.D. y D.B. y à eſte genero de Superficies llamamos planos, para mayor brevedad.

20. A flat surface is one that is equally between its lines and is caused by the lateral movement of the straight line when it moves equally: for example, let’s suppose that the line A.B. moved entirely from where it is, to D.C. equally, and directly by the shortest path, it will have caused a flat surface, bounded by four straight lines A.B.-A.C.-C.D. and D.B. and we call this type of Surfaces flat, for brevity.

Of the Flat Surface

En la Deſtreza ſe demueſtra eſta Superficie en el plano inferior, y tambien en el ſuperior, imaginando de los ombros de vn combatiente à otro linea de tocamento, pues quando ſe miran roſtro à roſtro los cuerpos de quadrado, y las Eſpadas en lineas paralelas, mirando las puntas à los ombros izquierdos, y conſiderando en ellas punto de tocamento, avràn formando vna Superficie plana.

In Fencing, this Surface is demonstrated in the lower plane, and also in the upper one, imagining a contact line from one combatant’s shoulders to another’s, for when they face each other, the bodies are squared, and the Swords in parallel lines, pointing the tips to the left shoulders, and considering in them a point of contact, will have formed a flat Surface.

Tambein las ſuperficies, ſi ſe componen de lineas curvas, como los arcos de los Edificios, les diremos por la parte inferior, que ſeñala la letra A. concava; y por la parte ſuperior, que eſtà la letra B. convexa, como ſe demueſtra en la figura del numero 19.

Also, the surfaces, if composed of curved lines, like the arches of Buildings, we will call them concave on the lower part, which is indicated by the letter A, and convex on the upper part, where the letter B is, as shown in figure number 19.

En la Deſtreza ſe conſidera eſta ſuperficie quando el cuerpo ſe extrema àzia adelante, ò atràs, ò à los lados, y tambien en la deſcripcion de los compaſes curvos, que ſi el pie dexàra raſtro, ſe viera formada eſta ſuperficie con perfeccion.

In Fencing, this surface is considered when the body leans extremely forward, or backward, or to the sides, and also in the description of curved compasses, which if the foot left a trail, this surface would be perfectly formed.

Tambien la linea recta cauſarìa ſuperficie plana, ſi eſtando firme la vna extremidad, la otra ſe movieſſe al rededor de ella; como en eſta figura en ſe ſigue, que ſe llama circulo, y le define Euclides de eſta manera.

Also, the straight line would cause a flat surface, if, with one end fixed, the other moved around it; as in the following figure, which is called a circle, and is defined by Euclid in this way.

Definicion del Circulo

21. Circulo es vna figura contenida de vna ſola linea, que ſe llama circunferencia, ſeñalanla las letras A.D. à la qual deſde vn punto que eſtà dentro de ella, todas las lineas que ſalen à la circunferencia, ſiendo rectas, ſon iguales entre sì; y eſte punto ſe llama centro, como ſeñala la letra B. en el numero 20.

21. A circle is a figure contained by a single line, which is called circumference, indicated by the letters A.D. to which, from a point within it, all the lines that go to the circumference, being straight, are equal to each other; and this point is called the center, as indicated by the letter B. in number 20.

Definition of the Circle

En la Deſtreza conſideramos diferentes circulos: como ſon, el ſuperior, el inferior, comun, particular, y otros, ſegun las poſiciones de los combatientes, y las proporciones que ſe hazen, como adelante explicarèmos.

In Fencing, we consider different circles: such as the upper, the lower, common, particular, and others, according to the positions of the combatants, and the proportions that are made, as we will explain later.

De Diametro

22. Diametro del circulo es vna linea, que paſſando por el centro, ſe termina por ambas partes en la circunferencia, y divide el circulo en dos partes iguales, como ſeñalan las letras A. y D. y la mitad del Diametro ſe llama Semidiametro, numero 21.

22. The diameter of the circle is a line, passing through the center, terminates on both sides in the circumference, and divides the circle into two equal parts, as indicated by the letters A. and D. and half of the Diameter is called the Radius, number 21.

Of Diameter

En la Deſtreza conſideramos la linea del Diametro, aſsi en el plano inferior, como en el ſuperior, y en el cuerpo del que combate.

In Fencing, we consider the line of the Diameter, both in the lower plane and in the upper one, and in the body of the combatant.

Del Semicirculo

23. Semicirculo es lo miſmo que medio circulo, por ſer figura contenida del Diametro, y de la mitad de la circunferencia, ſeñalaſe con la L.Y. numero 22.

23. A semicircle is the same as half a circle, being a figure contained by the Diameter, and half of the circumference, indicated by L.Y. number 22.

Of the Semicircle

En la Deſtreza le conſideramos, aſsi en la formacion de muchas tretas, como en la execucion de algunas heridas.

In Fencing, we consider it, both in the formation of many tricks and in the execution of some blows.

Porcion de circulo

24. Porcion de circulo es vna figura contenida de vna linea recta, y de vna parte de la circunferencia mayor ò menor, que la mitad de ella: llamaſe porcion, ò ſegmento mayor, la ſeñalada con A.B.y porcion, ò ſegmento menor, la ſeñalada con C.D. numero 23.

24. A portion of a circle is a figure contained by a straight line and part of the circumference, greater or smaller than half of it: it is called a larger portion or segment, indicated with A.B. and a smaller portion or segment, indicated with C.D. number 23.

Portion of Circle

En la Deſtreza tambien conſideramos eſtas porciones, para la formacion de tretas, y execucion de heridas, ſegun el viage que anduviere la Eſpada, como todo ſe manifieſta en el Tratado de las tretas, y en otras muchas propoſiciones, en que ſe verà ſu aplicacion.

