CAPITVLO TREINTA Y QUATRO

Aviendo dado razo de la univerſalidad del Angulo recto, ſu juriſdiccion, naturaleza, diferencias, y propriedades en orden à ſu alcance, reſta demonſtrar tenerle mayor que el obtuſo, ni agudo.

CHAPTER THIRTY-FOUR

Having given an account of the universality of the Right Angle, its jurisdiction, nature, differences, and properties in terms of its reach, it remains to demonstrate that it has a greater reach than either the obtuse or acute angles.

Demueſtraſe mathematicamente ſer el Angulo recto de mayor alcance, que el obtuſo, ni agudo.

Mathematically demonstrating that the Right Angle has greater reach than either the obtuse or acute angle.

Vitr. lib 3

Veg lib. 1 de Re militari

Para la verdadera inteligencia de la explicacion de eſta demonſtracion, me es preciſo fundarla primero en la Philoſophia Mathematica, que trata de las coſas que tiene cantidad numerable, ò menſurable; eſto ſupueſto, yo (à imitacion de Euclides, Principie de los Geometras, que pidiò de gracia le concedieſſen poder con qualquiera centro deſcribir vn circulo, y tirar de vn punto à otro vna linea recaa) pedì à los aficionados, al principio de eſte ſegundo Libro, en las Peticiones del regimen de la Eſpada, me concedieſſen, que la altura del hombre mas perfecta fueſſe de dos varas, pues Vitrubio en el libro tercero, dize: Que el pie del hombre es la ſexta parte de ſu cuerpo; y Claudio Vegecio, libro 1. de Re militari, hablando de què eſtatura ſe han de eſcoger los hombres para la guerra, dize: Que el Conſul Mario eſcogia los Soldados nuevos de ſeis pies de altura; y por eſto, conforme à Vegecio, debe entenderſe à Vitrubio en la lugar arriba citado, diziendo, que el pie del hombre, es la ſexta parte de ſu cuerpo, ò eſtatura; pero ſe debe entender, que el pie que dize es geometrico, que ſe compone de quatro palmas de mano, y cada mano tiene quatro dedos, que hazen 16. dedos, que es vna tercia de vara: con que ſiendo la altura de ſeis pies de eſtos, ſe hallarà tener dos varas de altura el hombre, ſegun los Autores citados; ſiguiendo eſtos miſmos à otros muchos antiguos, y modernos, que de eſto tratan, y lo afirman, ſegun regla de ſapientiſsimo Marco Barron, que tambien afirma debe ſer de ſeis pies geometricos la buena proporcion, ò altura del hombre; aunque riguroſamente el hombre galàn debe tener ms pequeño pie, que el que ſuponen eſtos Autores, no me pongo à contradezirlos, por eſtar tan recibidas ſus doctrinas, y afirmarlo Alberto Duràn, y Juan Darſe, en ſus Symmetrias, como tan diligentiſsimos eſtatuarios, y tienen por mas perfecta la de los dichos ſeis pies geometricos, que ſon las dos varas.

For a true understanding of the explanation of this demonstration, I must first base it on Mathematical Philosophy, which deals with things that have a countable or measurable quantity; given this, I (following the example of Euclid, the principal of geometers, who asked for the grace to be allowed to describe a circle with any center, and to draw a straight line from one point to another) asked enthusiasts at the beginning of this second book, in the Petitions of the sword’s regimen, to grant me that the most perfect height of a man would be two yards since Vitruvius in book three says: That a man’s foot is the sixth part of his body; and Claudius Vegetius, in book 1 of De Re Militari, discussing what stature men should be chosen for war, says: That Consul Marius chose new Soldiers of six feet in height; and for this, in accordance with Vegetius, Vitruvius should be understood in the cited place, saying that the foot of a man, is the sixth part of his body, or stature; but it should be understood that the foot he mentions is geometric, which is composed of four palm widths, and each palm has four fingers, making 16 fingers, which is a third of a yard: with the height being six of these feet, one will find that a man, according to the cited authors, will be two yards tall; following these same authors and many others, both ancient and modern, who discuss this and affirm it, according to the rule of the most wise Marco Vitruvius, who also states that six geometric feet is the proper proportion, or height of a man; although strictly speaking, a handsome man should have a smaller foot than what these authors suppose, I do not set out to contradict them, as their doctrines are so accepted, and affirmed by Alberti Durero and Juan De Arfe, in their Symmetries, as very diligent statuaries, and they consider the said six geometric feet to be the most perfect, which are the two yards.

