CAPITVLO DECIMONONO

En que por medio de la extatica se demueſtra como ſe tendrà conocimiento de los grados de fuerça, y flaqueza del brazo, y Eſpada por el que la rige, reſpecto de ſu potencia.

CHAPTER NINETEEN

Demonstrating through Statics How to Understand the Degrees of Strength and Weakness of the Arm and Sword, as Managed by Its Wielder, in Relation to Its Power.

Explicacion de la figura primera

Explanation of the First Figure

Coſa aſſentada es entre los Mathematicos que la propriedad de las libras (que es vna de las cinco potencias de la extatica) es proporcionar los peſos à las diſtancias en que ſe ponen reſpecto de el hipomoclio (à quien comunmente llaman Fiel) como ſe vè por la primera figura de la Lamina F.G. en la qual ſuponemos que la linea A.D. repreſenta el brazo, y Eſpada recto, como ſale del ombro paralelo al Orizonte, cuya longitud es de ſeis pies Geometricos, los dos deſde el centro del brazo A. haſta la linea raceta, ò muñeca O. y los otros quatro deſde la muñeca O. haſta la punta de la Eſpada D. que es la longitud que tiene deſde el pomo haſta la punta, la qual ſe divide en ſeis partes iguales, las dos tocan al brazo, y las otras quatro à la Eſpada.

It is a well-established concept among Mathematicians that the property of the scales (one of the five powers of statics) is to proportion weights to the distances at which they are placed in relation to the fulcrum (commonly called the Balance), as seen in the first figure of Plate F.G. In this figure, we suppose that the line A.D. represents the arm and sword extended straight, as it comes from the shoulder parallel to the Horizon, whose length is six Geometric feet, two from the center of the arm A. to the wrist line, or wrist O., and the other four from the wrist O. to the tip of the sword D., which is the length from the pommel to the tip, divided into six equal parts, two pertaining to the arm and the other four to the sword.

Para eàta demonſtracion ſe ſupone, que en punto B. que eſtà diſtante del centro del brazo A. quatro pies, eſtà pediente vn peſo de ſeis libras en F. y que la linea A.B. eſtà producida en C. deſuerte, que la A.C. ſe igual à la A.B. y que del punto C. eſtà pendiente otro peſo de ſeis libras en E. es evidente que quedaràn en equi librio, por eſtàr igualmente apartados del hipomoclio A. en que eſtà el centro del a gravedad de eſtos dos peſos, por el qual paſſa la linea de direccion H.I. como lo demueſtra Guido Baldo, propoſicion primera de la Leva.

For this demonstration, it is supposed that at point B., four feet away from the center of the arm A., a weight of six pounds is hanging at F., and that the line A.B. is extended to C., so that A.C. is equal to A.B., and from point C., another weight of six pounds is hanging at E. It is evident that they will be in equilibrium, as they are equally distanced from the fulcrum A., where the center of gravity of these two weights is, through which passes the line of direction H.I., as demonstrated by Guido Baldo, proposition one of the Lever.

De que ſe ſigue, que ſi el Dieſtro eſtuviere afirmado en Angulo recto con ſu brazo, y Eſpada, y ſuſtentaſſe pendiente del punto B. vn peſo de ſeis libras, tendrà la miſma potencia que tiene el peſo de ſeis libras pendiente del punto C.

From this, it follows that if the Fencer is affirmed at a right angle with his arm and sword and sustains a weight of six pounds hanging from point B., he will have the same power as the weight of six pounds hanging from point C.

Para regular eſta potencia con otro peſo que eſtè pendiente de la punta de la Eſpada D. formaſe vn regla de tres, poniendo por primer termino ſeis, que es la longitud del brazo, y Eſpada D.A. y por ſegundo termino la A.C. que es de quatro pies, y por tercero el peſo pendiente del punto C. de ſeis libras; y haziendo la operacion, reſulta por cociente el peſo G. de quatro libras, pendiente de la punta de la Eſpada. y eſtaràn tambien eſtos dos peſos en equilibrio, por la dicha propoſicion primera de Guido Baldo, en que mueſtra, que la proporcion que tiene la longitud D.A. à la A.C. que aqui es ſexquialtera, eſſa miſma tiene reciprocamente el peſo E. pendiente de C. de ſeis libras, al peſo G. pendiente de D. de quatro libras.

To regulate this power with another weight hanging from the tip of the sword D., a rule of three is formed, taking as the first term six, which is the length of the arm and sword D.A., as the second term A.C., which is four feet, and as the third term the weight hanging from point C. of six pounds; and performing the operation, the result is the weight G. of four pounds, hanging from the tip of the sword. These two weights will also be in equilibrium, by the aforementioned first proposition of Guido Baldo, where he shows that the proportion that the length D.A. has to A.C., which here is sesquialtera, is reciprocally the same that the weight E., hanging from C. of six pounds, has to the weight G., hanging from D. of four pounds.

