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Arquitecto Hist. exacta SCI. 60 (2006) 157 – 207 Identificador de objeto (DOI) digital 10.1007/s00407-005-0107-z

Interpretación de Newton de la segunda ley de Newton Bruce Pourciau Comunicado por G.E. Smith Introducción Una larga línea de newtonianos eruditos y comentaristas – de Hopkins, Thomson y Tait y Maxwell a mediados de 1800, a través de Dijksterhuis, Ellis, Herivel, Dolby, Perl, Hankins, Cohen, Truesdell, páramos de Poniente y Nicholas en la década de 1960 y 70, hasta Harman, Erlichson, Blay, Chandrasekhar y Cohen (otra vez) en la última década y media – ha argumentado que la segunda ley de Newton del movimiento, según se recoge en el Principia, se aplica sólo a una "fuerza impulsiva", es decir, a la representación matemática de una "instantánea impacto,"y no, al menos no directamente, a una"fuerza constante". El presente estudio, funcionamiento contrario a esta visión predominante, toma la posición de que la interpretación sólo impulso es una misinterpretación, hecho que en la interpretación de Newton la ley sec-ond se aplica y pretende que se aplique directamente a fuerzas impulsivas y continuadas. Como evidencia para este cargo, en primer lugar demostramos que el caso de que la interpretación sólo impulso es débil, presentando counterarguments a los argumentos que han sido ofrecidos en los años en apoyo de la interpretación sólo impulso. Luego hacemos el caso más directamente contra el impulso sólo para ver, como que llevar a cabo las diferentes pruebas, corolarios, definiciones, Escolios, ejemplos, revisiones y observaciones en la obra de Newton dynami-cal que aparecen confunden, incongruente o incorrecta cuando asumimos la segunda ley se aplica sólo a los impulsos. Pasar de la crítica a la constructiva y con dos ejemplos concretos para guiarnos, observaciones de Newton sobre colisiones oblicuas en el "libro basura" y su análisis del movimiento elíptico en el "manuscrito de Locke", desarrollamos a continuación lo que llamamos la segunda ley de la Com-libra , una interpretación alternativa de la segunda ley que se aplica directamente a las fuerzas impulsivas y continuadas. Después de una pausa para hacer varios comentarios sobre la segunda ley compuesta, comenzamos a presentar pruebas de que esta interpretación es interpretación de Newton de su segunda ley. Para este caso, mostramos cómo el compuesto segunda ley explica, simplemente y naturalmente, todos los pasajes en la obra de Newton que apareció tan problemático bajo una interpretación sólo impulso. (De hecho, la segunda ley de la com-libra es consistente con cada referencia a la segunda ley, o sus antepasados, en el trabajo de Newton en "racional mecánica" – en el "libro de residuos", en las versiones más últimas de "De motu", Principiay el de Newton prevista revisiones de los Principia en el 1690s temprano.) El poder explicativo de la interpretación compuesta, la sencillez y la armonía revela en el Principia y el trabajo de Newton en mecánica general, consideramos que esta convincente evidencia de que la segunda ley compuesta es en realidad de Newton interpretación. Por supuesto en los debates sobre la interpretación del texto, los argumentos son raramente decisivos, pero

nuestro caso termina con los testigos de más autoritario: en un manuscrito del 1690s temprano, Newton escribe con cuidado el significado exacto de su segunda ley, y lo que escribe es la segunda ley del compuesto.

Argumentos en contra de los argumentos para una interpretación sólo impulso 1. Abajo comenzamos a revisar los argumentos expuestos por los newtonianos eruditos y comentaristas que han apoyado una interpretación sólo impulso de la segunda ley, y como revisamos también se refutan, proponiendo counterarguments para cada argumento. ¿El punto de? Para demostrar que el caso de que se ha hecho una interpretación sólo impulso es débil, que los argumentos, incluso cuando se toman todos juntos, en última instancia, no persuadir. Pero antes de que podemos empezar este examen y refutación, debemos conocer

más acerca de la interpretación sólo impulso de sí mismo. Qué significado estos eruditos y comentaristas dan a declaración de Newton en los Principia de la segunda ley: Ley 2 Un cambio de movimiento es proporcional a la fuerza motriz impresionado y se lleva a cabo a lo largo de la línea recta en la que la fuerza que está impresionado. [32, 416]

En particular, ¿qué significado que dan a "un cambio en el movimiento"? ¿A la "fuerza motriz impresionado"? ¿A "lleva a cabo a lo largo de la línea recta en la que se impresiona la fuerza"? Los partidarios de la interpretación sólo impulso generalmente toman el "cambio en mo-ción" para ser el cambio en la "cantidad de movimiento," donde según la definición de Newton la "cantidad de movimiento es una medida de la propuesta que surge de la la velocidad y la cantidad de materia conjuntamente. " [32, 404] Por lo tanto el "cambio en el movimiento" se toma para ser el cambio M V = M V en lo que llamamos impulso lineal, donde M está parado para "Cantidad del neutonio de materia" y AV denota el cambio en la velocidad. Algunos comentaristas-tators (por ejemplo, vea [12, 472; 8, 154; 9, 164 – 165; 25, 243]) sugieren que la V tiene una dirección, presumiblemente "a lo largo de la línea en la que [la] fuerza está impresionada" y para indicar −→

Esto escriben V o v en lugar de V , pero estos autores no siempre dejar claro lo que habría sido el significado de tal un "dirigido el cambio en velocidad o velocidad" a Newton. El "motivo impresionado de la fuerza" se supone que la medida de un "impulso" , una "fuerza actuando en una sola instantánea," cambiar bruscamente la velocidad o dirección del movimiento. Estos comentaristas no ofrecen una definición independiente o forma de medir este impulso; así que la interpretación sólo impulso de la segunda ley, oM∝

∝ MV

Me

V

I,

se convierte, como Cohen admite, "básicamente un tipo de definición," [8, 158] en lugar de una ley física. Ahora, ¿qué argumentos se ofrecen en apoyo de esta interpretación sólo impulso? Tamizar a través de la literatura – de Hopkins, Kelvin y Tait en la década de 1800 a Blay y Cohen en 2001 – nos encontramos con un total de nueve argumentos justificativos. Nos contrarrestar estos argumentos que consideramos:

Argumento 1: ancestros de la segunda declaración de la ley en los Principia aparecen en el "libro basura" (un libro común que contiene los primeros trabajos de Newton en lo que más tarde llamó "mecánica racional"), y estos antepasados emplean la cantidad de movimiento para definir o medir una fuerza impulsiva. [8, 192; 21, 5 y 30 – 31; 33, 111 y 115; 44, 429] Esto es cierto. Por ejemplo, en el manuscrito etiquetado IId por Herivel [21, 141], vemos la siguiente afirmación en una larga lista de lo que Newton llama "Axiomes y propuestas": 4. SOE mucho la fuerza es necesaria para destruir cualquier cantidad de movimiento en un cuerpo, soe mucho se requiere generar; y soe tanto como es necesaria para generar el soe es alsoe necesaria para destruir lo

Por otro lado, este "axiome," quizás más una definición que una ley de la naturaleza, podría ser un antepasado, no de la Ley 2 en los Principia, pero de la definición de Newton de "cantidad de motivo de fuerza" en los Principia. (Más sobre el "fuerza motriz" más adelante.) Pero incluso suppos-ing este "axiome" es un antepasado de la Ley 2, es un antepasado muy temprano , habiendo sido escribió en el libro"residuos" en 1665 o 1666, cuando Newton era todavía un estudiante en Cam-bridge. Dos ancestros más recientes de la segunda ley do no apoyo el impulso sólo de vista. Casi veinte años después de Newton registró su "axiomes" en el "libro de residuos"

– Whiteside sugiere la fecha diciembre de 1684 [29, VI, 74]) – Newton inserta en una revisión de "De motu" este antepasado:

Lex 2.

El cambio de estado de movimiento o reposo durante un tiempo proporcional a lafuerza aplicada y se hace en línea cuando la fuerza está impresionado.

Este "Lex 2", escribe Herivel [21, 31], es "empleado allí [en"De motu"] en la demostración del lema 1 y el teorema 4, donde se asegura que las fuerzas iguales producen desviaciones igual (en tiempos iguales)." Estas "desviaciones" son desviaciones o desviaciones de la tangente a la trayectoria producido por un continuo de fuerza, no una fuerza impulsiva. Así este segundo antepasado de ley, en una versión aumentada de las vías originales "De motu", emplea la desviación de la tangente, no la "cantidad de movimiento," para medir la fuerza centrípeta y la ley es aplicada directamente (es decir, sin ningún aproximación por impulsos) a una fuerza continua . Un segundo antepasado de los Principiade ley 2, otra vez más reciente que nada en el "libro de residuos," se puede encontrar en un manuscrito de Newton enviada en marzo de 1690 a John Locke, quien antes había pedido a Newton si la "verdad de las dos proposiciones fundamentales es decir, las proposiciones 1 y 11 en el libro uno, no se podría demostrar de manera sencilla algunos más. " [5, 176] (Este manuscrito"Locke" es creído por algunos eruditos para ser una copia de un manuscrito original compuesto por principios, tan pronto pos-camélidos como 1679. La datación es incierta.) En la página de apertura del manuscrito de Locke [21, 246] encontramos: HYP. 2. la alteración del movimiento es siempre proporcional a la fuerza por la que se modifica.

Este antepasado de la segunda ley se aplica por Newton directamente (sin cualquier aproxim-ción por impulsos) a la fuerza centrípeta continua asociada a un cuerpo en movimiento elíptico sobre un foco [21, 255, nota d1 ], y se utiliza como "Lex. 2" en"De motu", para asegurarse de que las fuerzas iguales generan desviaciones igual desde la tangente.

Así, mientras que temprano "libro basura" las entradas reflejan los estudios de Newton de choques y los impactos y la medida (o definición) de un impulso por la cantidad de movimiento

(cuando la fuerza está en la dirección del movimiento), hemos visto que más reciente ances-Tor de los Principiade segunda ley refleja interés más maduro de Newton por problemas de fuerza continua y estos antepasados más recientes medir esa fuerza continua directamente, no por un cambio en la cantidad de movimiento, sino por la desviación de la tangente (que se desarrolla en un momento dado). Tomando una vista sólo por el impulso de la segunda ley en los Principia nos aprieta por lo tanto en una situación muy incómoda: nos vemos obligados aasume que la segunda ley en los Principia es menos como sus antepasados más recientes (que se aplica directamente a las fuerzas continuas) y más parecida a sus ancestros menos recientes (que se aplican sólo a los impactos). Argumento 2: por el "cambio en el movimiento" en la segunda ley, Newton debe significar el cambio en la cantidad de movimiento, es decir, M V = M V , donde V está parado para el cambio de velocidad, y M V puede ser una medida razonable de la fuerza sólo cuando la fuerza es impulsiva. [32, 111 10, 65-66; 2, 226; 14, 275; 19, 46] Aquí están cinco counterarguments a esta afirmación: (a) "La segunda ley dice que un 'cambio en movimiento' es proporcional a la ' motivo fuerza impresionada,"' escribe Cohen. [10, 65; véase también 8, 144] "Algunos comentaristas," él continúa, ha agregado una palabra o frase a la ley de Newton para que lo lea que la tasa de "cambio en el movimiento" (o el cambio de movimiento por unidad de tiempo) es proporcional a la fuerza. Esta alteración sería segunda ley de Newton leer como el encontrado en los libros de física de hoy.