Vitruvius’s “De Architectura”, Book 3

Vegetius’ “De Re Militari”, Book 1

Tambien pedì, que la Eſpada con que ſe han de regular las diſtancias, y menſuraciones, aya de ſer ſegun la marca de eſtos Reynos de Caſtilla, pues por Ley, y Pragmatica de ella eſtà prohibido que no ſe pueda traer ſino es de cinco quartas, menſurandoſe deſde los gavilanes à la punta, pues con el puño, y pomo, que es toda ſ magnitud, tiene quatro tercias, que hazen quatro pies geometricos. Eſto ſupueſto, y concedido (como ſe debe) fundo la regularidad de las demonſtraciones debaxo de eſtas medidas, por ſer las mas recibidas de todos los hombres ſcientificos, aſsi Eſtatuarios, como Pintores, que tratan de la Symmetria del hombre; porque ſi huvieſſe de ſeguir las medidas, que trae Albtero Durero, cauſarian confuſion, por ſer las partes aliquotas tan minimas, y prolixas para eſta ſciencia; y aſsi me valgo de medir, ò menſurar el cuerpo del hombre por pies, y por dedos, y lo miſmo la Eſpada, pueſto que es el inſtrumento con que ſe ha de hazer la defenſa, y ofenſa (ſi conviniere) menſurando, regulando, y proporcionando con ellas las diſtancias, los movimientos, los Angulos, las aciones, las tretas, los perfiles, y aſpectos, &c. con tal direccion, que llegue à gozar el cientifico Dieſtro de los admirables, y favorables efectos de eſta ſciencia.

I also asked that the sword, with which distances and measurements are to be regulated, must be according to the standard of these kingdoms of Castile, since by law and pragmatic sanction, it is prohibited to carry any but of five quarters, measuring from the guard to the tip, because with the hilt and pommel, which is its entire magnitude, it has four thirds, which make four geometric feet. Given and granted this (as it should be), I base the regularity of the demonstrations under these measures, as they are the most accepted by all scientific men, both Statuaries and Painters, who deal with the Symmetry of man; because if I had to follow the measures brought by Albert Durer, they would cause confusion, due to the aliquot parts being so minimal and lengthy for this science; thus, I make use of measuring or gauging the body of a man by feet, and by fingers, and the same with the sword, since it is the instrument with which the defense and offense (if convenient) are to be made, measuring, regulating, and proportioning with them the distances, movements, angles, actions, strategies, profiles, and aspects, etc., with such direction, that the knowledgeable Fencer may enjoy the admirable and favorable effects of this science.

Cuerpo 96. dedos de longitud

Eſto ſupueſto, y concedido que el cuerpo del hombre bien proporcionado tiene dos varas de altura, que es toda ſu longitud, que hazen ſeis pies geometricos, y que cada vno tiene de longitud diez y ſeis dedos, ſi ſe multiplican los diez y ſeis por ſeis, haràn noventa y ſeis dedos por toda la altura de la figura.

Given and granted that the well-proportioned human body is two yards tall, which is its total length, making six geometric feet, and that each one is sixteen fingers in length, if the sixteen are multiplied by six, they will make ninety-six fingers for the total height of the figure.

Body 96 fingers in length

Brazo 32. dedos de longitud

El brazo tiene deſde la linea racepta de la mano haſta la ſangria vn pie, y de alli al centro del brazo otro, que hazen dos; y reducidos à dedos, ſon treinta y dos de longitud, que tiene el braço deſde la muñeca al ombro, como queda dicho.

The arm, from the wrist line to the elbow, has one foot, and from there to the center of the arm another foot, making two feet in total; when converted to fingers, this equals thirty-two in length from the wrist to the shoulder, as stated.

Arm 32 fingers in length

La Eſpada deſde la punta al pomo, ſiendo de la marca, tiene quatro tercias, que hazen quatro pies, que multiplicados 16. por 4 hazen 64. dedos por toda ſu longitud deſde el pomo à la punta.

The sword, from the tip to the pommel, being of the standard size, measures four thirds, which make four feet, which, multiplied by 16, results in 64 fingers in total length from the pommel to the tip.