In Fencing, we also consider these portions for the formation of strategies and execution of strikes, depending on the path that the Sword follows, as everything is manifested in the Treatise of strategies, and in many other propositions, where its application will be seen.

Quarta parte de circulo

25. Quarta parte de circulo es vna de las quatro en que ſe divide toda la circunferencia, ſeñalaſe con las letras A.B.C. en el numero 24.

25. A quarter of a circle is one of the four parts into which the entire circumference is divided, indicated by the letters A.B.C. in number 24.

Quarter Circle

En la Deſtreza firve quando tomando la Eſpada cel contrario por la parte de adentro, ſe hiere con eſtocada en la vertical derecha: y quando tomandola por la defuera, ſe hiere por encima de la Eſpada en la Diametral del pecho: tambien ſe conſidera eſta quarta parte de circulo en la formacion de otras heridas, y en el plano inferior, pues ſe divide el circulo comun en quatro, como tambien el conſiderado en el pecho del hombre.

In Fencing, it is useful when, by taking the opponent’s Sword from the inside, one strikes with a thrust in the right vertical: and when taking it from the outside, one strikes above the Sword in the Diametral of the chest: this quarter of a circle is also considered in the formation of other strikes and in the lower plane, as the common circle is divided into four, as is the one considered in the man’s chest.

Circulos concentricos

26. Circulos concentricos ſon los que ſe deſcriben con ſoo vn centro, como A. y B. que ſe deſcriben con el centro M. num 25.

26. Concentric circles are those described with only one center, like A. and B. that are described with the center M. num 25.

Concentric Circles

En la Deſtreza ſe vèn patentes eſtos circulos, pues los maximos comprehenden à los que ſe cauſan de la revolucion, que cauſan las diviſiones de los brazos, y Eſpadas, y ſon todas concentricos; los quals firven de norte ò guia, aſsi para regula las medidas de los medios de proporcion, como proporcionados, dando por ellos cantidad fixa à los compaſes, como ſe explicarà en ſu lugar.

In Fencing, these circles are evident, as the maximums include those that are caused by the revolution, which is caused by the divisions of the arms, and Swords, and all are concentric; which serve as a guide or compass, both to regulate the measures of the means of proportion, as proportioned, giving them a fixed quantity to the compasses, as will be explained in its place.

Circulos excentricos

27. Circulos excentricos ſon los que tienen diverſos centros, y el mayor comprehende al menor en eſta figura, porque el mayor tiene por centro al punto A. y el menor al punto B. el qual intervalo es comprehendido del mayor, num. 26.

27. Eccentric circles are those that have different centers, and the larger one includes the smaller one in this figure, because the larger one has the point A. as its center and the smaller one has the point B. which interval is included by the larger one, num. 26.

Eccentric Circles

En la Deſtreza ſe conſideran quando fueren las Eſpadas, y cuerpos deſiguales que el circulo que deſcribiere el mayor, y con la mayor comprehenderà al que deſcribiere la menor: y tambien ſe conſider en el plano inferior, pues el circulo, à quien damos ombre de Maximo, comprehenderà al comun, que ſe imagina entre los pies derechos de los combatientes, t tienen diferentes centros.

In Fencing, they are considered when the Swords, and bodies are unequal, as the circle described by the larger one, and with the larger one, will include the one described by the smaller one: and it is also considered in the lower plane, as the circle, to whom we give the name of Maximum, will include the common one, which is imagined between the right feet of the combatants, and they have different centers.

Circulos contingentes

28. Circulos contingentes ſon aquells que tocandoſe en vn punto, no ſe cortan como los circulos A. B. del numero 27.

28. Tangent circles are those that, touching at one point, do not intersect, like the circles A. B. of number 27.

Tangent Circles

In Fencing, these circles are manifested in the Orbs, which, from the divisions of the arms, and Swords of the combatants, we consider in the lower plane; thus the Orb or circle described by the right-handed fencer with the tip of his Sword, is tangent to the one described by the opponent’s arm; and in this regard, in the other divisions of the Sword, as will be seen in its proper place.

En la Deſtreza ſe manifieſtan eſtos circulos en los Orbes, que de las diviſiones de los brazos, y Eſpadas de los combatientes, conſideramos en el plano inferior; pues el Orbe, ò ciruclo que deſcribe el Dieſtro con ſu punta de Eſpada, es contingente al que deſcribe el brazo del contrario; y à eſte reſpecto en las demàs diviſiones de la Eſpada, como ſe verà en ſu proprio lugar.

Tambien ſe vèn com perfeccion eſtos circulos en la formacion del tajo, y revès verticales, que en ſu execucion ſe tocan en la vertial de enmedio de el Zenit E. que baza por la frente, haſta el pecho, y fuelo F. como ſe demueſtra en dicho numero 27.

These circles are also seen with perfection in the formation of the vertical cut and reverse cut, which in their execution touch in the vertical middle of the Zenith E., going through the forehead to the chest, and to the ground F., as shown in said number 27.

Y tambien ſe llama linea de la contingencia, ò tangente, à la recta, que toca en la convexidad de eſtos circulos, y la ſeñalan las letras C.D. del miſmo numero 27.

Also, the straight line touching the convexity of these circles is called the line of contingency, or tangent, marked by the letters C.D. of the same number 27.

Qualquiera linea recta, que ſe ſaque dentro del circulo, que no ſe diametro, ſe llamarà Cuerda, como la A.B. de la figura 28. y la proporcion de circunferencia que ſe ſubtende, ſe llamarà Arco, como la C.D. y la Linea que divide por medio al Arco, y à la linea recta ſe llamarà Sagita, que ſeñala la E.

Any straight line drawn within the circle, which is not a diameter, is called a Chord, like the A.B. of figure 28, and the portion of the circumference it subtends is called an Arc, like the C.D., and the Line that bisects the Arc and the straight line is called the Sagitta, marked by the E.

