CAPITVLO DEZIMOSEPTIMO

Proposicion I. Theorema I.

Demuestrase que siempre que en el Dieſtro eſtuviere afirmado ſobre Angulo recto, y en Angulo recto en ſu plano vertical derecho, y ſu contrario le acometiere para herirle de eſtocada por el miſmo plano, podrà con vn guarnicion de dos dedos de ſemidiametro defender la profundidad de ſu cuerpo; pero ſerà neceſſario que los centros de las guarniciones eſtèn en la comun ſeccion de eſte plano vertical, y del plano ſuperior.

CHAPTER SEVENTEEN

Proposition I. Theorem I.

It is demonstrated that whenever the Fencer is positioned at a right angle, and at a right angle in his vertical right plane, and his opponent attacks him to strike with a thrust through the same plane, he can defend the depth of his body with a guard of two fingers in semi-diameter; however, it is necessary that the centers of the guards are in the common section of this vertical plane and the superior plane.

Sea el circulo A.M.T. la comun ſeccion del cilindro del Dieſtro, y del plano ſuperior, y el circulo B.N.K. la del cilindro del contrario con el miſmo plano, y la linea A.B. la comun ſeccion del plano vertical derecho del Dieſtro, y del miſmo plano ſuperior, y que eſta diſtancia de la A.B. ſea la del medio de proporcion de ocho pies Geometricos.

Let the circle A.M.T. be the common section of the Fencer’s cylinder and the superior plane, and the circle B.N.K. be that of the opponent’s cylinder with the same plane, and the line A.B. be the common section of the Fencer’s vertical right plane and the same superior plane, and let this distance A.B. be that of the middle of proportion of eight Geometric feet.

La A.E. de dos pies, es la longitud del brazo del Dieſtro deſde ſu centro haſta la muñeca, y la A.F. de does pies y quarto, es la longitud, deſde el centro del brazo, haſta el centro de la guarnicion, y lo miſmo ſe conſidera en la B.L. y B.H. para con el contrario: con que el centro de ſu guarnicion quedarà diſtante de la ſuperficie del cilindro del Dieſtro ſeis pies menos quarto, y la punta E. llegarà à dos pies de diſtancia del cuerpo del Dieſtro; y deſde eſta poſicion para poder ofender, ſe imagina que dà el contrario vn compàs de dos pies y medio: deſuerte, que ſi la guarnicion de el Dieſtro no ſe lo impidiera, llegàra con la punta à herirle de eſtocada por la miſma linea en punto D. que repreſenta el axis del cilindro, y el centro de la guarnicion de ſu Eſpada ſe quedarà en punto P.

A.E. of two feet is the length of the Fencer’s arm from its center to the wrist, and A.F. of two feet and a quarter is the length from the center of the arm to the center of the guard, and the same is considered in B.L. and B.H. for the opponent: thus, the center of his guard will be six feet less a quarter distant from the surface of the Fencer’s cylinder, and the tip E. will reach two feet distance from the body of the Fencer; and from this position to be able to offend, it is imagined that the opponent takes a step of two and a half feet: thus, if the guard of the Fencer did not prevent it, the opponent would reach with the tip to strike him with a thrust through the same line at point D. which represents the axis of the cylinder, and the center of the guard of his Sword will be at point P.

Determinacion que determina el queſito.

Digo que en eſta poſicion, teniendo el Dieſtro el centro de ſu guarnicion en punto F. cuyo Semidiametro F.G. ſe de dos dedos y dos quincehavos, defenderà la profundidad de ſu cuerpo, aunque tenga vn pie de Diametro, que es mas cantidad de lo que ha de tener vn hombre bien proporcionado.

I say that in this position, having the Fencer the center of his guard at point F, whose Semi-diameter F.G. is of two fingers and two fifteenths, he will defend the depth of his body, even though it has a diameter of one foot, which is more than what a well-proportioned man should have.