Yà queda dicho arriba, que el brazo deſde la linea recepta al ombro tiene dos pies geometricos, y la Eſpada quatro; eſtando empuñada en la mano, llega el pomo à la linea recepta, con que juntando à los 64. dedos, que tiene la Eſpada de longitud, 32. que tiene el brazo, ſe hallaràn que la ſuma del brazo, y Eſpada ſon 96. dedos, que es lo miſmo que tiene de alto la figura de el hombre (que dexo ſupueſo) con que evidentementeſe reconoce la proporcion que el hombre tiene con la Eſpada, y que haziendo centro principal del brazo al ombro, podrà deſcribir con el brazo, y Eſpada vna ſuperficie eſpherica, ſirviendo el brazo, y la Eſpada de ſemidiametro de la dicha eſphera.

As mentioned above, the arm from the wrist line to the shoulder is two geometric feet, and the sword is four feet; when held in hand, the pommel reaches the wrist line. Thus, adding the 64 fingers, which is the length of the sword, to the 32 fingers of the arm, it is found that the combined length of the arm and sword is 96 fingers, which matches the height of the man’s figure (as I have assumed), clearly recognizing the proportion between the man and the sword. By making the shoulder the principal center of the arm, one can describe a spherical surface with the arm and sword, with the arm and sword acting as the semi-diameter of said sphere.

Supueſtas todas eſtas medidas, y proporcionadas debaxo de eſta afinidad, ponog las demonſtraciones ſiguientes, para probar, aſsi con la Philoſofia, como con las evidencias Mathematicas de Arithmetica, Geometira, y Aſtrologia, qual de los Angulos es de mayor alcance; y por los grados de la eſphera ſe ſabrà el valor de ellas, ajuſtandolo por la Arithmetica con toda evidencia; y por la Symmetria las menſuraciones del hombre, para el vſo, y perfeccion de evta ſciencia, como queda dicho.

Given all these measurements and proportions under this affinity, I present the following demonstrations to prove, both philosophically and with the mathematical evidence from Arithmetic, Geometry, and Astronomy, which angle has the greatest reach. Through the degrees of the sphere, their value will be known, adjusted by Arithmetic with clear evidence; and by Symmetry, the measurements of a man for the use and perfection of this science, as mentioned.

Demoſtracion

Angulos de la eſphera

Viniendo à las demonſtraciones digo, que afirmado el hombre con ſu brazo, y Eſpada, como lo ſeñala A.B. ſiendo el centro A. y de intervalo haſta B. ſi el brazo, y Eſpada deſcribieſſe vn circulo ſobre el centro A. y moviendoſe la punta de la Eſpada que ſeñala B. paſſando por F.Q.E. haſta bolver à la B. ſe hallarà formada vna ſuperficie eſpherica, que ſeñala B.F.Q.E. y la eſphera, ſegun la conſideran los Philoſophos Aſtronomos, ſe divide en quatro quadrantes, y à cada vno ſuponen de 90. grados, que multiplicados pro quatro hazen 360. grados, que es la graduacion, y valor que la dieron los Philoſophos, como lo ſeñala la demonſtracion con las letras B.F.Q.E.

Proceeding to the demonstrations, I say that with the man positioned with his arm and sword, as indicated by A.B., with A being the center and interval up to B, if the arm and sword were to describe a circle over the center A, and moving the tip of the sword indicated by B, passing through F.Q.E. until returning to B, a spherical surface indicated by B.F.Q.E. would be formed. The sphere, according to philosophers and astronomers, is divided into four quadrants, each assumed to be 90 degrees, which, multiplied by four, make 360 degrees, the graduation and value given by philosophers, as indicated by the demonstration with the letters B.F.Q.E.

Demonstration

Angles of the sphere

Valor de los Angulos de la eſphera

Tambien la conſideran con ocho Angulos ſemirectos, y cada vno de ellos de à 45. grados, los quales multiplicados por 8. hazen los miſmos 360. grados, como lo demueſtran las letras BO. OF. FT. TQ. QZ. ZE. ED. DB. en que ſe hallan los miſmos 360. grados, que tiene los quatro quadrantes.

They also consider it with eight right angles, each of 45 degrees, which multiplied by 8 make the same 360 degrees, as demonstrated by the letters B.O., O.F., F.T., T.Q., Q.Z., Z.E., E.D., D.B., where the same 360 degrees are found that the four quadrants have.