Corolario

Corollary

De que ſe ſigue, que ſiendo A.B. igual à A.C. y el peſo F. igual al peſo E. la miſma proporcion que tiene D.A. con A.C. tiene D.A. con A.B. por la nona del quinto de Euclides: luego ſerà como D.A. à A.B. aſsi el peſo E. al peſo G. pero el peſo E. ſiendo igual al peſo F. por la propoſicion ſeptima, tambien del quinto, la linea D.A. à la linea A.B. tendrà la miſma proporcion, que el peſo F. al peſo G. reciprocamente ſexquialtera, que es lo que convenis demonſtrar, y lo demueſtra la primera figura de la Lamina F.G.

From this, it follows that since A.B. is equal to A.C. and weight F. is equal to weight E., the same proportion that D.A. has with A.C., D.A. has with A.B., according to Euclid’s ninth proposition of the fifth book: then as D.A. is to A.B., so is weight E. to weight G. But since weight E. is equal to weight F., according to proposition seven, also of the fifth book, the line D.A. to the line A.B. will have the same proportion, reciprocally sesquialtera, as weight F. to weight G., which is what was required to be demonstrated, and is demonstrated by the first figure of Plate F.G.

Explicacion de la figura ſegunda

Explanation of the Second Figure

En eſta ſegunda figura ſe dà la miſma longitud que en la antecedente al brazo, y Eſpada A.D. de ſeis pies, y la A.C. de tres, y el peſo G. de quatro libras pendiente de la punta de la Eſpada D. y ſe quiere examinar, reſpecto de eſta potencia el peſo que ſe podrà ſuſtentar en punto N. diſtante tres pies del centro del brazo A.

In this second figure, the same length as in the previous one is given to the arm and sword A.D. of six feet, and A.C. of three feet, and weight G. of four pounds hanging from the tip of the sword D., and it is desired to examine, in respect to this power, the weight that can be sustained at point N., three feet away from the center of the arm A.

Dizieſe por regla de tres la proporcion que tiene la longitud A.N. de tres pies, con la longitud A.D. de ſeis, eſſa miſma tendrà el peſo G. de quatro libras, pendiente de punto D. à otro, y haziendo la operacion, reſultarà por cociente en proporcion dupla el peſo F. de ocho libras, pendiente de punto N. que es lo que ſe podrà ſuſtentar ſegun eſta potencia, que repreſenta el peſto E. de ocho libras, pendiente de punto C. que es lo que convenia demonſtrar en la ſegunda figura.

By rule of three, the proportion that the length A.N. of three feet has to the length A.D. of six feet is the same as the weight G. of four pounds, hanging from point D., to another; and performing the operation, the result is, in double proportion, the weight F. of eight pounds, hanging from point N., which is what can be sustained according to this power, represented by the weight E. of eight pounds, hanging from point C., which is what was required to be demonstrated in the second figure.

Explicacion de la figura tercera

Explanation of the Third Figure

En eſta tercera figura ſe dà la miſma longitud que en las antecendentes, de ſeis pies al brazo, y Eſpada A.D. y la A.C. de dos pies, y el peſo G. de quatro libras, pendiente de la punto de la Eſpada D. y reſpecto de eſta potencia, ſe quiere examinar el peſo , que ſe podrà ſuſtentar en punto P. en la muñeca, diſtante dos pies del centro del brazo A.

In this third figure, the same length as in the previous ones of six feet is given to the arm and sword A.D., and A.C. of two feet, and weight G. of four pounds, hanging from the point of the sword D., and, in respect to this power, it is desired to examine the weight that can be sustained at point P. on the wrist, two feet away from the center of the arm A.

Dizieſe por regla de tres la proporcion que tiene la longitud P.A. de dos pies, à la longitud A.D. de ſeis, eſſa miſma tendrà el peſo de quatro libras G. pendiente de punto D. à otro; y haziendo la operacion, reſultarà por cociente el peſo F. de doze libras, pendiente de la muñeca punto F. y eſtarà en proporcion tripla, con el peſo G. pendiente de la punta de la Eſpada D. que es lo que convenia demonſtrar.

By rule of three, the proportion that the length P.A. of two feet has to the length A.D. of six feet is the same as the weight of four pounds G. hanging from point D. to another; and performing the operation, the result is weight F. of twelve pounds, hanging from the wrist point F., and it will be in triple proportion to the weight G. hanging from the tip of the sword D., which is what was required to be demonstrated.

Corolario

Corollary

De eſtas tres demonſtraciones ſe ſigue, que en las diviſiones, y grados de la Eſpada, que eſtuvieren mas cerca de la guarnicion, y brazo, ſe tendrà mayor fuerça, que en las diviſiones, y grados mas remotos; y que para exmainar la potencia que ſe tendrà en cada vno, proporcionando las longitudes mayores, y menores por la orden que ſe ha tenido en eſtas demonſtraciones, qualquiera podrà examinar la fuerça, y flaqueza, que tendrà, ſegun ſu potencia, en qualquiera grado de ſu Eſpada.

From these three demonstrations, it follows that in the divisions and degrees of the sword that are closer to the guard and arm, there will be greater strength than in the divisions and degrees that are more remote; and to examine the power that will be had in each one, proportioning the greater and lesser lengths in the order that has been maintained in these demonstrations, anyone can examine the strength and weakness that will be had, according to their power, in any degree of their sword.

Lamina treze del Libro ſegundo.

Plate thirteen of Book Two.