Newton, sin embargo, no hizo un error aquí. Escogió sus palabras cuidadosamente. En su formulación de la segunda ley, Newton fue declarando explícitamente una ley para las fuerzas impulsivas, no para las fuerzas continuas. Así segunda ley de Newton afirma acertadamente que una fuerza impulsiva, es decir, una fuerza que actúa instantáneamente . . . produce un cambio en la "cantidad de movimiento" o impulso.

Así Cohen reprende a comentaristas, como Rouse Ball y Max Jammer (véase [10, 83, Nota 17; 13, 109, nota 4]), que sería injustificadamente Inserte "tasa de" redacción de cuidado-ful de Newton de la segunda ley. Sin embargo en su siguiente frase Cohen inserta "cantidad de" la segunda ley con casi tan poca justificación. Si Newton era cuidadoso con palabras y fue, tal vez él realmente significa exactamente lo que dice: "cambio en el movimiento" y no "cambio de cantidad de movimiento". (b) La segunda ley aparece por primera vez, en algo parecido a la versión en los Principia, en una revisión de "De motu". 29, VI, 188ff La tercera (1727) edición de la Principia contiene el aspecto final de la segunda ley. En el medio, la segunda ley aparece varias veces más, en la impresión y entre los papeles de Newton: en otra refundición de "De motu", 29, VI, 92ff en el manuscrito de Locke [21, 246-256], en la primera (1687) edición de los Principia, en ocho diferentes recunstrucciones de la segunda ley como parte de planes de Newton en el 1690s temprano para una revisión radical de la primera edición [8, 160, 177] y luego en la segunda edición (1713). En conjunto, Newton tenía catorce oportunidades (que conocemos) para volver a escribir la segunda ley, y no una vez eso wordsmith cuidado reemplazar siempre "movimiento" con "cantidad de movimiento". Esto sugiere fuertemente que Newton realmente significa "cambio de movimiento" en su segunda ley y no "cambio de cantidad de movimiento." (c) "Def. 2 [en los Principia para la"cantidad de movimiento"]," dice Cohen [32, 95], "afirma que la medida del movimiento adoptado por Newton es presentarse uno de masa y velocidad, nuestro impulso. Aunque Newton no dice específicamente . . . es esta cantidad de movimiento que quiere decir cuando, como sucede a menudo, escribe simplemente "movimiento." " Pero no es el caso de que por "movimiento" Newton siempre significa "cantidad de movimiento". De hecho Newton utiliza la palabra "movimiento" en tres distintos sentidos técnicos. Utilizar en algunos lugares como una abreviatura para "cantidad de movimiento," como señala Cohen. Newton tiene un significado muy diferente en mente, sin embargo, en el título de la sección 6, libro 1: "para encontrar movimientos en dado órbitas." [32, 510] Aquí por el "movimiento" Newton significa una correspondencia entre tiempos y lugares como un cuerpo atraviesa una curva dada en el espacio. Él también utiliza "movimiento" en un tercer sentido, en el sentido de un "desplazamiento". Por ejemplo, en un libro de cuarenta páginas en la mano de Newton catalogada como Universidad biblioteca Cambridge manuscrito añadir. 4003 – la fecha de composición está en disputa, encontramos esta definición: "El movimiento es cambio de lugar". [21, 226]. Vea también [21, 208, 220, 227, 232]. En los Principia (como parte de la famosa scholium en "tiempo, espacio, lugar y movimiento"), Newton define movimiento absoluto y relativo como cambio de absolutos y relativos de posición [32, 409], respectivamente, y luego, en una oración como la Declaración de la segunda ley (que aparece sólo cuatro páginas más adelante), escribe de Newton: "el verdadero movimiento siempre es cambiado por fuerzas en un cuerpo en movimiento". [32, 412] Por el cambio en el "movimiento real" Newton manifiestamente significa un cambio de posición, es decir, una traducción o desplazamiento. De hecho, a lo largo de este cual scho claramente "cambio en el movimiento" significa una traducción de un lugar a otro. ¿Y qué es lo que sigue esta scholium en la página siguiente? Por qué la declaración de Newton de la segunda ley: "un cambio de movimiento es proporcional a la fuerza motriz impresionada . . .". [32, 416] Esto parece sugerir que el "cambio en el movimiento" en la segunda ley puede tener menos que ver con un cambio de velocidad y mucho más que ver con un cambio en la localización. (d) Sin duda hay ejemplos en el libro"residuos" y en otros manuscritos donde Newton explícitamente escribe que la cantidad de movimiento (es decir, M V = M V , donde V es un cambio en la velocidad) mide la fuerza (por ejemplo [21, 141 Axiome 4 y 158 Axiome 115), hay lugares en los Principia donde la segunda ley se cita expresamente a garantizar que el cambio de velocidad puede usarse para medir la fuerza (por ejemplo, libro 2, las proposiciones 3 y 8 [32, 635 y 650]), y hay muchos lugares en los Principia (en el libro 2 en particular) donde, con no segunda citación de la ley, el cambio en velocidad se utiliza para medir una fuerza. Pero todos estos casos implican una situación muy especial: la fuerza se mide por el cambio en velocidad es siempre una fuerza paralela a la direc-ción del movimiento. Claramente Newton hace citar la segunda ley en estas situaciones para garantizar que el incremento en la velocidad es proporcional a la fuerza. Pero lo hace no sigue de esto que debe leerse la declaración de la segunda ley diciendo que el incremento en la velocidad es proporcional a la fuerza. Todo lo que podemos deducir es que para el caso especial de una fuerza paralela a la dirección del movimiento, la segunda ley implica que el incremento en la velocidad es proporcional a esa fuerza. (Como veremos más adelante, la interpretación de la segunda ley que propondremos de hecho tiene como corolario que el incremento en la velocidad es proporcional a la fuerza paralela a la dirección del movimiento). De hecho un cambio V de velocidad (en contraste con un cambio en algún tipo de dirección "velocity") puede ser una medida razonable de la fuerza en dos casos: (i) la fuerza que se mide es paralela a la dirección del movimiento (tal vez porque un fuerza original ha sido resuelto en los componentes antes de la aplicación de la segunda ley) o (ii) la frase "V el cambio en la velocidad" se usa para significar el "cambio de velocidad en la

dirección del centro de fuerza" en lugar del cambio de velocidad a lo largo de la trayectoria del movimiento. (En caso de que (ii) la fuerza debe ser centrípeto y el centro de la fuerza debe ser conocido. Ver [46,

162 B. Pourciau 119-120].) En consecuencia, si el "cambio de movimiento" en la segunda ley significa "cambio de cantidad de movimiento", la segunda ley se aplicaría únicamente a los casos (i) y (ii). Una segunda ley nos puede decir nada sobre la resolución o capitalización (acciones de) las fuerzas y esto a su vez implicaría que 1 corolario de las leyes (que se refiere a la capitalización de las acciones de los impulsos) no es en realidad un corolario de la segunda ley. Pero Newton considera manifiestamente 1 corolario es el corolario de las leyes: de ahí su nombre, el corolario 1. Newton suponiendo que es correcto en esta creencia, entonces "cambio en el movimiento" en la segunda ley no puede significar un "cambio en la cantidad de movimiento". En efecto, Newton parece ser muy cuidadoso en su redacción, generalmente usando "el cambio de cantidad de movimiento" o "incremento en la velocidad [velocidad]" como una medida de una fuerza (o fuerza componente) sólo que cuando la fuerza es paralela a la dirección del movimiento y "cambio de movimiento"o"alteración de movimiento"cuando la fuerza puede ser oblicua a la dirección del movimiento. (e) En una revisión de "De Motu", en el manuscrito de Locke y en la Proposición 6 de la 1687 Principia la segunda ley es aplicada directamente a una fuerza continua para asegurar que la desviación de la tangente (generada en un tiempo determinado) proporcional a la fuerza. (Ver [21, 31 y 255; 5, 252; 30, 103].) Quizá esto indica que el "cambio de movimiento" en la segunda ley de las medidas de la "desviación de la tangente" y no un cambio en la velocidad de.

3 argumento: en su declaración de la segunda ley en los Principia, Newton no menciona un intervalo de tiempo durante el cual ocurre el cambio de"en movimiento". Por lo tanto, un intervalo de tiempo no participa en la segunda ley y resulta que la fuerza debe ser un impulso.[3, 56; 2, 226; 10, 66, 19, 46] Ofrecemos tres counterarguments: (a) Esto es como afirmar que la declaración hecha por un granjero que "la altura del maíz varía con la temperatura media" no puede implicar un intervalo de tiempo porque hace la declaración no menciona un intervalo de tiempo. Pero por supuesto la "temperatura media" por su definición depende sobre el cual se calcula el intervalo de tiempo y por lo que cualquier declaración que la "temperatura media" también depende de ese intervalo de tiempo. La similitud con la segunda ley es clara: la "fuerza motriz impresionado," por su propia definición como el "movimiento que [la fuerza] genera en un momento dado," [32, 407] depende manifiestamente en el intervalo de tiempo dado que se calcula y así la Declaración de la segunda ley, que contiene el "motivo impresionado de la fuerza", también debe depender de ese determinado intervalo de tiempo. (b) Un scholium final sección 1, libro 2, de los Principia cita la segunda ley en una explicación para las resistencias proporcionales al cuadrado de la velocidad [32, 641]: "en medios totalmente carente de rigidez," escribe Newton, las resistencias que encuentran los cuerpos son como el cuadrado de las velocidades. Por la acción de un cuerpo más rápido, un movimiento que es mayor en proporción a esa velocidad mayor es commu-nicated a una cantidad dada de la media en un menor tiempo; y así en un tiempo igual, porque una mayor cantidad del medio se altera, se comunica un movimiento mayor en proporción con el cuadrado de la velocidad (por las segunda y terceros leyes de movimiento) la resistencia es como el movimiento comunicado.

Esta explicación aclara dos cosas: el "movimiento comunicado" es el "cambio en el movimiento" de la segunda ley y el "movimiento comunicado" se comunica en un determinado intervalo de tiempo. Se deduce que el "cambio en el movimiento" en la declaración de la segunda ley de también se computa en un intervalo dado de tiempo.

Interpretación de Newton de la segunda ley de Newton

163

(c) En al menos dos lugares diferentes – hacia el final del manuscrito de Locke [16, 255, nota d 1] y en el argumento de la Proposición 6 en el 1687 Principia [5, nota 103, 252; 30, 26; 32, 454] – Newton cita la segunda ley para garantizar que la desviación de la tangente es proporcional a la fuerza (continua). Esto

sugiere que el "cambio de movimiento" en la segunda ley tiene algo que ver con la desviación, y claramente depende de la desviación de la tangente en el intervalo de tiempo sobre el cual se desarrolla esa desviación. Algunos eruditos han ofrecido una variación del argumento "sin intervalo de tiempo, por tanto, impulso": Argumento 4: por el "cambio en el movimiento" en la segunda ley, Newton significa los tiempos de "quan-tity de la materia" el cambio de una "velocidad direccional". Además, este cambio se calcula sin tener en cuenta un intervalo de tiempo, y esto nos dice que la fuerza es impulsiva. [9, 164 – 165; 13, 112; 12, 472; 25, 221] Decimos algo vagamente, "cambiar en una ' velocidad direccional,"' porque aparte de [25] estos autores no definen lo que significa este cambio! Lo que aparece en el sentido de este cambio en la velocidad de"direccional" es lo que vamos a escribir como v ≡ v1 − v0 , es decir, el moderno cambio en velocidad del vector. (Aquí v 1 es el vector cuya longitud y dirección igual a la velocidad instantánea y dirección, respectivamente, del cuerpo antes del impulso. El vector v2 registra la misma información, pero después del impulso.) No hay intervalo de tiempo es necesaria para calcular el vector diferencia v. Por lo tanto, no hay intervalo de tiempo le

se requiere para calcular el "cambio en el movimiento" siempre y cuando se fuera a interpretar el cambio en el movimiento como Mv = M vy esto sería consistente con el hecho de Newton no menciona un intervalo de tiempo en su declaración de la segunda ley. Esta coherencia se ve entonces por estos comentaristas como evidencia para una interpretación sólo impulso. Pero suponiendo que el "cambio en el movimiento" implica una "velocidad direccional" de algunos especie, Newton nunca definir o calcular este cambio en la manera moderna, sub−→

contratante los vectores v 0 y v 1 . Más bien él sería calcular la desviación o desviación LQ de uniforme generado movimiento de línea recta en un intervalo de tiempo dado h. (Aquí L es el cuerpo habría venido a través del movimiento de línea recta uniforme en tiempo h había el cuerpo sin obstáculos, Q es el lugar donde el cuerpo realmente viene en el tiempo de h −→

debido a la fuerza y LQ está parado para el segmento de línea dirigido desde L a Q.)