Value of the Angles of the sphere

Lo miſmo ſe hallarà ſi la eſphera ſe dividiere con dos Angulos obtuſos, y dos agudos, que valen los miſmos 360. grados, que los quatro rectos, ò los ocho ſemirectos, como lo demueſtran las letras B.G.Q.L. y cada vno de eſtos obtuſos los ſupongo de à 130. grados, y los agudos à 50. cada vno, que ambos valen 100. grados; y los dos obtuſos 260. que con los 100. de los agudos, ſe hallan los miſmos 360. grados; con lo qual ſe vè, que los dos obtuſos, y los dos agudos valen lo miſmo que los quatro rectos, ò los ocho ſemirectos: con que en virtud de la graduacion de la eſphera, ſe conoce el valor de los Angulos, ſu cantidad, y como ſe hereden los vnos à los otros los grados de ſu valor, como ſe verà en la forma ſiguiente.

The same will be found if the sphere is divided with two obtuse angles and two acute angles, which equal the same 360 degrees as the four right angles or the eight semi-right angles, as shown by the letters B.G., G.Q., Q.L., and each of these obtuse angles I suppose to be 130 degrees, and each acute angle to be 50 degrees, both equalling 100 degrees; and the two obtuse angles 260 degrees, which with the 100 degrees of the acute angles, make the same 360 degrees; with which it is seen that the two obtuse angles and the two acute angles equal the same as the four right angles or the eight semi-right angles. Thus, by virtue of the sphere’s gradation, the value of the angles, their quantity, and how they inherit degrees of their value from one another, as will be seen in the following form, is understood.

El Angulo E.A.B. es de 90. grados. Pongo exemplo: el brazo, y Eſpada declina haſta la letra L. perdiendo 40. grados, y los miſmos ſu opoſito, pues cada vno de ellos ſe hazen tener à 50. grados, y en Angulo E.A.B. ſe elevò deſde la B. à la K. deſuerte, que teniendo en B. 90 grados, ſe halla, por aver paſſado à la letra K. adquiriò 40. grados, y los miſmos ſu opoſito; con que ſe hallarà cada vno de 130. grados; y eſte es el valor que adquiriò cada vno de los Angulos obtuſos, pues ſiendo rectos de à 90. grados, ſe hizieron rectos à 90. ſ hallan de à 50. grados; con que vienen à tener los dos obtuſos, y dos agudos los miſmos 360. grados, que los 4. Angulos rectos referidos.

The angle E.A.B. is 90 degrees. For example: the arm and sword decline to the letter L, losing 40 degrees, and the same for its opposite, since each of them becomes 50 degrees, and in angle E.A.B. it rose from B. to K. so that having 90 degrees in B, it is found, by having moved to the letter K, it acquired 40 degrees, and the same for its opposite; thus, each will be found to be 130 degrees; and this is the value that each of the obtuse angles acquired, since being right angles of 90 degrees, they became right angles of 90 and are found to be 50 degrees; with which the two obtuse and two acute angles come to have the same 360 degrees as the 4 referred right angles.

La regla que ſeñala R.S. repreſenta el cuerpo del hombre, eſtà dividida en ſeis partes iguales, que hazen ſeis pies geometrico, y cada vno de à 16. dedos de longitud; por eſta regla ſe ha de menſurar el cuerpo, el brazo, y la Eſpada, ſirviendo de pitipie; tambien ſe divide el brazo, y Eſpada en otras ſeis partes, como lo demueſtran los numeros que eſtan à las perpendiculares, que caen ſobre la ſuperficie plana, quadriculando las diviſiones del cupero del hombre, que ſe vè en las propoſiciones que tiene tan realmente ajuſtadas à la Symmetria, y compoſicion ſuya.

The ruler that marks R.S. represents the man’s body, is divided into six equal parts, making six geometric feet, and each one of 16 fingers in length; by this ruler, the body, the arm, and the sword are to be measured, serving as a yardstick; the arm and sword are also divided into six parts, as shown by the numbers that are at the perpendiculars, falling on the flat surface, gridding the divisions of the man’s body, which is seen in the propositions so accurately adjusted to its Symmetry and composition.