Antes de entrar en la declaracion de las demàs Superficies, tratarèmos de algunas figuras neceſſarias para ſu inteligencia, que ſe cauſan de la Superficie plana; y por quanto à cada paſſo hablamos de figuras, pondremos aqui la definicion que dà de ellas Euclides lib. 1. Defin. 14.

Before entering the declaration of the other Surfaces, we will discuss some necessary figures for their understanding, which are caused by the flat Surface; and since we talk about figures every step, we will put here the definition given by Euclid book 1. Defin. 14. for them.

Definicion de la Figura

Figura es la Superficie contenida debxo de vno, ò muchos terminos; la de vn termino es el Circulo: la de dos terminos es el Semicirculo, ò ſegmento mayor ò menor: la de tres terminos, ſerà Triangulo; la de quatro, Quadrilatero, &c.

A figure is the surface contained beneath one or many boundaries; one with a single boundary is a Circle; one with two boundaries is a Semicircle, or a larger or smaller segment; one with three boundaries is a Triangle; one with four, a Quadrilateral, etc.

Definition of Figure

Figuras Rectilineas, y Trilateras

Figuras Rectilineas ſon las que eſtàn contenidas de lineas rectas. Figuras Trilateras ſon las que eſtàn contenidas de tres lineas rectas.

Rectilinear Figures are those contained by straight lines. Trilateral Figures are those contained by three straight lines.

Rectilinear and Trilateral Figures

Multilateras

Figuras Multilateras, ò Poligonas, ſon las que ſe contienen de muchas lineas rectas.

Multilateral, or Polygonal Figures, are those contained by many straight lines.

Multilateral

Definicion del Triangulo

27. Triangulo ex vna ſuperficie cerrada de tres lineas, y tres Angulos, toma ſu nombre ſegun las lineas, y Angulos; y aſsi quando fuere de lineas rectas, ſe dize Triangulo rectilineo, ſi de curvas, Curvilineo, y ſi de dos rectas, y vna curva, Mixto, como ſe dixo de los Angulos.

27. A Triangle is a closed surface with three lines and three angles, it takes its name according to the lines and angles; thus, when it consists of straight lines, it is called a Rectilinear Triangle, if of curved lines, Curvilinear, and if of two straight and one curved line, Mixed, as was said of the angles.

Definition of Triangle

Rectilineos

De los Triangulos rectilineos ay tres eſpecies, los quales ſe nombren por la grandeza de ſus lineas, ò por al abertura de ſus Angulos, en eſta manera.

There are three kinds of Rectilinear Triangles, which are named by the length of their sides, or by the opening of their angles, in this way.

Rectilinear

De las figuras Trilateras el Triangulo Equilatero es el que ſe contiene de tres lineas rectas iguales, como lo demueſtran las letras A.B.D. en la figura num. 29.

Among the Trilateral Figures, the Equilateral Triangle is the one contained by three equal straight lines, as demonstrated by the letters A.B.D. in figure num. 29.

En la Deſtreza nos valemos de èl en muchas coſas, aſsi para la defenſa propria, como para la ofenſa del contrario; y particularmente para el acometimiento perfecto por l a parte de adentro, conſiderando por vno de ſus lados el brazo, por otro la linea de la contingencia del pecho, y el otro imaginado deſde el pomo de la Eſpada, haſta concurrir en el ombro izquierdo.

In Fencing, we use it for many things, both for self-defense and for offending the opponent; particularly for the perfect attack on the inside, considering one of its sides as the arm, another one as the line of contingency of the chest, and the other imagined from the hilt of the Sword, to converge on the left shoulder.

Triangulo Iſoſceles

28. Triangulo Iſoſceles es el contenido debaxo de dos lados iguales, y la baſis deſigual, como parece en la figura, porque C.X. es igual à X.R. y deſigual de los dos lados C.R. como ſe vè en el numero 30.

28. An Isosceles Triangle is contained beneath two equal sides and a base that is unequal, as appears in the figure, because C.X. is equal to X.R. and unequal to both sides C.R. as shown in number 30.

Isosceles Triangle

En la Deſtreza es de iportancia en muchos caſos, aplicandole mas ò menos, como en la figura paſſada: y en el Plano inferior es preciſa ſu conſideracion para el verdadero acierto de los compaſes, pues ſi ſe huivere de herir, ſe avrà de caminar por vno de ſus lados, ò Perpendiculo, que en èl ſe conſidera, como ſe verà en nueſtra demonſtracion vniverſal del Circulo.

In Fencing, it is important in many cases, applying it more or less, as in the previous figure: and in the lower Plane, its consideration is crucial for the true understanding of the steps, for if one is to strike, one must walk by one of its sides, or Perpendicular, that is considered in it, as will be seen in our universal demonstration of the Circle.

Triangulo Eſcaleno

29. Triangulo Eſcaleno es el contenido debaxo de todos tres lados deſiguales, como P.T. que es mayor que T.R. y eſte menor que R.P. y ſe maniſieſta en el numero 31.

29. A Scalene Triangle is contained beneath all three unequal sides, like P.T. which is greater than T.R. and this is less than R.P. and is manifested in number 31.

Scalene Triangle

En la Deſtreza nos valemos dèl, aſsi en el Plano inferior, como en el ſuperior, pues quando tiramos el acometimiento riguroſo por de dentro, mediante atajo, conſideramos por linea mayor à la que imaginamos ſalir de la punta del pie derecho, haſta concurrir con la tangente del contrario; y à la menor, la que ſale del talon derecho, haſta concurrir por el mas breve camino con la miſma tangente, y eſte tangente es el tercer lado.