Así que para Newton un "cambio de velocidad direccional" no sería la moderna v ≡ v 1 −v0, −→ −→

pero algo la desviación en un tiempo fijo determinado: V ≡ LQ / h. (Nosotros usamos un signo igual con tres barras en el sentido de "es igual por definición.") Dado un impulso (y cualquier intervalo de tiempo −→ −→ h para la informática V ), se puede calcular v y V por separado y por supuesto que −→ −→ −→ −→ −→ −→

V = v: P L = v0h y P Q = v1h, de modo que LQ = Q P − P L = (v1 −v0) h = v· h . Pero estos dos cambios de "velocidad direccional" se definen y calculan, por tanto, de muy diferentes maneras, el punto es que para calcular (lo que necesita para ser) de Newton −→ −→

cambio en la "velocidad de direccional" V uno debe calcular primero la desviación LQy uno

164 B. Pourciau no se puede calcular esta desviación sin ser dado un intervalo de tiempo específico h. Si de hecho el "cambio de movimiento" en la segunda ley es la "cantidad de materia" veces un cambio en "velocidad direccional", como creo que es – más sobre esto más adelante – entonces realmente el −→ −→

opción razonable sólo para este "cambio de velocidad direccional" es V ≡ LQ / h, la deflexión generada en un momento dado. Se deduce que las dos afirmaciones de argumento 4 – el "cambio en el movimiento" implica un cambio en la velocidad de"direccional" que no implica un intervalo de tiempo – son incompatibles. Por otra parte, una vez estamos de acuerdo en cambio de que Newton en −→ −→

"velocidad direccional" debe ser V ≡ LQ / h, entonces el argumento de "no hay tiempo por lo tanto mencionado impulso" para la interpretación sólo impulso de la segunda ley se evapora, para la deformación por unidad de tiempo puede ser utilizado como una medida de fuerzas impulsivas ycontinua!

5 argumento: comentando la segunda ley, escribe Newton, "si algunos fuerza gen porta cualquier movimiento, dos veces la fuerza va a generar dos veces el movimiento, y tres veces la fuerza generará tres veces el movimiento de... ," y este comentario es coherente con una fuerza impulsiva. [Véase, por ejemplo, 32, 112; 10, 66] Es cierto este comentario de Newton es constante con una fuerza impulsiva. No es, sin embargo, necesariamente incompatible con una fuerza continua, dependiendo por supuesto en cómo Newton mide una continua fuerza. En este trabajo de hecho vamos construir una cierta interpretación de la segunda ley, que se aplica igualmente bien a impulsivo y contin-superfluo las fuerzas y entonces sostienen que esta interpretación del"compuesta" es de Newton lo que significa de la segunda ley de la intención. Desde Newton "dos veces la fuerza de" observación resulta para ser completamente consistentes con esta interpretación compuesta de la segunda ley, es difícil ver esta observación como evidencia para una interpretación sólo impulso. 6 argumento: Newton de "dos veces la fuerza" comentario sigue: "... Si la fuerza se impresionó a la vez o sucesivamente grados." La descripción "de una vez" ("simul & semel) sugiere un impulso y"sucesivamente por grados"(" gradatim et sucesivos "), porque"gradatim"significa"paso a paso", indica, no es una fuerza continua, sino más bien una serie de impulsos. [32, 112; 8, 148] Por otro lado, entre actualizaciones varias de Newton de la segunda ley en escrito en el 1690s temprano, nos encontramos con dos casos. [8, 168] En el caso 1 que la fuerza es impresionada en una vez ("simul & semel"), mientras que en el caso 2 la fuerza es impresionados continuamente ("por-petuo"). En este segundo caso, el "modo de acción," escribe Cohen, "debe ser ' gradatim et sucesivas,' que ahora se describe como ' perpetuo."' Esto sugiere que Newton, como él considera modificaciones en la redacción de la segunda ley en el 1690s, decidió que la división "en conjunto y a la vez" ("simul & semel'') versus"continuamente"("perpetuo") mejor refleja su intención de que la división original "en conjunto y a la vez" versus "suc-cessively por grados" ("gradatim et sucesivas"). Y además esto sugeriría que Newton 1 pensó de la Ley 2 que se aplica directamente al impulsivo y directamente a las fuerzas continuas.

1

Cohen [8, 148] y Cohen Whitman [32] traducen "perpetuo" como "continuamente" en lugar de "continuamente". En mi opinión esto es una traducción inexacta, para "perpetuo" no permite interrupciones, mientras que "continuamente" en uso moderno. Cohen [32, 41] defiende su interpretación de "perpetuo", señalando por ejemplo que el manuscrito escrito en inglés que Newton enviada a John Locke 1690 utiliza "continuamente" en exactamente las situaciones donde Newton utiliza "perpetuo" en manuscritos en latín. Pero este argumento pierde su fuerza cuando descubrimos que en la de Newton

Interpretación de Newton de la segunda ley de Newton

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7 argumento: ver que la fuerza en la segunda ley debe ser un impulso, solo necesitamos mirar 1 corolario de las leyes. Este corolario refiere manifiestamente a impulsos sólo. [10, 66; 12, 472; 14, 274]

Sí y no. Sí, corolario 1 se refiere a impulsos sólo. Pero no, esto no es evidencia de una impulso sólo interpretación de la segunda ley. Nos dice que Newton pensaba que de la segunda ley que se aplica a un impulso. Que no nos dicen que Newton pensaba que de la segunda ley que se aplica a sóloimpulsos. (De hecho otros corolarios de las leyes, como veremos más adelante, parecen implicar una fuerza continua .) 8 argumento: en la demostración de la 1 de la Proposición fundamental de los Principia, Newton pretende producir la acción de una fuerza centrípeta continua construyendo el límite de una secuencia de acciones impulsivas de corolario 1 de las leyes. ¿Por qué se utiliza una secuencia de impulsos y corolario 1 (que se aplica sólo a los impulsos) si su segunda ley se aplica directamente a una fuerza continua? (Ver, por ejemplo, [12, 472]). De la forma de la demostración de la Proposición 1 podemos seguramente concluir en primer lugar que Newton ve tomando un límite de impulsos como al menos una forma de construir una fuerza que "ininterrumpidamente" y en segundo lugar lo tiene la intención de 2 de la ley a aplicar por lo menos a los impulsos. Pero sin duda no podemos concluir que la segunda ley se pretende aplicar a impulsos sólo. Newton puede bien han optado por tener una de impulsos en la prueba de la Proposición 1, no porque la segunda ley se aplica sólo a los impulsos, sino porque con esta estrategia podría construir una corta demostración que halló convincentes. (Leído [38] para un análisis extendido de los Principiade argumento por la Proposición 1. Vea también [28]. En la construcción de una "fuerza constante" de límite de los impulsos, vea [39]). Argumento 9: en la sección 11 del libro 1 [32, 561], Newton escribe que le es "considera-ción centrípeta de las fuerzas como atractivos, aunque tal vez, si hablamos en el lenguaje de la física – podrían más verdaderamente llamar impulsos." Esto ilustra que en los Principia fuerza impulsiva es primario, mientras que fuerza continua es derivado de la. [32, 113] Aviso de condición de Newton: "si hablamos en el lenguaje de la física." Y continúa en la siguiente frase para recordarnos que "aquí estamos interesados en las matemáticas; y por lo tanto, dejando a un lado cualquier debate sobre física, estamos utilizando lenguaje familiar con el fin de ser más fácilmente entendido por lectores matemáticos." Newton claramente desea subrayar la diferencia entre física y fenómenos físicos, por un lado y matemáticas y representaciones matemáticas en la otra. Bien puede creer la fuerza centrípeta de la gravitación física a ser impulsivo, de hecho él puede ser imposible-sible concebir otra posibilidad, pero una convicción sobre la naturaleza física de la gravitación ciertamente no impediría De Newton matemáticas representaciones de fuerza centrípeta de ser representaciones de fuerza continua, como la fuerza que "actúa en forma ininterrumpida" en la discusión de la Proposición 1 del libro 1. Tampoco evitaría la segunda ley de ser directamente aplicables a las fuerzas continuas. Las proposiciones del libro 1 son las proposiciones matemáticas , que involucra las representaciones matemáticas de

tiempo "continuamente" había tenido ningún significado secundario, como lo hace hoy en día, que permitió interrupciones. (En Un diccionario [11], por ejemplo, publicó en 1684, Elisha Coles define "continua" como "sin interrupción.") Esto sugeriría que a lo largo de Cohen y de Whitman fina nueva traducción de los Principia cada "continuallly" derivan "perpetuo" debería sustituirse por "continuamente" o "ininterrumpidamente".

166 B. Pourciau fuerza centrípeta y seguramente no sorprende, aunque cree que Newton gravitación física a ser impulsivo, que sus representaciones matemáticas de la atracción son representaciones de fuerza continua. Para un movimiento generado por una fuerza continua debe ser lo que llamamos más "suave" movimiento – donde la velocidad y la dirección varían continuamente, mientras que debe ser un movimiento generado por una fuerza impulsiva "nonsmooth" o "desigual", con cambios bruscos de velocidad o dirección, y Herramientas matemáticas de Newton – por ejemplo el var-pagarés geométrico limitantes técnicas en la sección 1, libro 1 – requieren cierta suavidad antes de que se pueden aplicar. En otras palabras, si Newton no había asumido la continuidad de sus fuerzas (matemático representaciones de), para que los movimientos generados sea lisos, entonces su capacidad de establecer teoremas significativos habría sido disminuido grandemente.