La linea F.A.E. es la perpendicular, que divide la eſphera en dos partes iguales, ſirve de vertical derecha del coſtado del hombre. La letra A. es el centro de la eſphera, y del brazo, y Eſpada. La F. repreſenta el Zenith. Y la letre E. el Nadir. B.Q. la linea del orizonte, que tambien paſſa por centro A. cauſando con ſu ſeccion quatro Angulos rectos en èl, como los ſeñalan las letras B.A.F. y F.Q.A. y Q.A.E. y E.A.B. que todos ſon rectos: con lo qual queda demonſtrado el plano à la altura en que ha de traer el Dieſtro ſu brazo, y Eſpada, para eſtar afirmado en Angulo recto, como ponga el cuerpo, y los pies en la forma que lo demueſtra la figura.

The line F.A.E. is the perpendicular that divides the sphere into two equal parts, serves as the man’s right vertical side. The letter A is the center of the sphere, and of the arm and sword. F represents the Zenith. And the letter E, the Nadir. B.Q. is the horizon line, which also passes through center A, causing with its section four right angles in it, as indicated by the letters B.A.F., F.Q.A., Q.A.E., and E.A.B., all of which are right angles: with which it is demonstrated the plane at the height at which the Diestro must carry his arm and sword to be affirmed in a right angle, as he places the body and feet in the form demonstrated by the figure.

Ya que dexo dado conocimiento de las medidas mas porporcionales del cuerpo del hombre, ſegun la meyor Symmetria, y la longitud de la Eſpada (ſegun la marca deſtos Reynos) con el conocimiento de los grados de la eſphera, y el valor de los Angulos; ſerà bien examinar ſus alcances, conforme à la Geometira, en la manera ſiguiente.

Having already provided knowledge of the most proportional measurements of the human body, according to the best Symmetry, and the length of the sword (according to the standard of these kingdoms), along with the understanding of the degrees of the sphere and the value of the angles; it is good to examine their reaches, according to Geometry, in the following manner.

Sea la linea C.D. vna de las verticales, ò colaterales, que ſe conſideran en el contrario, y en el centro del brazo del Dieſtro A. y haſta la B. la diſtancia que tiene con ſu brazo, y Eſpada, afirmado en Angulo recto; ſaqueſe del punto A. vna linea recta perpendicular à la C.D. por la 12. del primero del libro de los Elementos el punto A. la Y.A.J. paralela à C.D. y ſerà A.J. vertical del Dieſtro, afirmado, como ſe ſupone, en Angulo recto, y ſobre Angulo recto; con que por la propoſicion 29. del miſmo libro de Euclides, ſerà tambien recto el Angulo B.A.J. que haze la linea del Brazo, y Eſpada del Dieſtro con ſu vertcal derecha: por ſer alterno de el Angulo J.A.B. iguales à dos rectos, centro A. intervalo A.B. veſe deſcripto por el plano vertical la porcion de circulo K.L. digo, que ſubiendo, ò baxando del Angulo recto A.B. ſu brazo, y Eſpada el Dieſtro, en qualquiera parte de la eſphera, cauſando con la linea vertical Y.J. Angulo obtuſo, ò agudo, tendrà en cada vno de eſtos Angulos menor alcance que en el Angulo recto A.B. Supongaſe que la Eſpada, y brazo baxa del Angulo recto A.B. al Angulo agudo de 50. grados en punto L. deſuerte, que la linea A.L. repreſente la del brazo, y Eſpada; produzcaſe haſta concurrir con la C.B.D. en el punto D. digo alcançarà menos en eſta poſicion, que en el Angulo recto, en la cantidad D.L. y aunque la propoſicion es evidente, y clara, por la figura ſe demueſtra geometricamente en dos maneras.

Let the line C.D. be one of the verticals, or collaterals, considered on the opponent, and at the center of the Diestro’s arm A. up to B. the distance he has with his arm and sword, affirmed at a right angle; draw from point A. a straight line perpendicular to C.D. by the 12th of the first book of Elements, the point A., the Y.A.J. parallel to C.D., and A.J. will be the Diestro’s vertical, affirmed, as supposed, at a right angle, and upon a right angle; with which by proposition 29 of the same book by Euclid, the angle B.A.J. made by the Diestro’s arm and sword line with his right vertical will also be right: being alternate to the angle J.A.B. equal to two rights, center A. interval A.B. it is seen described by the vertical plane the portion of circle K.L. I say, that rising or lowering from the right angle A.B. his arm and sword, the Diestro, in any part of the sphere, causing with the vertical line Y.J. an obtuse or acute angle, will have in each of these angles less reach than in the right angle A.B. Suppose that the sword and arm lower from the right angle A.B. to the acute angle of 50 degrees at point L., so that the line A.L. represents the arm and sword; it is extended to meet C.B.D. at point D., I say it will reach less in this position than in the right angle, by the quantity D.L. and although the proposition is evident and clear, by the figure it is geometrically demonstrated in two ways.