Determination that settles the question.

Conſtruccion

Saqueſe deſde punto P. centro de la guarnicion la linea P.I. que toque al circulo del Dieſtro en punto I. y la linea D.I. deſde el centro D. al punto del contacto I. ſaqueſe por la 31. del primero la F.G. deſde el centro F. de la guarnicion del Dieſtro paralela à la D.I.

Draw from point P, the center of the guard, the line P.I. that touches the circle of the Fencer at point I, and the line D.I. from the center D to the contact point I. Draw the line F.G. from the center F of the Fencer’s guard parallel to D.I by the 31st of the first book of Euclid’s Elements.

Construction

Demõſtracion

Porque en el triangulo B.P.D.I. ſe ha ſacado la F.G. paralela à la baſa D.I. dividirà los lados P.D. y P.I en la miſma proporcion por la ſegunda del ſexto de los Elementos de Euclides, y quedaràn formados los dos triangulos ſemejantes I.P.D. y G.P.F. por tener los Angulos P.G.F. y P.F.G. igaules à los Angulos P.I.D. y P.D.I. eternos à internos por la 29. del primero, y el Angulo en P. comun, con que los tres Angulos del vno ſon iguales à los Angulos del otro, cada vna cada vno.

Since in the triangle B.P.D.I. the line F.G. is drawn parallel to the base D.I., it will divide the sides P.D. and P.I. in the same proportion by the second proposition of the sixth book of Euclid’s Elements, and will form the two similar triangles I.P.D. and G.P.F. for having the angles P.G.F. and P.F.G. equal to the angles P.I.D. and P.D.I., external to internal by the 29th of the first book, and the angle at P common, thus the three angles of one are equal to the angles of the other, each to each.

Demonstration

Luego por la quarta propoſicion del ſexto tendràn los lados homologos proporcionales, y ſerà como P.D. à D.I aſsi P.F. à F.G. y de eſtas quatro cantidades proporcionales, las tres ſon conocidas, la P.D. de 60. dedos, la D.I. de 8. y la P.F. de 16. y formando con eſtos numeros la regla de tres, reſultarà por cociente de dedos, y dos quincehavos de otro dedo, que es la propoſion. Luego, &c. que es lo que convenia demonſtrar en la figura 1.

Therefore, by the fourth proposition of the sixth book, the homologous sides will be proportional, and as P.D. to D.I. so P.F. to F.G. and of these four proportional quantities, three are known, the P.D. of 60 fingers, the D.I. of 8, and the P.F. of 16. And forming with these numbers the rule of three, the result will be a quotient of two fingers and two fifteenths of another finger, which is the proportion. Therefore, &c. which is what was to be demonstrated in figure 1.

Advertencia

Note

Porque es dificil, que los centros de las guarniciones de los dos contrarios eſten en la linea A.B. ò comun ſeccion de los dos planos vertical, y ſuperior, no ſolo ſe ha dado en Eſpaña doblado ſemidiametro à las guarniciones, pues es muy cerca de quatro dedos, mas tambien es neceſſario valernos del vſo de nueſtras Pyramides, como lo demonſtramos en la idea de nueſtro Fuerte; y para que ſe ſepa la defenſa que ſe tiene en las guarniciones de Eſpaña, moſtrarèmos lo que cubren del cuerpo en los tres planos mas principales, que ſon el vertical derecho, colateral derecho, y plano vertical, que paſſa por la vertical del pecho.

Because it is difficult for the centers of the guards of the two opponents to be on the line A.B. or common section of the two vertical and superior planes, not only has the semi-diameter of the guards been doubled in Spain, as it is very close to four fingers, but it is also necessary to make use of our Pyramids, as we demonstrate in the idea of our Fortress; and to know the defense that is provided by the guards in Spain, we will show what they cover of the body in the three most principal planes, which are the vertical right, right collateral, and vertical plane that passes through the chest vertical.