Argumentos en contra de una interpretación sólo impulso

2. Una vez más, el plan para el presente estudio es presentar el caso contra una interpretación sólo impulso de la segunda ley, entonces para desarrollar, desde un estudio de la obra de Newton, una particular interpretación, la segunda ley compuesto como lo llamamos, que se aplica por igual a fuerzas impulsivas y continuo y finalmente argumentar que esta interpretación del"compuesta" es en realidad la interpretación de Newton. En la sección anterior, al comenzar el caso contra una interpretación sólo impulso, tenemos reunidos los argu-momentos que han sido propuestos sobre los años para apoyar la afirmación de que Newton tiene la intención de la segunda ley para aplicar a las fuerzas impulsivas sólo y para cada uno tal argumento nos ofrece nuestras propia counterarguments. Hasta ahora, entonces, nos hemos estado argumentando contra la evidencia para un impulso sólo de interpretación. Pero ahora comenzaremos discutiendo directamente contra una interpretación sólo impulso, señalando las aplicaciones de la segunda ley, com-ments en escolios, definiciones, ejemplos, corolarios, demostraciones y las revisiones, en la Principia y revisiones previstas de Newton en su segunda edición – que parecen ser demasiado compleja, confusa, inconsistente, antinatural o incorrecta cuando asumimos una interpretación sólo impulso de la Ley 2. Empezamos con los ejemplos que Newton utiliza para ilustrar el concepto de fuerza centrípeta. Según la declaración de la segunda ley, "un cambio de movimiento es adicional proporcional a la fuerza motriz impresionada. . .". [32, 416] Pero la única "fuerza de motivo" Newton define es el "motivo cantidad de centrípeta fuerza" [32, 407] en 8 de definición anteriores a las leyes y todos cuatro ejemplos dados fuerza centrípeta implican fuerzas continua : gravedad, fuerza magnética, la fuerza "por que los planetas están continuamente [es decir,"cont-uously,"– véase la nota 1 anterior] retirado de movimientos rectilíneos," y la "fuerza que [a] Honda continuamente [continuamente] dibuja la piedra hacia atrás, hacia la mano". [32, 405] (De hecho en lo Principia de Newton nunca llama a un impulso centrípeto. Las fuerzas centrípetas son siempre fuerzas continuas. Una serie de impulsos podría llamarse centrípeta, si cada impulso en la serie se dirige hacia el mismo punto fijo, pero sería tonto llamar a un único impulso centrípeto, puesto que un solo impulso tiene una sola dirección.) Así debe el "fuerza motriz" que aparece en la segunda ley "Motivo cantidad de fuerza centrípeta" de Newton porque es la única "fuerza motriz" que él define y los únicos ejemplos de fuerza centrípeta que él ofrece son ejemplos de fuerza continua.

Interpretación de Newton de la segunda ley de Newton

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Bajo estas condiciones parece difícilmente creíble que Newton tiene la intención de la segunda ley para aplicar a las fuerzas impulsivas sólo. Hablar de la fuerza centrípeta, nos permite mirar más cuidadosamente 8 definición: "la cantidad de motivo de fuerza centrípeta es la medida de esta fuerza que es proporcional al movimiento que genera en un momento dado". [32, 407] Como nos vamos más adelante, Newton tiene en mente la siguiente medida de fuerza: suponga que la fuerza mueve un cuerpo, inicialmente en reposo, desde el lugar P al lugar G (a lo largo de la línea a través P y el centro de la fuerza) en un momento dado h . Entonces la "cantidad de motivo de fuerza centrípeta" P G (o simplemente la "fuerza motriz") es la cantidad M h , donde M es la cantidad de"materia" del cuerpo. (La cantidad de −→

M

P G h se puede pensar en como teniendo la dirección del segmento de línea dirigido P G. Por lo tanto, −→

P G vamos a menudo a elegir, anacrónicamente, a escribir la fuerza motriz como M h .) Observar que esta medida de fuerza centrípeta aplica igualmente bien impulsivo y fuerzas continuas. (Por supuesto en el caso la P G medida M h tiende a cero como el tiempo transcurrido h tiende hacia cero. Pero esto no es un problema para Newton, por gener-aliado compara las medidas por la formación de su relación, y cuando la parte superior e inferior de una relación de cada enfoque cero, su proporción no es necesario.) Sin duda sería extraño si la segunda ley se pretende aplicar a fuerzas impulsivas sólo, cuando la medida de fuerza en la segunda ley, es decir, la "fuerza impresionado, motivo" se aplica igualmente bien a las fuerzas continuas. De hecho en su discusión siguiendo la definición 8, Newton hace evidente que la "cantidad de fuerza de motivo" puede utilizarse para medir la magnitud de una fuerza continua, pues observa que la "fuerza motriz [surge] de la fuerza despertada y la cantidad de materia conjuntamente." [32, 407] Por la "fuerza despertado él significa la" velocidad [es decir, la "velocidad" – ver Nota 2 abajo] que [la fuerza] genera en un momento dado, "[32, P G definición 7, 407] es decir, h en nuestra notación, una medida que se aplica igualmente bien a impulsivas y continuadas las fuerzas. Pero el término "despertado" es generalmente utilizado por Newton sólo cuando la velocidad cambia, no bruscamente, pero continuamente con el tiempo. Sigue que la fuerza de Newton"motiva" debe aplicarse a las fuerzas continuas , y esto a su vez sugiere que la segunda ley, que prevé el "cambio en el movimiento" de la "fuerza motriz", debe aplicarse también a las fuerzas continuadas.

3. El término "despertado" aparece otra vez en corolario 6 de las leyes [32, 423]: Corolario 6 Si los cuerpos se están moviendo en forma alguna con respecto a uno con el otro y se insta por igual fuerzas despertadas a lo largo de líneas paralelas, todas siguen a mover con respecto a otros de la misma manera como lo harían si no eran actuados sobre por esas fuerzas.

Demostración corta de Newton hace un llamado a la segunda ley: "para esas fuerzas, actuando por igual (proporcionalmente a las cantidades de los cuerpos a moverse) y a lo largo de líneas paralelas, voluntad (Ley 2) mover todos los cuerpos igualmente (con respecto a la velocidad) y por lo tanto nunca cambiar sus posiciones y movimientos con respecto a uno con el otro". Newton nunca habla de un impulso como generando una aceleración. Solamente continua fuerzas aceleran cuerpos. Sigue que Newton tiene una fuerza continua presente en corolario 6. Pero puesto que él invoca Ley 2 en la prueba, sólo podemos concluir que la segunda ley se aplica a las fuerzas continuas. Así, una interpretación sólo impulso de la segunda ley parece ser incompatible con la demostración de Newton de corolario 6.

168 B. Pourciau 4. En el scholium después corolario 6, encontramos famoso (o infame) Galileo atribución del neutonio [32, 424]: Por medio de las dos primeras leyes y los dos primeros corolarios Galileo encontró que el descenso de los cuerpos pesados está en la relación cuadrada del tiempo y que el movimiento de proyectiles se produce en una parábola, como experimento confirma, excepción en la medida en que estos movimientos son un poco retardado por la resistencia del aire.

Eruditos que apoyan una interpretación sólo impulso de la segunda ley toman una mala opinión de esta atribución. Viendo como el "cambio en el movimiento" en la segunda ley como un instantáneo cambio en momentum (lineal o direccional), estos estudiosos concluyen reclamación de que Newton – que Galileo 2 utilizó la segunda ley para derivar el t Ley de caída y de la trayectoria parabólica de los proyectiles, es mejor en serio exagerado y en el peor com-pletamente incorrecta. (En nuestra opinión, por supuesto, es su interpretación de la segunda ley que es incorrecto.) "Rastros inconfundibles de la génesis del [segundo] ley," escribe Herivel [21, 40, 41], se encuentran en [el "libro de residuos"]. Allí Newton define la fuerza como el cambio en el movimiento producido. Asimismo, el cambio en la Dirección del movimiento producido en un cuerpo está en la dirección de la actuación de la fuerza en él, "un cuerpo debe mover así que es presionado". Pero esta definición de fuerza, su conexión con el movimiento y su efecto en dirección surgió directamente de su discusión sobre el problema de las colisiones. La influencia, si fue por lo tanto, de Descartes, no de Galileo. . .. . . . Por último, no hay ninguna indicación en las primeras investigaciones de cualquier influencia de Galileo . . . en la génesis de la segunda ley del movimiento. . . . la posibil idad de cualquier influencia efectivamente es descartado por la forma real de la ley y su origen indudable en el estudio de Newton impulso y colisiones.

En un artículo dedicado a la mención de Galilea de Newton [7, XXXIX], Cohen afirma que suponer que Galileo sabía que la relación cuantitativa (proporción exacta) entre impulso y cambio en la cantidad de movimiento . . . o que Galileo había utilizado la segunda ley para encontrar la ley de la caída de los cuerpos . . . sería malinterpretar tanto la Diálogo y los Discorsi o imaginar lo que Galileo "debe haber hecho."

Cohen es contundente también en [8, 176], donde llama atribución de Newton una "mala representación completa del procedimiento de Galileo." Ellis [14, 227] es justo como directo: "Newton simplemente podría no tienen los conceptos que él utilizó en particular Ellis significa el cambio en el movimiento producido por un impulso en la interpretación de un sólo impulso de la segunda ley] del trabajo de Galileo." (Por supuesto los que objeto la atribución de las dos primeras leyes Galileo podrá objetar por razones distintas a su interpretación de la segunda ley, el hecho de que Galileo no habla de fuerzas impresionadas ser un.)

Según éstos y otros eruditos, entonces, una interpretación sólo impulso de la legislación sec-ond es incompatible con la exactitud de la atribución de Galileo de Newton. Creer en una interpretación sólo impulso es por lo tanto creo que Newton no sabía lo que estaba hablando cuando le atribuye a Galileo. Consideramos esto algo fuerte evi-anza contra una sólo impulso de segunda ley. Más adelante veremos que cuando la segunda ley es lo que sostenemos es la interpretación de Newton, una interpretación que llamamos la segunda ley compuesto, entonces segunda ley de Newton se convierte en una progresión natural de la hipótesis de Galileo sobre el movimiento y de Newton Galileo atribución luego

Interpretación de Newton de la segunda ley de Newton

169

se convierte en el más apropiado. De hecho la segunda ley compuesta abarca una reafirmación de la capitalización Galileo: en palabras de Galileo, "movimiento estable" (es decir, uniforme movimiento de línea recta sobre un plano horizontal) compuestos de forma independiente con el "movimiento naturalmente acelerado" de caída cuerpos para producir el movimiento parabólico real. (Véase [17, 268].) 5 . En la tercera edición (1727) de la Principia, Newton amplifica su Galileo atribuye-Tor, insertando ahora famoso "caer cuerpo" paso [32, 424]: Cuando un cuerpo cae, la gravedad uniforme, actuando igualmente en partículas individuales iguales de tiempo, impresiona igual fuerzas sobre ese cuerpo y genera velocidades iguales [velocidad]; y en el tiempo total que impresiona una fuerza total y genera una velocidad total [velocidad] proporcional al tiempo. Y los espacios descritos en tiempos proporcionales son como los tiempos y las velocidades [velocidad] conjuntamente, es decir, en la relación de cuadrados de los tiempos. . . . Y cuando un cuerpo es pro-jected a lo largo de cualquier línea recta, su movimiento derivados de la proyección se agrava con el movimiento de gravedad. . . . y la línea curva . . . que el cuerpo describirá una parábola. . ..

Newton considera claramente este pasaje "cayendo el cuerpo" como una aplicación sencilla y directa de las primeras y segunda leyes de movimiento. Sin embargo, mediante una interpretación sólo impulso de la legislación sec-ond, esta aplicación parece ser ni simple ni directa. De hecho, no un erudito, entre los muchos que han comentado en este pasaje, ha demostrado cómo la ley de la caída y la trayectoria parabólica se siga de las leyes de movimiento (por ellos mismos). (Lea, por ejemplo, [12, 476; 7; 8; 13; 14; 44, 521-522].) Esta circunstancia puede verse sólo como evidencia en contra de una interpretación sólo impulso de la segunda ley. Por el contrario, debemos demostrar en §30, t -ley cuadrada de la caída y la trayectoria parabólica siga fácilmente y naturalmente de la segunda ley compuesta (junto con 1 de la ley). Este éxito constituye una sola pieza, pero una pieza llamativa, de la evidencia de que la segunda ley compuesta es interpretación de Newton de la segunda ley.