In Fencing, we use it, both in the lower Plane and in the upper one, for when we make the rigorous assault from inside, through a block, we consider the longest line to be the one we imagine coming from the tip of the right foot, converging with the opponent’s tangent; and the shorter one, the one that comes from the right heel, converging on the shortest path with the same tangent, and this tangent is the third side.

En el Plano ſuperior conſideramos en eſte acometimiento por linea mayor à la que ſe imagina deſde la punta de la Eſpada, haſta el ombro haſta la linea racepta, ò empuñadura; y la otra linea, que cierra el triangulo, es la Eſpada.

In the upper Plane, we consider in this assault the longest line to be the one imagined from the tip of the Sword, to the shoulder to the received line, or hilt; and the other line, which closes the triangle, is the Sword.

Triangulo Rectangulo

30. Triangulo Rectangulo es el que tiene vn Angulo recto, como ſe maniſieſta en el numero 32. con la letra B. y la linea que tiene opueſta C.D. donde concurren ſus extremos, cauſando Angulos agudos: ſe llama Ipotenuſa, por oponerſe al Angulo recto B. por cuyo deſcubrimiento ofreciò Pitagoras el ſacrificio de las cien vacas à la Dioſa Minerva.

30. A Right Triangle is the one that has a right angle, as manifested in number 32. with the letter B. and the line that is opposite C.D. where its extremes converge, causing acute angles: is called Hypotenuse, being opposed to the right angle B. for whose discovery Pythagoras offered the sacrifice of a hundred cows to the Goddess Minerva.

Right Triangle

In Fencing, this Triangle is formed in many propositions of the Sword and Dagger, when fighting from the nearest extreme, and with the Sword alone it is seen when the opponent retracts and lowers the arm to the acute angle, and the tip of the Sword rises to the obtuse, and the fencer, applying his sword with the necessary gradation, either from within or from without, directs a thrusting wound, his sword serving as the Hypotenuse line.

En la Deſtreza ſe forma eſte Triuangulo en muchos propoſiciones de la Eſpada, y Daga, quando ſe combate deſde el extremo propinquo, y con la Eſpada ſola ſe vè quando el contrario encoge, y baxa el brazo al Angulo agudo, y la punta de la Eſpada ſube al obtuſo, y el Dieſtro aplicando ſu eſpada con la graduiacion neceſſaria, ſea por de dentro, ò por de fuera, encamina herida de eſtocada, ſirviendo ſu eſpada de linea Ipotenuſa.

Triangulo ambligonio

31. Triangulo ablogonio, ò obtuſiangulo, es el que tiene vn Angulo obtuſo; demueſtrale la letra A. en el num 33. La aplicacion que tiene en la Deſtreza queda hecha en los antecedentes.

31. An oblique triangle, or obtuse-angled triangle, is one that has an obtuse angle; this is shown by the letter A. in number 33. Its application in Fencing is detailed in the preceding sections.

Oblique Triangle

Trioangulo Oxigonio

32. Trioangulo Oxigonio, ò acutiangulo, es el que tiene todos los tres Angulos agudos, como demueſtran las letras M.C.P. en el numero 34.

32. An acute triangle, or acute-angled triangle, is one that has all three angles acute, as is shown by the letters M.C.P. in number 34.

Acute Triangle

Tambien en los Triangulos antecedentes queda hecha ſu aplicacion en la Deſtreza, por el conocimiento de las lineas que le componen.

Its application in Fencing is also covered in the preceding triangles, through the knowledge of the lines that compose them.

Definicion del Quadrado

33. De las figuras quadrilateras, el quadrado es la que tiene los quatro lados, y los quatro Angulos iguales, como ſe demueſtra con las letras P.B.C.D.

33. Among the quadrilateral figures, the square is the one that has all four sides and all four angles equal, as is demonstrated with the letters P.B.C.D.

Definition of a Square

Si de vno de ſus Angulos ſe retiraſſe vna linea recta al Angulo opueſto, ſe dize Diagonal, y dicha linea dividirà el quadrado en dos triangulos rectangulos de ſuperficies iguales, como demueſtra la tirada deſde C. à B. y todo en el numero 35.

If a straight line is drawn from one of its angles to the opposite angle, it is called a Diagonal, and this line will divide the square into two right triangles of equal surfaces, as shown by the line drawn from C. to B., all in number 35.

En la Deſtreza conſideramos el Quadrado en el pecho de cada vno de los combatientes; y tambien la linea Diagonal, que lo divide de vn Angulo à otro; como al Quadrangulo, ò PAralelogramo, que conſideramos en el roſtro, en cuya linea ſe executantodas las heridas Diagonales del plano ſuperior.

In Fencing, we consider the Square in the chest of each of the combatants; and also the Diagonal line that divides it from one angle to another; as well as the Quadrangle, or Parallelogram, which we consider on the face, where all the Diagonal wounds of the upper plane are executed.

Del Paralelogramoo

34. Paralelo es vna figura plana, contenida de quatro lineas, de las quales, las opueſtas ſean paralelas, cuyas eſpecies ſon el Quadrado (de que ya queda hecha mencion) el Rectangulo, el Rhombo, y Rhomboides.

34. A parallelogram is a flat figure, contained by four lines, of which the opposite ones are parallel, its species are the Square (already mentioned), the Rectangle, the Rhombus, and the Rhomboid.

Of the Parallelogram

Del Paralelogramo Rectangulo

Paralelogramo Rectangulo es el que tiene los quatro Angulos iguales, y ſolo los dos lados opueſtos, y iguales, como V.K. y C.D. en el numero 36.

A rectangular parallelogram is one that has all four angles equal, and only the two opposite and equal sides, like V.K. and C.D. in number 36.