Proposicion II. Theorema II.

Proposition II. Theorem II.

Dado lo miſmo qu een la figura atecedente, excepto el Semidiametro de la guarnicion del contrario, que en aquella era queſito, y en eſta ſe dà de quatro dedos, como la tienen las guarniciones ordinarias de Eſpaña, y el queſito es ſaber quanto cubrirà en la longitud, y latitud del cuerpo?

Given the same as in the previous figure, except for the semi-diameter of the opponent’s guard, which in that was a question, and in this is given as four fingers, as is common for guards in Spain, the question is to know how much it will cover in the length and width of the body?

Saqueſe deſde P. centro de la guarnicion del Dieſtro, la tangente P.G. y produzgaſe lo que fuere neceſſario; ſaqueſe deſde el punto F. centro de la guarnicion del contrario, la F.G. al punto del contacto, y deſde el punto D. la D.I. paralela à la F.G. que concurra con la P.G. en el punto I. digo que la D.I. ſerà de quince dedos, lo qual ſe demueſtra en eſta forma.

From P, the center of the Fencer’s guard, draw the tangent P.G. and extend it as necessary; from point F, the center of the opponent’s guard, draw F.G. to the contact point, and from point D, draw D.I. parallel to F.G. meeting P.G. at point I. I claim that D.I. will be fifteen fingers, which is demonstrated in this way.

Porque en el triangulo P.D.I. la linea D.I. es paralela à la F.G. ſeràn los lados P.D. y P.I. y del triangulo P.D.I. divididos en la miſma proporcion, y por ſer los Angulos en F.G. iguales à los Angulos D.I. y el Angulo en P. comun, ſeràn ſemejantes los triangulos P.I.D. y P.G.F. por la definicion primera del ſexto; y por la quarta propoſicion de el miſmo libro tendran los lados homologos proporcionales, y ſerà como P.F. à F.G. aſsi P.D. à D.I. pero la P.F. es de 16. dedos, y la F.G. de quatro, y la P.D. de ſeſenta; y formando la regla de tres, ſe hallarà por cociente la D.I. de quince dedos, que ſon cerca de vn pie: luego la guarnicion de la Eſpada, que tuviere de ſemidiametro quatro dedos, podrà cubrir treinta dedos de latitud, y longitud del cuerpo, que ſon dos pies menos dos dedos, que es lo que convenis demonſtrar en la figura ſegunda.

In the triangle P.D.I., since the line D.I. is parallel to F.G., the sides P.D. and P.I. of the triangle P.D.I. will be divided in the same proportion, and since the angles in F.G. are equal to the angles D.I., and the angle at P is common, the triangles P.I.D. and P.G.F. will be similar by the first definition of the sixth book; and by the fourth proposition of the same book, the homologous sides will be proportional, and as P.F. to F.G. so P.D. to D.I. But P.F. is 16 fingers, F.G. is four, and P.D. is sixty; forming the rule of three, the D.I. of fifteen fingers will be found, which is about one foot: therefore, the guard of the Sword, which has a semi-diameter of four fingers, can cover thirty fingers of breadth and length of the body, which are two feet less two fingers, which is what was to be demonstrated in figure two.

Proposicion III. Theroema III.

Proposition III. Theorem III.

Estando el Dieſtro en la miſma diſtancia de ſu contrario con el centro de ſu guarnicion en punto P. como en las dos propoſiciones paſſadas, y los dos centros de las guarniciones P.F. ocupando la comun ſeccion el plano colateral derecho del contrario con el plano ſuperior, en cuya poſicion ſu guarnicion ſe halla tres dedos mas cerca de ſu cuerpo, que en las dos antecedentes, cubrirà de latitud, y longitud de èl vn eſpacio en circunferencia de 25. dedos y vn tercio de diametro.