6 . Consideremos ahora la Proposición 6 (libro 1) de la 1687 Principia. Esta proposición, que aparece como corolario 1 de una nueva proposición 6 en las segunda y terceros ediciones, registra una fórmula para medir la magnitud de una fuerza centrípeta continua. En nuestra reproducción parcial de la figura de Newton que ilustran la Proposición 6, un cuerpo P, giran sobre un centro de S, describe la línea curva APQ, mientras que la línea recta ZPR toca la curva en cualquier punto P; y QR, paralelo a distancia SP, se dibuja al gent tan desde cualquier otro punto Q de la curva, y QT es dibujada perpendicular a esa distancia SP.

En estas condiciones, la Proposición 6 concluye que en el límite como Q tiende a P, la "fuerza centrípeta es como · · ·

SP 2×QT 2

inversamente. " [32, 454; 5, 252] La demostración en QR

170 B. Pourciau la primera edición se abre con una invocación de la segunda ley: "para"la figura indefinidamente pequeño QRPT el elemento de línea naciente QR, si se da el tiempo, es como la fuerza centrípeta (por la Ley 2). . .. [32, 454, nota bb; 5, 252] Claramente se trata de una aplicación directa de la segunda ley para una fuerza continua. Observe también que la segunda ley se utiliza, no para predecir un cambio instantáneo de velocidad o "velocidad direccional", sino a predecir un cambio en la posición que se produce en un momento dado. Este cambio de posición puede verse como la desviación de R (aproximadamente donde el cuerpo habría sido en ausencia de la fuerza centrípeta) a Q (donde el cuerpo realmente termina en presencia de la fuerza). Sería difícil construir una aplicación de la segunda ley más desacuerdo con la afirmación de que la segunda ley se aplica solamente a una fuerza impulsiva produciendo un cambio brusco en la velocidad con el tiempo no transcurrido. 7 . Mientras continuamos a recoger evidencia en contra de una interpretación sólo impulso de la la segunda ley, nos movemos en las porciones posteriores de los Principia y parar en la Proposición 24 (libro 2), en el movimiento de un péndulo simple. Esta proposición refiere a movimiento bajo la influencia de una fuerza continua , y en su manifestación pide a la segunda ley Newton y lo aplica directamente (es decir, no a través de cualquier aproximación por impulsos) para el movimiento del péndulo: " Para la velocidad que puede generar una determinada fuerza en un momento dado en una determinada cantidad de materia es como la fuerza y el tiempo directamente la cuestión inversamente. Cuanto mayor sea la fuerza, o mayor el tiempo, o al menos la materia, mayor será la velocidad que se generarán. Esto es manifiesto de la segunda ley del movimiento". [32, 700] Descripción y aplicación de aquí la segunda ley de Newton es incompatible con una interpretación sólo impulso de la ley. 8 . Para las pruebas finales que nos sometemos en nuestro caso contra un sólo impulso inter-interpretación de la segunda ley, nos movemos hacia fuera de la publicado Principia y en revisiones propuestas de Newton para la segunda edición, revisiones en escrito en el 1690s temprano. Además de algunas reestructuraciones bastante grave de los Principiade primeras secciones, estos planes para la segunda edición incluyen algunas alteraciones menos dramáticos, más estilísticos, en varios axiomas, lemas y propuestas. [29, VI, 538-567] De particular relevancia para nosotros son rewordings propuesta de Newton de la segunda ley que se encuentran en las páginas del manuscrito de la colección de Portsmouth de la biblioteca de la universidad Cambridge (U.L.C. MS Add.

3965). [29, VI, 539-542; 8, 160 – 169] uno de estos rewordings dice lo siguiente: Derecho II Todo el nuevo movimiento por el cual se modifica el estado de un cuerpo es proporcional a la fuerza motriz impresionado y se produce desde el lugar que el cuerpo lo contrario ocuparía el objetivo al que apunta la fuerza impresionada.

Para ilustrar la ley, Newton dibuja la figura de abajo (en el que hemos sustituido sus cartas A, a, b, B con nuestra propia P, Q, L, G) –

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– una figura absolutamente reñidas con cualquier interpretación sólo impulso de la segunda ley: la "trayectoria es curva y no una línea recta," preocupa a Cohen (un partidario sólo de impulso), así parece implicar que el nuevo movimiento es acelerado y uniforme no . . . que la 'fuerza' debe ser continua, produciendo una aceleración constante, como gravedad, en lugar de un impulso. . .. Sin embargo a pesar de la figura, allí puede no ser duda el texto sí mismo Newton 'impresionó a fuerza' es impulsivo. . .. [8 164-165]

¿Por no qué él tiene duda la fuerza impulsiva? Porque en el texto de Newton afirma que el "movimiento" es proporcional a la "fuerza" y "sólo por impulsos", dice Cohen, "puede el 'movimiento' (que él interpreta como 'cantidad de movimiento,' o impulso) ser proporcional a la ' fuerza." " (Por supuesto nos diría que el problema es no con Newton, pero con la toma de "movimiento" que significa "cantidad de movimiento.'') Cohen debe de alguna manera explicar la trayectoria curvada que Newton utiliza para ilustrar este cambio de redacción de la segunda ley, pero no sólo no simple resolución de este conflicto, un conflicto creado por la postura sólo impulso de Cohen. Su explicación es una construcción compleja y enrevesada: The parabola-like orbit is thus an infinitesimal orbital segment, produced by a first-order force-impulse which itself proves to be compounded of an infinite number of second-order infinitesimal force impulses, each of which we may consider to be acting instantaneously in a time interval which is not the whole interval d t but rather d t /n as n → ∞, itself infinitesimally small. [8, 181]

e If “truth is ever found in simplicity, & not in y multiplicity & confusion of things,” as Newton tells us [26, 6; 31], then Cohen’s characterization cannot be the truth. Manifestly Newton himself regards the curved trajectory as a simple and obvious illustration of the second law. When a commitment to an impulse-only interpretation of the second law forces commentators into an overly complex explanation for what ought to be a simple matter, it is time to question that interpretation. As we shall see later, from the point of view we defend in this study – that Newton’s own interpretation of the second law is what we call the compound second law, which applies equally well to impulsive and continuous forces – the explanation for the curved trajectory is simplicity itself: it merely illustrates the meaning of the second law for the case of a continuous force!

Developing the compound second law 9. It is time to be less critical and more constructive. We now begin to develop the compound second law, an interpretation of Newton’s second law that applies equally well to both impulsive and continuous forces. Later, after we have made several comments on this compound interpretation in §14–19, we will argue that the compound second law is in fact Newton’s own understanding of his second law. Here, once again, is the statement of the second law from the Principia [32, 416]: Law 2 A change in motion is proportional to the motive force impressed and takes place along

the straight line in which that force is impressed.

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Deciding what Newton means by the second law amounts to deciding what he means by “a change in motion,” “the motive force impressed,” and “takes place along the straight line in which that force is impressed.” We begin with “a change in motion.” As we have seen, some impulse-only supporters assume that Newton somehow forgot to insert “quantitas” into “mutationem motus” in his statement of the second law, for they take “a change in motion” to mean “a change in the quantity of motion,” that is, to mean an instantaneous change in (linear) momentum

– this despite the fact that Newton, a very careful writer, rewrote the second law at least fourteen times between 1684 and 1693 and never once inserted the word “quantitas.” Perhaps we should assume Newton has some other meaning in mind for “a change in motion.” To see whether we can uncover that meaning, we first turn to Axiom 122 of the “Waste Book,” an axiom that involves the oblique collision between two (hard) bodies [21, 159]: Axiom 122 Therefore if the body p come from c and the body r from d soe much as p’s motion is changed towards w, so much the motion of r will be changed towards v.

To illustrate this assertion, Newton draws the figure below:

For the moment, let us consider this collision from a more modern stance: Suppose the two bodies, with (modern) masses m and n, move before the collision with constant (modern vector) velocities v0 and u0 and move after the collision with constant (mod-ern vector) velocities v1 and u1, respectively. (We strive in our notation, by choosing lower or upper case letters, to preserve the distinction between modern and Newtonian conceptions of similar notions. If, for example, by m we mean the modern mass of a body, then by M we would mean Newton’s “quantity of matter” in that body.) By the conservation of momentum, mv0 + nu0 = mv1 + nu1, and rearranging we have m(v1 − v0) = −n(u1 − u0). But we can write the lefthand side (as well as the right) in terms of the “deflection” that develops in a given time interval h:

Newton’s Interpretation of Newton’s Second Law −→

m(v m LQ . v ) m hv1 −hv0 h h = 1− 0 =

173

−→

Here LQ stands for the directed line segment from L to Q. We call this directed line segment the deflection (generated in time h) from the place L (where the first body would have been in time h in the absence of the collision) to the place Q (where the first body actually ends up in time h after the collision). (We might note here that although the −→

notation LQ for a directed line segment is certainly anachronistic, the notion of a line segment with a given direction, which occurs repeatedly in the Principia, is certainly −→

LQ not.) If we were to call the product m h the “change in motion” of the first body and if we were to define the change in motion for the second body similarly, then the above equation m(v1 − v0) = −n(u1 − u0), written in terms of deflections, asserts that “soe much as [the first body’s] motion is changed toward w, so much the motion of [the second body] will be changed toward v,” [21, 159] just as Newton claims in Axiom 122! Replacing m, standing for the modern mass, by M, standing for Newton’s “bulk” (moles, used in his earlier work, such as the “Laws of Motion Paper” [21, 208]) or his “quantity of matter” (massa, used in the Principia), it is natural then to ask whether the expression −→

LQ M h could be what Newton means by the “change in motion,” not just in Axiom 122 of the “Waste

Book,” but in the Principia’s second law as well. 10. For more evidence that the intended meaning of the “change in motion” in the second law involves the deflection (in a given time) from uniform straight line motion, we consider a second example of Newton’s work, this one a continuous rather than impulsive force example: Newton’s use of the second law in the so-called “Locke manu-script.” After the English philosopher John Locke in 1689 asked him whether the “truth of the two fundamental propositions, namely, Propositions 1 and 11 in Book One, could not be demonstrated in some more simple way,” Newton responded by sending Locke a manuscript written in English entitled “A Demonstration that the Planets by their gravity towards the sun may move in Ellipses.” [5, 176; 4; 18, 293–301; 21, 246–254] In this manuscript, found among Locke’s papers, Newton gives the following statement of his second law [21, 246]: Hyp. 2 The alteration of motion is ever proportional to the force by which it is altered.