Of the Rectangular Parallelogram

En la Deſtreza le conſideramos en el plano inferior, y ſe verà en nueſtra demonſtraction vniverſal del circulo, y tambien en el roſtro de cada vno de los combatientes.

In Fencing, we consider it in the lower plane, and it will be seen in our universal demonstration of the circle, and also in the face of each of the combatants.

Definicion del Rhombo

35. Rhombo es vna figura equilatera, mas no rectangula, componeſe de quatro lados iguales, como el quadrado: difieren en que ſus Angulos ſon deſiguales, y ſi de vn Angulo à otro ſe tiraſſen dos lineas Diagonales, no ſeràn iguales, y ſus lados ſon equidiſtantes, y de poſita contrariedad, porque los Angulos A.B. obtuſos, y los C.D. ſon agudos, y ſe verifica en el numero 37.

35. A Rhombus is an equilateral figure but not a right-angled one, composed of four equal sides, like the square: they differ in that their angles are unequal, and if two Diagonal lines are drawn from one angle to another, they will not be equal, and its sides are equidistant and of opposite contradiction, because the angles A.B. are obtuse, and C.D. are acute, as verified in number 37.

Definition of Rhombus

En la Deſtreza ſirve eſta figura para algunas propoſiciones de la doctrina Franceſa, pues quando ſe afirman, doblan las rodillas para deſplomar el cuerpo igualmente; y en eſta manera ſe vè con perfeccion, ſi juntaren ambos talones, como muchas vezes ſe executa en la proſecucion de la batalla.

In Fencing, this figure serves for some propositions of the French doctrine, since when they affirm, they bend the knees to balance the body equally; and in this manner, it is seen perfectly if both heels are joined, as it is often executed in the progression of the battle.

Del Rhomboide

36. Rhomboide es vna figura que tiene los lados, y Angulos, opueſtos iguales, mas no es equilatera, ni rectanguloa; pareceſe al Rhombo, ſino que los lados opueſtos ſon mayores, como A.B. es mayor que C.D. y à la linea que ſeñalan las OO. dividiendole de vn Angulo à otro, le llamamos Diagonal; y à la linea que le divide en dos Rhomboides, como ſeñalan las letras M. y M. le llamamos Diametral, numero 38.

36. A Rhomboid is a figure that has its sides, and angles, equal to the opposite ones, but it is neither equilateral nor rectangular; it is similar to the Rhombus, except that the opposite sides are longer, like A.B. is longer than C.D. and to the line designated by the O.O. dividing it from one angle to another, we call it Diagonal; and to the line that divides it into two Rhomboids, as indicated by the letters M. and M. we call it Diametrical, number 38.

Of the Rhomboid

Y para la Deſtreza quando ſe afirman los dos combatientes en vn participio de ambas poſturas (quadrado, y perfil) y las Eſpadas dirigidas à los ombros izquierdos, ò colaterales del miſmo lado, ſe vè eſta figura.

And in Fencing, when the two combatants affirm themselves in a part of both postures (square, and profile) and the Swords are directed to the left shoulders, or collaterals of the same side, this figure is observed.

Definicion del trapecie

37. Las demàs figuras de quatro lados, que no fueren las referidas, ſe llamaràn Trapecias; de cuya eſpecie ay vnas mas irregulares que otras, la mas regula es la que tiene dos lados iguales, como S.V. y los otros dos opueſtos paralelos, como P. y N. en la figura del numero 39.

37. The remaining four-sided figures, which are not the aforementioned, are called Trapeziums; of this species, some are more irregular than others, the most regular one has two equal sides, like S.V. and the other two opposite sides parallel, like P. and N. in the figure of number 39.

Definition of Trapezium

Para la Deſtreza quando los combatientes ſe afirman de quadrado, y las puntas de las Eſpadas miran la vna à la Diametral del pecho, y la otra à la Colateral izquierda.

In Fencing, when the combatants affirm themselves in square, and the tips of the Swords are pointing, one to the Diametrical of the chest, and the other to the left Collateral.

Trapecio irregular

38. Trapecio irregular es la figura que demueſtra la letra Q. en la figura del numero 40.

38. An Irregular Trapezium is the figure demonstrated by the letter Q. in the figure of number 40.

Irregular Trapezium

Y para la Deſtreza ſirve en la doctrina de Bella Eſpañola; pues quando nos afirmamos en ella, ſe deſploma algo el cuerpo, doblando de la rodilla de genero, que el muslo ſe vn lado, y la pierna ſea otro; el tercer lado ſerà la diſtancia que ha de aver deſde el pie izquierdo, que ſuſtenta el cuerpo, haſta el derecho; y el quarto lado ſerà la pierna, y muslo derecho, que eſtarà ſin hazer Angulo: tambien ſe conſidera eſta figura en otros caſos, que ſe referiràn quando tratemos de eſta doctrina en el tercero Libro, que ſe trata de todas las tretas.

And in Fencing, it is useful in the doctrine of Beautiful Spanish; for when we affirm ourselves in it, the body slightly collapses, bending the knee in such a way that the thigh forms one side, and the leg forms another; the third side will be the distance that should exist from the left foot, which sustains the body, to the right one; and the fourth side will be the right leg and thigh, which will be without forming an Angle: this figure is also considered in other cases, which will be referred to when we discuss this doctrine in the third Book, which deals with all the strategies.

Figuras de muchos lados

39. Otra figuras ay de muchos lados, que tiene ſus nombres particulares: como la de cinco lados, que ſe llama Pentagono, la de ſeis lados Exagono, la de ſiete lados Eptagono, la de ocho Octagono, ò de ocho lados, y aſsi las demàs en infinito.

39. There are other figures with many sides, each having its particular names: like the one with five sides, which is called a Pentagon, the one with six sides Hexagon, the one with seven sides Heptagon, the one with eight sides Octagon, and so forth, infinitely.