With the Fencer in the same distance from his opponent with the center of his guard at point P, as in the past two propositions, and both centers of the guards P.F. occupying the common section of the opponent’s right collateral plane with the superior plane, in which position his guard is three fingers closer to his body than in the previous ones, will cover a space in circumference of 25 fingers and one-third in diameter of the breadth and length of his body.

Sea A.B. la comun ſeccion del plano colateral del contrario con el plano ſuperior, con las miſmas diſtancias que en las antecedentes entre los dos combatientes, excepto que el centro F. de la guarnicion del contrario, en eſta poſicion eſtà mas cerca de ſu cuerpo tres dedos. Digo, que ſolo cubrirà de la longitud, y latitud de ſu cuerpo vn eſpacio en circunferencia de 25. dedos, y vn tercio de Diametro, cuyo Semidiametro ſerà D.I. de doce dedos, y dos tercios.

Let A.B. be the common section of the opponent’s right collateral plane with the superior plane, with the same distances as in the previous ones between the two fighters, except that the center F of the opponent’s guard, in this position, is three fingers closer to his body. I say that it will only cover a space in circumference of 25 fingers and one-third in diameter of the breadth and length of his body, whose semi-diameter D.I. will be twelve fingers and two-thirds.

Saqueſe deſde punto P. centro de la guarnicion del Dieſtro, la tangente P.G. producida haſta I. ſaqueſe la F.G. deſde punto F. centro de la guarnicion del contrario, al punto del contacto, y deſde el punto D. la de I. paralela à la F.G.

From point P, the center of the Fencer’s guard, draw the tangent P.G. extended to I. From point F, the center of the opponent’s guard, draw F.G. to the contact point G, and from point D, draw D.I. parallel to F.G.

Porque en el triangulo P.D.I. la linea D.I. es paralela à la F.G. ſeràn los lados P.D. y P.I. del triangulo P.D.I. divididos en la miſma proporcion, y por ſer los Angulos en F.G. iguales à los Angulos D.I. y el Angulo en P. comun, ſeràn ſemejantes los triangulos P.I.D. y P.G.F. por la definicion primera del ſexto, y por la quarta propoſicion del miſmo libro, tendràn los lados homologos proporcionales, y ſerà como P.F. à F.G. aſsi P.D. à D.I. pero la P.F. es de 19 dedos, y la F.G. de 4. y la P.D. de 60. y formando la regla de tres, ſe hallarà por cociente la D.I. de 12. dedos, y dos tercios: luego la guarnicion de la Eſpada que tuviere de Semidiametro quatro dedos, pueſta en el plano colateral derecho en la forma referida, cubrirà 25. dedos, y vn tercio de Diametro de latitud, y longitud del cuerpo, que es lo que convenia demonſtrar, y ſe manifieſta en la figura tercera.

In the triangle P.D.I., since the line D.I. is parallel to F.G., the sides P.D. and P.I. of the triangle P.D.I. will be divided in the same proportion; and since the angles in F.G. are equal to the angles D.I., and the angle at P is common, the triangles P.I.D. and P.G.F. will be similar by the first definition of the sixth book; and by the fourth proposition of the same book, the homologous sides will be proportional, and as P.F. to F.G. so P.D. to D.I. But P.F. is 19 fingers, F.G. is four, and P.D. is 60. Forming the rule of three, the D.I. of 12 fingers and two-thirds will be found: therefore, the guard of the Sword, with a semi-diameter of four fingers, placed in the right collateral plane as described, will cover 25 fingers and one-third in diameter of the breadth and length of the body, which is what was to be demonstrated, as shown in figure three.

Proposicion IV. Theorema IV.

Proposition IV. Theorem IV.

Estando el Dieſtro en la miſma diſtancia de ſu contrario con el centro de ſu guarnicion en punto P. como en la tras Propoſiciones antecedentes, y los dos centros de las guarniciones P. y F. ocupando la comuno ſeccion del plano vertical del pecho del contrario con el plano ſuperior, en cuya poſicion ſe halla ſu guarnicion mas cerca de ſu cuerpo ocho dedos, que en las de las dos primeras figuras, cubrirà de latitud, y longitud de èl vn eſpacio en circunferencia de veinte dedos de diametro.