Newton applies this version of the second law to measure the magnitude of the con-tinuous centripetal force in the case of a “body . . . attracted towards either focus of

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any Ellipsis and by that attraction . . . made to move in the Perimeter of the Ellipsis.” [21, 251] (In the figure below and in the subsequent lines from his demonstration, we have replaced Newton’s X, Y, and x, y with the letters L, Q and l, q, respectively, just −→

to be consistent with the notation LQ that we have already used for the deflection.) We reproduce part of the figure he draws to illustrate his argument:

“Let P be the place of the body in the Ellipsis at any moment of time,” writes Newton to begin his demonstration, and PL the tangent in which the body would move uniformly were it not attracted and L the place in that tangent at which it would arrive in any given part of time and Q the place in the perimeter of the Ellipsis at which the body doth arrive in the same time by means of the attraction. [21, 252] −→

The directed line segment LQ from L to Q is clearly the deflection we encountered ear-lier in Newton’s “Waste Book” analysis of an oblique collision, namely the deflection from L, where the body would have arrived in the given time “were it not attracted,” to Q, where the body actually arrives in the same given time “by means of the attraction.” “And because the attraction in P is made towards F,” the argument continues, and diverts the body from the tangent PL, . . . translat[ing the body] from L to Q: the line LQ generated by the force of attraction in P must be proportional to that force and parallel to its direction to PF. . . . And in like manner if pl be the tangent of the Ellipsis at p and lq [be the line generated in the same given time by the force of attraction in p] . . .

then “LQ [is] to lq therefore as the attraction in P [is] to the attraction in p by Hypoth: 2 and 3.” [21, 252, and 255 Note d1] (Hypothesis 3 is the parallelogram rule for “com-pounding motions.”) Thus Newton compares the magnitude of the force at P to the magnitude of the force at p by comparing the deflection LQ (generated in a given time) at P to the deflection lq (generated in that same given time) at p, and in support of this proportionality between deflection and force he cites Hypothesis 2, his English wording of the second law, that the “alteration of motion is ever proportional to the force.” Clearly Newton regards his citation of the second law (in the form of Hypothesis 2) as a simple and direct application of this law. Of course no scholar wedded to an impulse-only interpretation of the second law, with the “alteration of motion” taken to be the change in (linear) momentum due to an impulse, could ever see an application of the second law to a continuous force as either simple or direct, but because we do not share this impulse-only view, we are free to accept the simple and obvious explanation for this application of the second law in the Locke manuscript: Newton’s definition for the “alteration of motion” must make

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175

−→

LQ it manifestly proportional to the deflection h generated in a given time. Our earlier example, the “Waste

Book” analysis of an oblique collision, points to the “quantity of matter” M as the correct proportionality constant, which leads us to believe that New-ton’s “alteration of motion” in Hypothesis 2 of the Locke manuscript, and by natural

−→

LQ extension Newton’s “change in motion” in Law 2 of the Principia, is by definition M h . 11. Newton’s second law – “A change in motion is proportional to the motive force impressed. . .” – relates two different quantities: the “change in motion” and the “motive −→

LQ force.” We have just been arguing that by a “change in motion” Newton means M h . Now we ask, what does he mean by the “motive force”? The “motive force” is short for the “motive quantity of centripetal force,” where, according to Definition 8 of the Principia, “the motive quantity of centripetal force is the measure of this force that is proportional to the motion which it generates in a given time.” [32, 407] But how are we intended to understand the “motion which it generates in a given time”? In his comments that follow Definition 8, Newton says the “motive force [arises] from the accelerative force and the quantity of matter jointly.” And what does Newton mean by the “accelera-tive force”? The “accelerative force” is short for the “accelerative quantity of centripetal force,” which by Definition 7 is the “measure of the force that is proportional to the velocity which it generates in a given time.” [32, 407]. The meaning of the “motive force” in the second law therefore depends finally on the meaning of the “velocity which [the centripetal force] generates in a given time.” But the meaning of this ‘generated velocity’ is less clear than it might seem. Why? Because according to Newton the “accelerative force” refers to the “places of bodies [seeking a center],” [32, 407] and consequently the “accelerative force” cannot depend on the speed or direction of the body at the time when the force is applied, but only the location of the body. Yet whatever Newton might mean by the “velocity which [the force] generates in a given time,” without evidence to the contrary this ‘generated velocity’ might well vary with either the speed or direction of the body – unless of course we are guaranteed otherwise by some axiom or law. Since no such axiom or law has been stated at this point in the Principia (the definitions naturally precede the laws), we have to assume that Newton intends the “velocity which [the force] generates in a given time” to be measured on a body having some ‘standard’ speed and ‘standard’ direction. But the only speed that could reasonably serve as a ‘standard’ speed would be speed zero, and the only direction that could reasonably serve as a ‘standard’ direction would be along the line which passes through the center of force. (As Newton has restricted himself in these definitions to a centripetal force, we do in fact have a center of force.) In other words, the “velocity which [the force] generates in a given time” must be measured on a body initially at rest and subsequently moved by the force along a ray from the center of force. Without this assumption, Definition 7 (for the “accelerative force”) and hence Definition 8 (for the “motive force”) are rendered unusable by the a priori possibility that the “velocity which [the force] generates in a given time” might vary with the speed or direction which the body has when the force is applied. We have been arguing abstractly, not historically, but Newton himself confirms this reading of Definition 7. Just after Definition 7, he writes that the “force that produces gravity . . . is everywhere the same at equal distances, because it equally accelerates all

P G/ h h

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falling bodies. . ..” [32, 407] And much later in the Principia, in his discussion of Propo-sition 19 (Book 3) on the shape of a planet, he measures the motive force of gravity at the latitude of Paris by calculating the distance a body falls from rest at that latitude in one second: a “body falling in a vacuum will describe a space of 2,174 lines in the time of one second.” [32, 822] Not only Newton, but scientists of the 17th century generally, from Galileo onward – Mersenne, Riccioli, and Huygens, just to mention the most prominent – used the distance fallen in the first second as the measure of surface gravity. Of course in Newton’s case it is not just gravity that he measures by the movement produced on a body at rest, but other forces as well. In Propositions 35 and 38 of Book 2, for example, on the resistance to a globe moving in a “fluid medium,” he applies a nondimensional measure of resistance forces which, as he puts it in the first edition, uses “the force that, uniformly impressed . . . in the time in which the globe by progressing described 2/3 of its own diameter, could generate the globe’s velocity in that body.” [40, 308 and Note 42 on 293] As George Smith notes, the Latin in the first edition is open to two inter-pretations: “progressing uniformly at its acquired velocity” or “progressing from rest to its acquired velocity.” Given our contention that Newton generally measures forces with the displacement produced on a body at rest, we would naturally vote for the latter interpretation, and indeed, when Newton rewrote this section for the second edition, his revision removed any doubt that he had the “from rest” interpretation in mind.

Let us agree then that the “accelerative force,” that is, the “velocity which [the force] generates in a given time,” is by definition always measured on a body initially at rest. Now suppose, under the influence of a centripetal force, that a body of “quantity of matter” M in a given time h falls from rest from P to G along the line through the center of force. How then does Newton intend us to understand the “velocity which [the force] generates” in this given time h? A reader today, thinking about the modern concept of acceleration, might be inclined to represent th is ‘generated velocity ’ by

, but New-

ton himself would surely represent it by P G/ h (or perhaps by 2P G/ h, in the case of a “uniform” centripetal force). For instance, in his argument for Proposition 10 (Book 2), we find the following sentence (where we have made only a notational change, replacing Newton’s N, I, and t with our P, G, and h): “In a body falling and describing in its fall the space PG, gravity generates a velocity by which twice that space could have been P G described in the same time, as Galileo proved, that is, the velocity 2 h . . ..”[32, 657] Thus, following Newton’s lead we shall take the “velocity which [the force] generates” in the given time h, that is, the “accelerative force,” P G to be the quantity h . But then, because the “motive force [arises] from the accelerative force and the quantity of matter −→

P G P G jointly,” we can deduce the meaning of “motive force”: M h , or, even better, M h , if we wish to register the direction of the force. 12. Moving on to the last piece of the second law, we have only to decide what New-ton means when he writes that the “change in motion” must “take place along the straight line in which that force is impressed.” But given our interpretation for the “change in motion” and the “motive force,” this can only mean that the direction of the “change in −→

−→

LQ P G motion” M h is the same as the direction of the “motive force” M h , or, equivalently, −→ that the directed line segment LQ has the same direction as the directed line segment −→

P G.

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177

13. Let us put the pieces together now, and write down our (and we believe Newton’s own) interpretation of the second law. We call it the Compound Second Law. Suppose a body at rest at P, acted on by a given force, moves along a line from P to G in time h. (This force may be impulsive, uniform, or continuously varying. In the continuously varying case, we require the force to be centripetal. Later we shall see how the Compound Second Law can be made to apply to noncentripetal forces, such as resistance forces.) Suppose the same body, this time in uniform straight line motion before arriving at P, when acted on by the same force at P, moves along a line or a curve from P to Q in the same time h. If this force had not acted, this body in uniform straight line motion, by the first law, would have continued its uniform straight line motion, moving from P to L, say, in time −→

h. We call the directed line segment LQ the moving deflection generated in time h or just the moving deflection for short, and we think of the force as deflecting the body from L (where it would have come to in the absence of the force) to Q (where it actually comes −→

to in the presence of the force). Similarly, we call P G the resting deflection generated in time h or just the resting deflection. If the body has “quantity of matter” M, we call

−→

M

LQ h the change in motion (generated in time h) and M in time h).

−→ PG

the motive force (generated

h

Compound Second Law

A change in motion equals the motive force: −→

−→

M

h

LQ

=M

PG

.

h

Equivalently, a moving deflection equals the resting deflection: −→

−→

LQ = P G . This law holds exactly for any h when the force is either impulsive or uniform and exactly in the limit as h tends toward zero when the force is centripetal and continuously varying.

Comments on the compound second law 14. The definition of impulsive, continuous, and centripetal force. Up to this point, we have been using the terms “impulsive force,” “continuous force,” and “centripetal force” rather loosely, depending on the reader’s own understanding and intuition to give our references to such forces some meaning. In so doing, because we have used these

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terms not just informally but now formally (in the statement of the compound second law above), we have violated a fundamental rule of correct reasoning: agree on the meaning of an assertion before making that assertion. Definitions of these kinds of force, from the logical point of view, should have appeared in this paper, as they do in the Principia, before the statement of the second law. But from an explanatory point of view, it is easier to motivate these definitions if we can refer to the second law. Better late than never, then, let us ask how one might characterize “impulsive force,” “continuous force,” and “centripetal force” in definitions that would appear before the statement of the second law. In the Principia, forces are characterized in terms of the motions generated in bodies. In particular, by the first law of motion it is

a deflection from uniform straight line motion which announces the presence of a force, and it is then the magnitude and direction of that deflection which is the natural candidate to indicate the character of that force. Can we use either deflection, the moving deflection −→

−→

LQ or the resting deflection P G? Well, after we have stated and assumed the compound second law, we can use either deflection to characterize the force, because the com-pound second law tells us that these two deflections are equal (or “equal in the limit,” an expression that we will make precise below). But before the compound second law is stated, the situation is different, because before the compound second law is assumed it is conceivable a priori that the “effect of a force” on a given body (that is, the deflection) might depend on the speed or direction of that body at the time the force is applied. This possibility makes it impossible to define the kind of force in terms of the moving −→

deflection LQ, for the kind of force (impulsive or continuous, say) cannot have any conceivable dependence on the speed or direction which a body has when the force acts. We just cannot have a definition that might tell us that a force is impulsive when it acts on a given moving body, but continuous when that same force acts on that same body moving with a different speed or direction. Yet it is only through the compound second law that we know the “effect of a force” is in fact independent of the speed and direction −→

−→

of the body’s motion: that the moving deflection LQ equals the resting deflection P G. It follows that any definitions for “impulsive force,” “continuous force,” and “cen-tripetal force” which appear before the statement of the second law must be given in terms of what the force does to a body at rest, that is, in terms of the resting deflection −→

P G. But these definitions must come before the second law, because the statement of the second law refers to these forces. So there is no choice: we must characterize the −→

kinds of force in terms of the resting deflection P G. Let us begin with the concept of an “impulsive force.” Intuitively, a body at rest which instantaneously achieves a uniform straight line motion would announce the presence of an impulse. We can easily turn this P G into a definition: If a body at rest at P begins to move along a line, reaching G in time h, and if h has the same value for all times h, we say an impulse or an impulsive force acts at P. Newton would say that an impulse is a kind of force, but not a “finite force,” where Newton’s meaning of “finite force” is revealed in Lemma 10 (Book 1):