Figures with Many Sides

De eſtas figuras, las vna ſe llaman regulares, y las otras irregulares; las regulares ſon las que tienen los lados, y Angulos iguales; y las irregulares ſon las que tienen los lados, y Angulos deſiguales.

Of these figures, some are called regular, and others irregular; the regular ones are those that have equal sides and angles; and the irregular ones are those that have unequal sides and angles.

Las lineas que contienen eſtas figuras ſe llaman lados, las que las dividen por medio, ſe llaman Diametrales; y las que paſſan por los Angulos opueſtos, ſe llaman Diagonales.

The lines that contain these figures are called sides, those that divide them in the middle are called Diametric; and those that pass through the opposite Angles are called Diagonals.

De eſtas figuras de muchos lados, ſolo nos ſervimos en la Deſtreza del Pentagono, aſsi del regular, como del irregular, en eſta manera.

Of these figures with many sides, we only use the Pentagon in Fencing, both the regular and the irregular, in this manner.

Del Pentagono regular, quando afirmandonos con el cuerpo en poſtura Franceſa, igualmente ſobre ambos pies, quedare diſtante vn talon del otro la miſma cantidad que tiene de longitud el muslo, ò pierna; porque deſde el nacimiento que haze el muslo en la orcajadura haſta la rodilla, ò corva, conſideramos el vn lado; ſeñalale la letra A. el ſegundo lado, igual à eſte, conſideramos deſde la rodilla, haſta el fuelo; ſeñalale la B. el tercero el otro muslo, que ſeñala la C. y el quarto la otra pierna D. el quinto, y vltimo lado, es el imaginado de vn talon à otro; ſeñalale la E. figura, numero 41. Y del Pentagono irregular, quando el intervalo de vn talon à otro, ſea de mayor, ò menor longitud, que los tamaños referidos.

For the regular Pentagon, when affirming ourselves with the body in the French posture, equally on both feet, one heel will be distant from the other the same amount as the length of the thigh or leg; because from the birth that the thigh makes in the fork to the knee or bend, we consider one side; it is indicated by the letter A. the second side, equal to this, we consider from the knee, down to the ground; it is indicated by the B. the third is the other thigh, indicated by C. and the fourth the other leg D. the fifth, and last side, is the imagined one from one heel to the other; it is indicated by the E. figure, number 41. And for the irregular Pentagon, when the interval from one heel to another, is of greater, or lesser length, than the sizes referred to.

De los lados en el Triangulo

40. Quando ſe trata de algun Angulo en vn triangulo, las lineas que le comprehenden ſe llaman lados, como demueſtran las dos CC. y la linea que ſe opone al Angulo A. ſe llama baſa, como la letra D.F. y la linea que ſe ſaca deſde el Angulo A. perpendicular à la baſa, ſe llama Perpendiculo; y las partes de la baſa ſegmentos, el mayor D. y el menor F. num. 42.

40. When dealing with any angle in a triangle, the lines that comprehend it are called sides, as demonstrated by the two C’s. And the line that is opposite to the angle A. is called the base, like the letter D.F. And the line that is drawn from the angle A. perpendicular to the base, is called the Perpendicular; and the parts of the base are segments, the greater D. and the lesser F. num. 42.

Of the Sides in the Triangle

Al conocimiento de las lineas rectas circulares, y mixtas, y de los Angulos recto, obtuſo, y agudo, y de la Superfie plana, que hemos dividio en diferentes figuras, aſsi de tres, como de quatro, cinco, ſeis, y mas lados; ſigue aora el de la Superficie curva, que ſe divide en eſpherica, y en mixta.

Following the understanding of straight, circular, and mixed lines, and of right, obtuse, and acute angles, and of the flat Surface, which we have divided into different figures, both of three, four, five, six, and more sides; we now move on to the understanding of the curved Surface, which is divided into spherical, and mixed.

De la Superſicie eſpherica

41. Superficie eſpherica, ſegun Euclides Defin. 12. del lib. 11. de ſus Elementos, ſe forma de la revolucion de la mitad de la circunferencia ſobre ſu diametro: no lo dize con expreſſas palabras, mas ſe infiere de la definicion que dà de la eſphera, en eſta forma.

41. Spherical Surface, according to Euclid Defin. 12. of book 11. of his Elements, is formed by the revolution of half the circumference over its diameter: he does not say it in express words, but it is inferred from the definition he gives of the sphere, in this form.

Of the Spherical Surface

Eſphera es vna figura ſolida, contenida de vna ſola ſuperficie, à la qual deſde vn punto, que eſtà dentro, todas las lineas que ſe ſacan à la circunferencia ſon iguales entre sì; y la deſcripcion, ò formacion de la eſphera, ſe haze con la revolucion de vn ſemicirculo, ſobre ſu diametro, que eſtè fixo, haſta que buelva al lugar donde empezò.

A sphere is a solid figure, contained by a single surface, to which, from a point that is inside, all lines drawn to the circumference are equal to each other; and the description, or formation of the sphere, is made with the revolution of a semicircle, over its fixed diameter, until it returns to the place where it began.

No declara eſpecificamente, que la ſuperficie ſe aya cauſado del movimiento de la linea circular; mas como eſtà dicho ſe infiere, porque la linea con ſu movimiento cauſa ſuperficie; y dize, que la eſphera, como cuerpo, eſtàn contenida de vna ſuperficie, que no puede ſer otra mas, que la que deſcribe la linea del ſemicirculo C. y C. que ſe mueve ſobre ſu diametro A.B. num. 43.