La primera, porque en el triangulo rectangulo B.A.D. los quadra dos de los lados A.B. y B.D. comprehende à la hypotenuſa A.D.Y. para mas claridad, la examino por numeros en la forma ſiguiente.

The first, because in the right-angled triangle B.A.D. the squares of the sides A.B. and B.D. contain the hypotenuse A.D.Y. For more clarity, I examine it by numbers in the following form.

Digo, que el brazo, y Eſpada tiene ſeis pies Geometricos, como queda dicho, que hazen 96. dedos, y la linea B.D. igual à la J.A. paralelas entre sì, de cino pies, que ſon 80. dedos, y quadrando el lado A.B. haze 9216. y quadrando B.D. 6400. y la ſuma de ambos 15616. cuya raiz quadrada mas proxima es de 125. dedos, que es el valor de la hypotenuſa A.D. de los quales reſtando 96. dedos, quedan 29. dedos para la linea L.D que ſon vn pie y treze dedos; y eſta cantidad es la que avia de tener mas longitud la Eſpada del Dieſtro para alcançar à ſu contrario en D. como lo alcança en B. y eſte miſmo calculo ſe ajuſta en todo con el antecendente.

I say that the arm and sword have six geometric feet, as previously mentioned, which make 96 fingers, and the line B.D. equal to J.A. parallel between them, of five feet, which are 80 fingers, and squaring the side A.B. makes 9216, and squaring B.D. 6400, and the sum of both 15616, whose nearest square root is 125 fingers, which is the value of the hypotenuse A.D., of which subtracting 96 fingers, there remain 29 fingers for the line L.D which are one foot and thirteen fingers; and this quantity is what the Diestro’s sword would need to be longer to reach his opponent in D. as it reaches in B., and this same calculation matches completely with the previous one.

Si el Dieſtro ſube ſu brazo, y Eſpada de la poſtura del Angulo recto A.B. al Angulo obtuſo de 130. grados en C. por el triangulo A.B.C. que ſe cauſa en eſta poſtura igual, y ſus lados al triangulo A.B.D. que cauſan el Angulo agudo, porque la miſma cantidad, que tiene B.D. la tiene la B.C. y los Angulos que cauſa el Dieſtro en el centro A. ſon iguales entre sì en ambas poſturas, ſerà la A.C. igual à la A.D. y ka A.K. igual à la A.L. y la C.K. igual à la D.L. y por eſta orden ſe podràn examinar otras qualeſquier poſturas, aſsi del Angulo agudo, como del obtuſo; y en qualquiera ſiempre ſe verifica, que el Dieſtro tendrà menos alcance en ellas, que en la poſtura del Angulo recto A.B.

If the Diestro raises his arm and sword from the posture of the right angle A.B. to the obtuse angle of 130 degrees at C., through the triangle A.B.C. caused in this posture being equal, and its sides to the triangle A.B.D. that causes the acute angle, because the same quantity that B.D. has is had by B.C. and the angles that the Diestro causes at the center A. are equal among themselves in both postures, A.C. will be equal to A.D. and A.K. equal to A.L. and C.K. equal to D.L. In this order, any other postures, both of the acute and the obtuse angle, can be examined; and in any, it always verifies that the Diestro will have less reach in them than in the posture of the right angle A.B.

La ſegunda demonſtracion en la antecendente ſe ſupuo, que los dos combatientes no ſe movian, y ſe examinò la cantaidad que avia de tender (de mas de la ordinaria) la Eſpada del Dieſtro, para alcançar en la poſtura del Angulo agudo à ſu contrario en D. y aora ſe averiguarà la cantidad que ſe ha de acercar el Dieſtro para herir à ſu contrario en la miſma poſtura de Angulo agudo con ſu brazo, y Eſpada, y ſe demueſtra de eſta ſuerte.

The second demonstration in the preceding supposed that the two combatants did not move, and examined the quantity that had to extend (more than the ordinary) the Diestro’s sword, to reach in the posture of the acute angle to his opponent in D. and now will be determined the quantity that the Diestro has to approach to strike his opponent in the same posture of the acute angle with his arm and sword, and it is demonstrated in this way.