With the Fencer at the same distance from his opponent with the center of his guard at point P, as in the previous three propositions, and both centers of the guards P. and F. occupying the common section of the opponent’s vertical chest plane with the superior plane, in which position his guard is eight fingers closer to his body than in the first two figures, will cover a space in circumference of twenty fingers in diameter of his breadth and length.

Sea la linea B.A. la comun ſeccion del plano vertical del pecho del contrario con el plano ſuperior, con las miſmas diſtancias que en las figuras paſſadas, entre los dos combatientes, excepto que el centro F. de la guarnicion de el contrario eſtà mas cerca de ſu cuerpo, que en las dos figuras primeras, ocho dedos: digu que ſolo cubrirà de la longitud, y la latitud de èl vn eſpacio, en circunferencia de veinte dedos de diametro, cuyo ſemidiametro D.I. en eſta poſicion ſerà de diez dedos.

Let the line B.A. be the common section of the opponent’s vertical chest plane with the superior plane, with the same distances as in the previous figures between the two fighters, except that the center F of the opponent’s guard is closer to his body than in the first two figures by eight fingers: I claim that it will only cover a space in circumference of twenty fingers in diameter of his breadth and length, whose semi-diameter D.I. in this position will be ten fingers.

Saqueſe deſde punto P. centro de la guarnicion de el Dieſtro, la tangente P.G. producida haſta I. ſaqueſe la F.G. deſde punto F. centro de la guarnicion del contrario, al punto del contrario G. y deſde el punto D. la D.I. paralela à la F.G.

From point P, the center of the Fencer’s guard, draw the tangent P.G. extended to I. From point F, the center of the opponent’s guard, draw F.G. to the contact point G, and from point D, draw D.I. parallel to F.G.

Porque en el triangulo P.D.I. la linea D.I. es paralela à la F.G. ſeràn los lados P.D. y P.I de el triangulo P.D.I. divididos en la miſma proporcion; y por ſer los Angulos en F.G. iguales à los Angulos D.I. y el Angulo en P. comun, ſeràn ſemejantes los triangulos P.I.D. y P.G.F. por la definicion primera de el ſexto; y por la quarta propoſicion del miſmo libro tendran los lados homologos proporcionales, y ſerà como P.F. à la F.G. aſsi P.D. à D.I pero la P.F. es de 24. dedos, y la F.G. de 4. y la P.D. de 60. y formando la regla de tres, ſe hallarà por cociente la de I. de 10. dedos: luego la guarnicion de la Eſpada, que tuviere de ſemidiametro quatro dedos, pueſta en la forma referida, cubrirà vn eſpacio de 20. dedos de Diametro de la longitud, y latitud del cuerpo, que es la que convenia demonſtrar, comoſe maniſieſta por la figura quarta.

In the triangle P.D.I., since the line D.I. is parallel to F.G., the sides P.D. and P.I. of the triangle P.D.I. will be divided in the same proportion; and since the angles in F.G. are equal to the angles D.I., and the angle at P is common, the triangles P.I.D. and P.G.F. will be similar by the first definition of the sixth book; and by the fourth proposition of the same book, the homologous sides will be proportional, and as P.F. to F.G. so P.D. to D.I. But P.F. is 24 fingers, F.G. is four, and P.D. is 60. Forming the rule of three, the D.I. of 10 fingers will be found: therefore, the guard of the Sword, with a semi-diameter of four fingers, placed as described, will cover a space of 20 fingers in diameter of the breadth and length of the body, which is what was to be demonstrated, as shown in figure four.

Lamina once del Libro ſegundo.

Plate eleven of Book Two