He does not specifically declare that the surface is caused by the movement of the circular line; but as said it is inferred, because the line with its movement causes surface; and he says, that the sphere, as a body, is contained by a surface, which cannot be other than the one described by the line of the semicircle C. and C. that moves over its diameter A.B. num. 43.

Mixta

42. Superficie mixta es la que ſe cauſa de linea recta, movida circularmente, ò de la linea circular, movida rectamente; como ſon las que ſe vèn en la ſuperficie del Cilindro, y del Cono, ò Piramide conica, y ſe infiere de la definicion 16. y 18. del Libro 11. de Euclides, que dize.

42. Mixed surface is the one caused by a straight line, moved circularly, or by the circular line, moved straight; as seen in the surfaces of the Cylinder, and the Cone, or conical Pyramid, and it is inferred from definitions 16. and 18. of Book 11 of Euclid, who says.

Mixed

El Cono es vna figura ſolida, que ſe forma de la revolucion de vn lado de vn triangulo, al rededor de otro que ſe eſtè quedo, como el triangulo A.B.C. que el lado A.B. ſe mueve al rededor de A. C. y cauſa le ſuperficie conica, y todo el triangulo, como es ſuperficie, cauſa el cuerpo contenido en la ſuperficie conica, figura numero 44.

The Cone is a solid figure, formed by the revolution of one side of a triangle, around another that remains still, like the triangle A.B.C. where the side A.B. moves around A.C. and causes the conical surface, and the whole triangle, being a surface, causes the body contained in the conical surface, figure number 44.

Lo miſmo ſe vè en la figura que ſe le figue, que ſe llama Cilindrica, la qual ſe cauſa de la revolucion del lado A.C. en el paralelogramo A.B.C.D. al rededor de B.D. cauſando eſte lado la ſuperficie Cilindrica, en la qual eſtà contenido el cuerpo del Cilindro; adonde ſe vè, que eſta ſuperficie ſe llamarà Mixta, por ſer compueſta de dos movimientos, ò de dos diferencias de lineas, la vna es circular, y la otra recta, formandoſe de la revolucion que haze la linea recta ſobre ò al rededor del otro lado.

The same is seen in the following figure, which is called Cylindrical, which is caused by the revolution of the side A.C. in the parallelogram A.B.C.D. around B.D. this side causing the Cylindrical surface, in which the body of the Cylinder is contained; where it is seen, that this surface is called Mixed, as it is composed of two movements, or two different types of lines, one is circular, and the other straight, being formed by the revolution that the straight line makes around the other side.

Tambien ſe cauſa del movimiento del circulo A.G.E. ſi ſe movieſſe rectamente de arriba abaxo, como ſe podrà conſiderar mas facilmente, ſi ſe moviera algun circulo rectamente por algun barro blando, en que ſe hallaria aver cauſado vn ſuperficie Cilindrica, que llaman Mixta, figura num 45.

It is also caused by the movement of the circle A.G.E. if it moved straightly from top to bottom, which can be more easily considered, if some circle moved straightly through some soft mud, in which it would have caused a Cylindrical surface, which they call Mixed, figure num 45.

Otras infintas diferencias ay de ſuperficies mixtas, mas eſtas dos ſon las mas neceſſarias en la Deſtreza, como ſe verà en ſu lugar.

There are other infinite differences in mixed surfaces, but these two are the most necessary in Fencing, as will be seen in their place.

Definicion de los Cuerpos ſolidos

43. Solido (ſegun Euclides, definicion 1. del Libro 11. de ſus Elementos) es el que tiene longitud, latitud, y profundidad, y l;os terminos ſon ſuperficies.

43. Solid (according to Euclid, definition 1. of Book 11 of his Elements) is what has length, width, and depth, and its limits are surfaces.

Definition of Solid Bodies

De la miſma ſuerte, que la linea ſe divide en recta, circuylar, y mixta, y la ſuperficie en plana, eſpherica, y mixta; tambien los cuerpos ſe dividen en tres diferencias: es aſaber, en los que ſe comprehenden de ſuperficies planas, como ſon los cinco cuerpos cque llaman Regulares, y otros infinitos, que ay en la naturaleza, en los que ſe comprehenden debaxo de ſuperficies mixtas, de que ay infinidad; mas entre todos, en la Deſtreza neceſsitamos de la conſideracion de tres de ellos, que ſon la Eſphera, el Cilindro, y la Piramide redonda, que aunque quedan demonſtrados, volverèmos à declararlos, en eſta manera.

In the same way that the line is divided into straight, circular, and mixed, and the surface into flat, spherical, and mixed; also, the bodies are divided into three differences: namely, those that are comprised of flat surfaces, such as the five bodies called Regular, and other infinite ones found in nature, those that are understood under mixed surfaces, of which there are many; but among all, in Fencing, we need to consider three of them, which are the Sphere, the Cylinder, and the round Pyramid, which although demonstrated, we will declare again, in this manner.

Definicion de la Eſphera

44. Eſphera es vna figura ſolida, contenida de vna ſola ſuperficie, à la qual deſde vn punto, que eſtà dentro de ella, todas las lineas, quye ſe ſacaren ſeràn iguales entre sì, como ſe maniſieſta por ella en el numero 46.

44. A sphere is a solid figure, contained by a single surface, from which, from a point within it, all the lines drawn will be equal to each other, as shown in figure number 46.

Definition of the Sphere

La deſcripcion, ò formacion de la Eſphera, ſe haze con la revolucion del Semicirculo, quedando firme, y inmobil ſu Diametro, haſta que vuelve al lugar à donde comenzò, como ya queda explicado.

The description, or formation of the sphere, is made by the revolution of the semicircle, keeping its diameter firm and immobile, until it returns to the place where it began, as already explained.