Saqueſe por el punto L. la linea K.L. perpendicular à la A.B. y paralela à la D.C. corteſe A.B. en el punto N. con que tendrèmos dos triangulos A.B.D. y A.C.B. rectangulos, y ſemejantes, pro la propoſicion dos de Euclides, y ſerà como A.L. à la L.D. aſsi A.B. à B.N. y quando de quatro cantidades proporcionales, las tres ſon conocidas, como en eſte caſo la A.D. y la D.L. y la A.B. la quarta ſe hallarà por linea, por la propoſicion 12. del libro 6. de Euclides, y por numeros ſe hallarà formada vna regla de tres, diziendo: Si 125. dedos que tiene la A.D. por la propoſicion antecendente, nos dan la D.L. de 29. dedos, què nos darà la A.B. de 96. dedos? y ſiguiendo la operacion, ſe hallarà la quarta proporcional B.N. de 22. dedos, que es la cantidad que el Dieſtro diſminuye de ſu alcance, eſtando en dicha poſtura de Angulo agudo en A.L. del que tiene afirmado en Angulo recto A.B. y eſta cantidad de 22. dedos, es con la que ſe ha de acercar à ſu contrario para que le pueda alcançar, dando compàs de la miſma cantidad de punto J. à punto M. con el centro del brazo A. paffarà à punto V. y ſu vertical en V.M. y la linea de la Eſpada de punto L. à punnto P. y eſta cantidad (que como ſe ha dicho es de 22. dedos) es la ventaja que tiene el Angulo recto A.B. en alcance à la poſtura en Angulo agudo A.L. que es lo que ſe avia demonſtrado.

Draw through point L. the line K.L. perpendicular to A.B. and parallel to D.C. cutting A.B. at point N. with which we will have two triangles A.B.D. and A.C.B. right-angled, and similar, by proposition two of Euclides, and it will be as A.L. to L.D. so A.B. to B.N. and when of four proportional quantities, the three are known, as in this case A.D. and D.L. and A.B. the fourth will be found by line, by proposition 12 of book 6 of Euclides, and by numbers will be formed a rule of three, saying: If 125 fingers which have the A.D. by the preceding proposition, give us the D.L. of 29 fingers, what will A.B. of 96 fingers give us? and following the operation, will be found the fourth proportional B.N. of 22 fingers, which is the quantity that the Diestro diminishes from his reach, being in said posture of the acute angle at A.L. than he has affirmed in right angle A.B. and this quantity of 22 fingers, is with which he has to approach his opponent so that he can reach him, giving compass of the same quantity from point J. to point M. with the center of the arm A. will pass to point V. and his vertical in V.M. and the line of the Sword from point L. to point P. and this quantity (which as said is of 22 fingers) is the advantage that the right angle A.B. has in reach to the posture in acute angle A.L. which is what was demonstrated.

Lo miſmo reſultaria ſi el Dieſtro eſtuivere afirmado en Angulo obtuſo A.K. ſi huviera alguna parte del cuerpo de ſu contrario en que le pudiera alcançar; porque el triangulo A.N.K. ſon iguales en todo à los triangulos A.B.D. y A.N.L. como ſe vè en la figura, que no ſe demueſtra por no repetir lo miſmo: y ſe haze advertencia, que ſi el Dieſtro variaſe las poſturas en Angulo obtuſo, y agudo, que ſean de mas, ò menos grados de la eſphera, cada vna de ellas ſe podrà ir examinando por la orden que aqui he tenido en las dos antecendentes, ajuſtando los Angulos, y proporciones y los demàs, ſegun fueren las poſturas, ò rectitudes en que el Dieſtro ſe puede hallar afirmado.

The same result would occur if the Diestro were affirmed in obtuse angle A.K. if there was any part of his opponent’s body he could reach; because the triangle A.N.K. are equal in everything to the triangles A.B.D. and A.N.L. as seen in the figure, which is not demonstrated to avoid repeating the same: and it is noted, that if the Diestro varied the postures in obtuse and acute angle, that are of more or less degrees of the sphere, each one of them can be examined by the order that I have had here in the two preceding ones, adjusting the angles, and proportions and the rest, according to the postures, or rectitudes in which the Diestro can be found affirmed.

Lamina treinta y tres, y vltima del Libro ſegundo.

Plate thirty-three, and last of Book two.