Lemma 10 The spaces which a body describes when urged by any finite force . . . are at the very beginning of the motion in the squared ratio of the times. [32, 437–438]

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We may restate Lemma 10 as a definition: if a body initially at rest moves from P to G in time h and the limit of 2 P G/ h is finite and nonzero, we say that a finite force acts

at P. (Read [36] for more on Lemma 10 and [37] for more on Newton’s surprisingly good understanding P G of limits.) In the case of an impulsive force, the ratio h remains a nonzero constant, which implies the P G ratio h 2 grows arbitrarily large as the time h tends toward zero. Thus for Newton an impulsive force is not a finite force. For that reason we might call it an “infinite force.” The finite forces addressed in the Principia’s Definitions 6, 7, and 8 – the definitions of the absolute quantity, the accelerative quantity, and the motive quantity of centrip-etal force – are measures of centripetal forces. In his Definition 5, Newton writes that a “centripetal force is the force by which bodies . . . tend toward some point as to a center.” [32, 405] Of course a body at rest would be said to “tend toward some point,” if it falls along a line through that fixed point. Thus we say a centripetal force acts in a given region of space provided for some fixed point S every body initially at rest in this region falls along a line through S. Given a finite centripetal force acting in a given

−→ PG

region, we say the force is continuous if the limit (as h tends to zero) of the ratio h2

varies continuously (in magnitude and direction) as P varies over the given region. If this limiting value is constant in magnitude and direction as P varies over the region, we say the force is uniform. Thus by a uniform force we mean a force which has constant magnitude and parallel “lines of force.” We may regard a uniform force as centripetal provided we imagine the center of force S as being “infinitely remote.” (A uniform force, as we have just defined it, where the “lines of force” are parallel, should not be confused with a force having constant magnitude but nonparallel “lines of force” directed toward a fixed point. Such “uniform centripetal forces” appear only in Section 10 of the Prin-cipia, where Newton studies “the motion of bodies on given surfaces and the oscillating motion of simple pendulums.”) We have given definitions for impulsive, centripetal, continuous, and uniform force in terms of the motion generated on a body at rest, because in any formal development these definitions would have to appear before the statement of the second law, that is, at a point in the exposition where it is conceivable that definitions in terms of a body in motion would be invalid. But after the second law has been stated and assumed, the situation changes, for the compound second law tells us the moving deflection equals (or equals in the limit) the resting deflection. In the wake of the compound second law, −→ therefore, the moving deflection LQ may be used to test for the impulsive, centripetal, −→

continuous, and uniform forces that we defined in terms of the resting deflection P G. To obtain these tests, we merely replace in our definitions the resting deflection by the moving deflection. −→

−→

15. The law LQ = P G holds in the limit. In our statement of the compound sec−→

−→

ond law, the equality LQ = P G between the moving and resting deflections is said to hold “in the limit as h tends toward zero” when the (centripetal) force is continuous −→ and varying. What does this mean? We mean that although the moving deflection LQ −→ and the resting deflection P G may not be exactly equal when the force is continuous and −→ −→

varying, still the difference LQ − P G becomes so small so fast, as h tends toward zero,

−→ −→ −→ −→

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that this difference is small even compared with h2, which itself is very small. More precisely, we mean that the ratio LQ − P G → → as h → 0. (In general, given two quantities

h2

0

which each tend toward zero, we can compare the rates at which they tend toward zero by studying their ratio. If their ratio tends toward zero, for example, then the quantity in the numerator “tends toward zero faster” than the 2 quantity in the denominator. See [36; 37].) Using the standard mathematical notation o(h ) to represent any quantity that 2

2 tends toward zero faster than h

o(h )

, that is, any quantity satisfying h −→ −→ 2

we could equivalently write our “equality in the limit” as LQ = P G + o(h −→

prefer, as an actual equality in the limit: lim h→ 0

), or, if we

−→

LQ h2

→ 0 as h → 0, 2

lim =h

→ 0

PG

.

h2

16. Alternate formulations of the compound second law. We have phrased the com-pound second law in terms of the moving and resting deflections, in part because Newton himself phrases his own interpretation of the second law in just this way – more about

this later – but we could have formulated the compound second law in other equivalent −→

ways. For example, the compound second law tells us the effect P G of the force on a body at rest is a vector, not just because it has both magnitude and direction, but also because −→ it combines according to the parallelogram law of vector addition with the effect P L −→

−→

−→

of uniform straight line motion to give the actual location of the body: P Q = P L + P G.

−→

(Of course for a varying continuous force this formula for P Q would not be exact, but would hold “exactly 2 in the limit,” and by this we mean it would hold approximately, with an error which is o(h ).) Naturally Newton does not use the words “vector” or “vector addition,” but he uses both these concepts when he interprets the second law, as shall see later when we examine a particular manuscript written in the early 1690s. Though he does not use the words “vector addition,” Newton does write of motions “compounding,” and in the language of compounding we can reword the compound second law: From the uniform straight line motion from P to L that would have occurred −→

in time h in the absence of a force and from the motion P G that would have been gen-erated by the force acting on the body at rest, there emerges the actual motion along −→

−→

a path from P to Q, compounded from the two independent motions P L and P G . In

this compounding, the motions P L and P G combine independently, with neither having any effect on the other. This law is exact for impulsive and uniform forces and exact in the limit for continuous varying forces. The independent compounding of motions (at least for horizontal “equable motion” and vertically downward “naturally acceler-ated motion”) appears as a fundamental assumption in Galileo’s Two New Sciences [17, 222], where he demonstrates that this compounding, in which these two motions “do not alter, disturb, or impede one another,” yields a parabolic trajectory. Huygens follows and

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slightly generalizes Galileo, as we see from reading the second and third hypotheses of motion in Horologium Oscillatorium: “By the action of gravity . . . bodies are moved by a motion composed both of uniform motion in one direction or another and of a motion downward due to gravity. These two motions can be considered separately, with neither being impeded by the other.” [22, 33] In the present study, we see the Principia’s statement of the second law as having everything to do with the compounding of motions, but Newtonian commentators generally see the second law and compounding as indepen-dent assumptions. (See, as just one example, [7, XXXVIII].) There have, though, been rare exceptions. The scientist and philosopher William Whewell, in his 1819 textbook, An Elementary Treatise on Mechanics [45], gives the following interpretation specifi-cally for continuous forces only: “The Second Law of Motion. When any force acts upon a body in motion, the motion which the force would produce in the body at rest is compounded with the previous motion of the body.” Speaking of Galileo (whose sandal prints are all over our interpretation of the second law, just as Newton himself claims – see §30–31), we may also formulate the second law in terms of an invariance principle: If an observer in one reference frame sees a body at

−→

rest begin to fall and measures the resting deflection P G, then an observer in a second frame moving with uniform straight line motion with respect to the first frame will see −→ the same body in motion and will measure a moving deflection LQ equal to (or equal in −→ the limit to) P G . In the particular case of a varying continuous force, for example, LQ −→ −→−→

is equal in the limit to P G, by which we mean that LQ = P G + o(h

that lim h →

0

−→

−→

LQ

PG

lim =h →

h2

−→ 2

) or, equivalently,

, and this implies that each observer will measure the same

0 h2

(modern vector) acceleration of the body at P. The frames will be “equivalent as far as dynamical experiments are concerned – either frame may be assumed stationary and the other frame in motion, with all the same laws of mechanics providing correct explana-tions for observed trajectories.” [16, 12–10] This is what physicists today call “Galilean relativity” or “Galilean invariance.” Honoring Galileo in this way has some historical justification, given his famous and colorful description in the Dialogo of gnats, flies, fish, and droplets moving in a room below deck of a ship first at rest and then in uniform straight line motion. Newton probably never read Galileo (see [7]), but in Newton’s lifetime this invariance principle appeared prominently in two places: as Corollary 5 to the Laws of Motion in the Principia and, in a paper well known to Newton, “On the Motion of Bodies Resulting From Impact” by Huygens, where the invariance principle took the following form: The motion of bodies and their equal and unequal speeds are to be understood respec-tively, in relation to other bodies which are considered at rest, even though perhaps both the former and the latter are involved in another common motion. And accordingly, when two bodies collide with one another, even if both together are further subject to another uniform motion, they will move each other with respect to a body that is carried by the same common motion no differently than if this motion extraneous to all were absent. [23, 1]

17. Where is f = ma? We have now phrased the compound second law in terms of deflections, vector addition, compounding, and Galilean invariance. None of these

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formulations has the f = ma form that physicists today call Newton’s second law. Nev-ertheless, let us show that by passing to the limit as the time interval h shrinks toward zero, we can derive from the compound second law a Newtonian version of the familiar

modern form of the second law.

−→

P G We may feel uncomfortable with the motive force M h as a measure of the force at P, for this measure in the continuous case tends to zero as h tends to zero. Yet for Newton this is not a problem, because he generally compares two forces by forming the ratio of their motive or accelerative forces. The numerator and denominator of such a ratio each will tend to zero, but the ratio of such “evanescent quantities” may well tend to a nonzero, finite limit which then serves to quantify the comparison of the two forces. As just one illustration of such a comparison, consider the demonstration of Proposition 10 (Book 2), where we see Newton compare the resistance of a given medium to gravity [32, 657]: In a body falling and describing in its fall the space NI, gravity generates a velocity by which twice that space could have been described in the same time, as Galileo proved, 2N I ; . . . And accordingly, . . .the resistance will be to gravity as that is, the velocity t

GH

T

HI

−

t

2M I ×N I

+

t ×H I

to

2N I

.

t

After some calculations and estimates, Newton then takes the limit of this ratio, finding 2 2

“the resistance will now be to gravity as . . . 3S 1 + Q to 4R .”

As we noted, however, the modern physicist does not proceed this way, but seeks instead a “stand-alone” or “absolute” measure of the force at a given point P. Newton’s −→

P G motive force M h is more a “relative” measure of force: it can be used to relate or compare one force with another, but since (in the continuous force case) it tends to zero as h tends to zero, we cannot take the limit and expect to get a useful measure of the −→

P G

force at P. On the other hand, the quantity M h

has no such problem, for by Newton’s

2

−→ Lemma 10 (Section 1, Book 1) the limit of P G is finite and nonzero: the space PG h2

“which a body describes when urged by any finite force . . . [is] at the very beginning of the motion [that is, in the limit as the time tends toward zero] in the squared ratio of −→

the times.” [32, 437–438] Let us call A 0

−→

PG

lim ≡ h→0

LQ

A

2 h2

and

lim 2 h2

the resting and

≡ h→0

moving accelerations at P, respectively. For a varying continuous force, the compound −→

−→

−→

second law says LQ

=

PG

2

+

−→

LQ

o(h

PG

), or, equivalently, lim

2 0 h

h →

lim

=

h

. A Now

→A

words, the moving acceleration at P equals the resting acceleration at P:

=

F

MA

define the force at P to be the product

. In other

2 0 h

0

of the “quantity of matter” M and the

0

resting acceleration. Then the equality A

A = , implied by the compound second law,

0

may be rewritten as

! Thus, although it is expressed in terms of “relative”

=

F

MA

measures of force, the compound second law (which we believe to be Newton’s own interpretation of the second law) implies a “limiting version” of the compound second law which has the form of the modern second law. (By the way, Newton appears to be quite aware of this limiting version of his second law. For example, in his demonstration for Proposition 24, Book 2, he asserts that the “velocity that a given force can generate in a given time in a given quantity of matter is as the force and the time directly and the matter inversely. . . .This is manifest from the second law of motion.” [32, 700].