Definicion del Cilindro

45. Cilindro es vna figura ſolida, que ſe haze de la revolucion de vna Paralelogramo, ſobre vno de ſus lados, que eſtè quedo, haſta tanto que vuelva al lugar donde empezò: ſu axis, ò exe, ſerà el lado que eſtuvo inmobil, ò baſis de los paralelogramos, como en eſta figura numero 47.

45. A cylinder is a solid figure, formed by the revolution of a parallelogram around one of its sides, which remains still, until it returns to the place where it started: its axis, or axle, will be the side that was immobile, or the base of the parallelograms, as in this figure number 47.

Definition of the Cylinder

El Paralelogramo A.D. moviendoſe al rededor de B.D. causò Cilindro A.E. ſu axis es la linea B.D. ſu baſas C.H.E. y A.G.F. que ſe cauſaron de las revoluciones de las lineas A.B.C.D. al rededor de los puntos B.D.

The parallelogram A.D., moving around B.D., caused the cylinder A.E. Its axis is the line B.D., its bases are C.H.E. and A.G.F., which were caused by the revolutions of the lines A.B.C.D. around the points B.D.

Definicion del Cono

46. Cono ex vna figura ſolida, contenida de dos ſuperficies, la vna plana, y la otra mixta; formanſe de la revolucion del triangulo A.B.C.ſobre ò al rededor del lado A.C. que eſtè inmobil, haſta que vuelva al miſmo lugar donde comenzò: el lado que queda inmobil, ſe llama axis, y baſa el circulo que deſcribe la baſa del triangulo, numero 48.

46. The cone is a solid figure, contained by two surfaces, one flat and the other mixed; it is formed by the revolution of triangle A.B.C. around or along the side A.C., which remains immobile, until it returns to the same place where it began. The side that remains immobile is called the axis, and the base is the circle that describes the base of the triangle, number 48.

Definition of the Cylinder

Otra Deſcripcion deſta miſma Piramide

Apolonio Pergeo pone otra deſcripcion de eſta Piramide, diziendo: Si deſde vn punto ſe ſacare vna linea recta à la circunferencia de vn circulo, que no eſtè en el miſmo plano en que eſtà el punto, y eſtandoſe el punto firme, ſe moviere la linea, por la circunferencia del circulo, haſta que vuelva al miſmo lugar de donde partiò, ò comenzò à moverſe: la ſuperficie, deſcripta de la linea recta, ſe llama Superficie conica; el vertice, el punto que no ſe mueve; axis la linea que ſe ſaca deſde el vertice al centro de el circulo; y Cono, que es lo que llamamos Piramide, lo que eſtà contenido de la ſuperficie conica, y del circulo, y el circulo le ſirve de baſa.

Apollonius of Perga provides another description of this Pyramid, saying: If from a point a straight line is drawn to the circumference of a circle, which is not in the same plane as the point, and the point remains fixed, the line moves along the circumference of the circle, until it returns to the same place where it started, or began to move: the surface described by the straight line is called a conical surface; the vertex, the point that does not move; the axis, the line drawn from the vertex to the center of the circle; and the Cone, what we call Pyramid, is what is contained by the conical surface, and the circle, and the circle serves as its base.

Another Description of the same Pyramid

Como por exemplo ſea el punto A. que eſtè en diferente plano que el circulo B.C.D. ſaqueſe vna linea recta deſde A. haſta B. vn punto de la circunferencia del circulo, y eſtando firme el punto A. muevaſe la linea A.B. ò la extremidad B. por la circunferencia del circulo B.C.D ella deſcribirà vna ſuperficie conica, ſu vertice ſerà A. ſu axis A.E. ſu baſa el circulo B.C.D y todo lo contenido entre el circulo, y la ſuperficie ſerà el Cono, ò Piramide conica, como lo manifieſta la figura del numero 49.

For example, let point A. be in a different plane than circle B.C.D., draw a straight line from A. to B., a point on the circumference of the circle, and with point A. fixed, move the line A.B., or the extremity B., along the circumference of the circle B.C.D, it will describe a conical surface, its vertex will be A., its axis A.E., its base the circle B.C.D., and everything contained between the circle and the surface will be the Cone, or conical Pyramid, as shown in figure number 49.

Eſtos tres cuerpos ſon de mucha conſideracion en la Deſtreza, como ſe dirà en ſu lugar: el primero, que es el eſpherico, no ſolo ſirve para conſiderar los movimientos, que ſe hazen dentro de la juriſdiccion de vna eſphera, que ſe forma, ò ſe conſidera formada con el movimiento de la Eſpada; mas tambien firve para conſiderar algunos Planos orizontales, verticales, y obliquos, por donde ſe ha de mover la Eſpada, para la formacion de las tretas, que tienen ſemejanza con los circulos que ſe conſideran en la eſphera Celeſte.

These three bodies are of great consideration in Fencing, as will be mentioned in their place: the first, which is spherical, is not only used to consider the movements made within the jurisdiction of a sphere, which is formed, or considered formed with the movement of the Sword; but it also serves to consider some horizontal, vertical, and oblique planes, along which the Sword must move, for the formation of strategies that resemble the circles considered in the celestial sphere.

The Cylinder will serve to represent a Castle, in which the fencer is considered placed, which will also be formed by the movement of the Sword.

El Cilindro nos ſervirà para repreſentarnos vn Caſtillo, en que conſideramos pueſto el Dieſtro, que tambine ſe formarà del movimiento de la Eſpada.

La Piramide tambien es muye neceſſaria, porque ſu conſideracion, y ſu formacion no ſolo le ſirve de defenſa al Dieſtro, mas le dà conocimiento para la ofenſa.

The Pyramid is also very necessary, because its consideration and its formation not only serve to defend the fencer, but also give him knowledge for the offense.

Lamina primero del Libro ſegundo

First Plate of the Second Book