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183

18. A physical law, but does the matter matter? In its impulse-only interpretation, the second law tells us an impulse produces a “change in motion” proportional to the magnitude of the impulse. But supporters of this interpretation never supply an inde-pendent way to gauge the magnitude of the impulse (that is, other than by observing the “change in motion”), and this turns the impulse-only second law into a definition. In contrast, the compound second law affirms Galilean invariance, a fundamental physical −→

−→

law: the effect LQ of a force on a moving body is the same as the effect P G of that force on the same body at rest. Observe, however, that the “quantity of matter” M cancels out −→

−→

in the compound second law, leaving only the basic equality LQ = P G. It follows that the “quantity of matter” plays only a ceremonial role in the compound interpretation of −→ PG

−→

the second law. Of course for a given motive , the resting deflection P force M h G(and therefore the moving deflection as well) varies inversely as the “quantity of matter” M. (Newton himself puts it this way: “To achieve the same motion [that is, to achieve the −→

LQ same value of M h ], the translation of a greater body by an impressed force is less, and of a lesser one greater.” [29, VI, 543]) But this fact contains no real information −→

−→

beyond the equality LQ = P G, for it derives from this equality plus a mere definition, −→

P G namely the definition of motive force as M h . The “quantity of matter” M can enter nontrivially only when an alternate way – besides the definition itself – of computing the motive force is available. For example, throughout Book 2 the deceleration produced by a given resistance force depends on the mass of the body. The resistance force then, often proportional to the speed or the speed squared, is a motive (not accelerative) force,

and the Compound Second Law would be applied in its “massful” rather than massless form. Apart from such cases, however, for instance in Book 1 generally, the actual physical −→

−→

content of the second law lies in the equality LQ = P G between the moving and resting deflections, and the “quantity of matter” plays an integral role – a role in the law itself and not just in a definition – only in the third law of motion. 19. Newtonian velocity. It is a common view among Newtonian scholars that the “change in motion” in the Principia’s statement of the second law should be understood as the “change in momentum,” where by the “change in momentum” some of these scholars [10, 65–66; 32, 111; 2, 226; 14, 275; 10, 46] mean M v = M v, with v being the change in speed, while others [9, 164–165; 13, 112; 12, 472] appear to mean Mv = M v with v being the change in modern vector velocity. We took note of such views in §1 (Arguments 2 and 4) and presented several counterarguments. Even accepting these counterarguments, however, we can in fact see the “change in motion” in the second law as what we might call a “Newtonian change in momentum,” provided we are careful to understand the “change in velocity,” not as the change in speed nor as the change in modern vector velocity, but rather in a way more consistent with Newton’s preference for using the deflection to measure the force. To see this, let us consider for simplicity the case of a uniform force. Given a body −→

P G initially at rest, it would of course be appropriate to call 2 h the “velocity generated” (by the force in the time h), because the body actually traverses the segment PG in time h. −→

On the other hand, for a body initially in motion, the moving deflection LQ equals the −→

−→

LQ

resting deflection P G, making it in this case natural also to call 2 184

h

the “velocity

B. Pourciau

generated” (by the force in time h) – even though the body does not actually traverse −→ the segment LQ. Provided we remember to use V ≡ 2

−→

−→

LQ

, this “Newtonian change in

h

−→

−→

velocity,” then our interpretation of the second law, M LQ = M h

−→ tonian change in momentum” equals the motive force: M V = M

PG h

, tells us the “New−→ PG h

! (Notice, by the

−→ V is not a “change in Newtonian −→ in M V in front of the M −→

way, that this “Newtonian change in momentum” M momentum,” – indeed it creates only nonsense to move the

– nor is the “Newtonian change in velocity” a “change in Newtonian velocity,” for V is not in fact a difference of “Newtonian velocities,” but rather a “deflection per unit time.”) We have just formulated the compound second law as “Newtonian change in momen-tum” equals motive force, but perhaps we should stress again that this “Newtonian change in momentum” is neither the change in linear momentum nor the change in vector momentum that we may be familiar with today. In particular, let us stress the difference between how we measure force today, through the change in modern vector velocity, v ≡ v(t0 + h) − v(t0), and how Newton measures force in the Principia, −→

−→ LQ

−→

through the “Newtonian change in velocity,” V ≡ 2 . It is true that v and V h yield equivalent directed line segments for impulsive and uniform forces, but even so they are defined and thus calculated quite differently. Indeed,

v ≡ v(t0 + h) − v(t0) = (what will be) − (what is),

while

−→

V

−→ 2 LQ

≡

h

2

(Q

L)

.

= (what will be) − (what would have been)

−

=h

(The grammatically inclined might say that Newton is more at home with the future perfect subjunctive.)

−→

Moreover, v and V are not equal for a varying continuous force. Indeed if the vector function r = r(t ) records the positions of a moving body in space, it turns out that

Newton’s Interpretation of Newton’s Second Law

185

1 v = a(t0)h +

2 2 r (t0)h + · · ·

−→

1

while V = a(t0)h + −→

so that v = V (although

+···,

−→

v h

3 r (t0)h

2

V

and

h

both tend to the modern vector acceleration a(t0)

as h → 0.) Of course when the force is uniform, all terms beyond the acceleration vanish, −→

making v and V come out to be equal (yet still different in the way they are defined).

Arguments for the compound interpretation 20. We began this study with a review and rebuttal of the main lines of argument that have appeared over the years in favor of an impulse-only interpretation for the second law. We then presented more direct evidence against an impulse-only interpretation, pointing out the many passages – demonstrations, propositions, corollaries, definitions, scholia, revisions, and remarks – that appear overly complex, incongruous, or just plain hard to explain, when we take an impulse-only view of the second law. Following this, two examples from Newton’s dynamical studies – one on oblique collisions in the “Waste Book,” the other on elliptical motion in the Newton-Locke manuscripts – helped us to develop the compound second law, an interpretation of the second law that applies directly to both impulsive and continuous forces. Here and there along the way we have professed our belief that this compound second law is in fact Newton’s own interpretation of the second law. It is finally time to back up this claim.

How should we proceed to make the case for our compound interpretation? More generally, how should one go about making the case that a particular interpretation of some text is the correct interpretation? Let us turn for advice to an expert, a careful, even obsessive, Biblical scholar who has given us sixteen hermeneutic rules to guide the interpretation of scripture. Here, for example, are rules eight and nine: ch

. . . reduce contemporary visions to y greatest har-mony of their parts.

ch

without straining reduce things to the greatest sim-plicity.

8. To choose those constructions w 9. To choose those constructions w

e

“The reason [for Rule 9],” elaborates this scholar, e

is manifest by the precedent Rule. Truth is ever found in simplicity, & not in y multiplicity e

ch

& confusion of things. As y world, w to the naked eye exhibits the greatest variety of objects, appears very simple in its internall constitution when surveyed by a philosophic understanding, and so much ye simpler by how much the better it is understood. . ..

This Biblical scholar – no surprise – is Newton himself, and these rules of interpretation appear in his “Treatise on the Apocalypse” (ca. 1672). [26, 6; 31] Of course Newton’s rules for textual analysis have been quoted here more for fun than for any belief that

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they support our argument in a serious way. We do not need Newton to tell us that an interpretation of the second law of motion which spreads both harmony and simplicity across the pages of the Principia would be a hard interpretation to resist. Believing with Newton that “truth is ever found in simplicity,” let us return to those places and passages in the Principia and elsewhere that appeared confused, overly com-plex, incongruous, or mistaken under an impulse-only interpretation of the second law and ask these questions: Does this complication and confusion turn into simplicity and harmony under the compound interpretation? Does the unexplainable become explain-able, even perhaps natural and obvious? The answer, again and again, is yes. Taking all these individual increments in simplicity and harmony and explanatory power together, the evidence that the compound second law is in fact Newton’s own interpretation be-comes overwhelming. We begin in the “Waste Book.” Recall that in one argument for interpreting the “change in motion” as the “change in quantity of motion” (that is, as M v = M v, where v is a change in speed), some commentators point to apparent ancestors in the “Waste Book” of the Principia’s statement of the second law, such as “Axiome 115”: Soe much force a[s] is required to generate any quantity of motion in a body, so much is required to destroy it, and e contra. [21, 158]

Clearly in this early version of the second law the force does generate a change in the “quantity of motion.” Blay [2, 226–227], as we have seen, makes a similar claim, based not on axioms in the “Waste Book”, but on the way Newton cites the second law in two demonstrations in Book 2 of the Principia: Two references to Law 2 Newton gives in the context of the proofs worked out in Book II confirm [the impulse-only] interpretation of the law. Proposition 3 reads thus: “will be as the absolute forces with which the body is acted upon in the beginning of each of the times, and therefore (by Law II) as the increments of the velocities;” and Proposition 8 reads: “for the increment PQ of the velocity is (by Law II) proportional to the generating force KC.” In neither case is time mentioned. Law II appears then to be associated with an impulse model of action (action by impact). . ..

Does an invocation of the second law (or one of its ancestors), in order to infer that the increment in speed is proportional to the force, necessarily imply that the “change in motion” in the Principia’s second law stands for the “change in quantity of motion,” as Blay, Cohen, and others claim? In §1 (see the discussions following Arguments 1 and 2), we have given counterarguments to this claim, and we have no wish to repeat those counterarguments here, but this claim does suggest a first test for our interpretation of the second law: Can the compound second law “explain” those instances where Newton cites the second law to ensure the proportionality between the increment in speed and the force? Yes. Can the compound interpretation even explain the many other cases of resisted motion in Book 2, where the second law may not be cited, but where Newton measures the resistance (or the resistance together with a component of an oblique force) with a change in speed? Yes again. In fact every instance where Newton measures a force with a change in speed involves a very special situation: the force being measured is parallel to the direction of motion. And for the particular case of a force parallel to the direction of motion, the compound second law implies that the increment in speed

Newton’s Interpretation of Newton’s Second Law

187

(generated in a given time) is in fact proportional to that parallel force (generated in that same given time)! Let us see why. Suppose a body with “quantity of matter” M moves under the influence of some force. We first suppose that the net force on this body is at all times parallel to the direction of motion, resulting in a straight line motion. When the body comes to P, assume it has acquired a speed which, in the absence of any force at P, would carry that body in uniform straight line motion from P to L, say, in time h. But in the presence of the force at P, suppose the body actu-ally moves from P to Q in the time h. Assume the force at P (impulsive or uniform

−→

P G for now) has an accelerative force equal to h . By definition this means, that had the body been at rest

at P, it would have moved from P to G in the time h under the influence of this force. Then according to the compound second law, the moving −→

−→

deflection LQ equals the resting deflection P G, and it follows that the actual −→

−→

−→

−→

−→

−→

displacement P Q of the body (in time h) is P Q = P L + LQ = P L + P G. But in straight line motion, this vector addition becomes scalar addition (or scalar subtraction in the case where the force is

opposed to the direction of motion, as with resistance forces, for example), so that (if v0 denotes the speed of the body at P) P Q = P L + P G = v0h ± P G, and dividing by h we see that

PQ h

= v0 ±

PG

.

h

P G Hence the increment in (average) speed is ± h , which is obviously proportional to the motive force P G M h . All this is exact when the force is either impulsive or uniform continuous. A varying continuous force would merely introduce a negligible error term:

PQ h

= v0 ±

PG

+ o(h),

h

where o(h) denotes a quantity which tends toward zero faster than h itself, in the sense −→

P G that o(h)/ h → 0 as h → 0. The increment in (average) speed would then be ± h +o(h) which is “proportional P G in the limit” to the motive force M h (by which we mean that the ratio tends toward a finite, nonzero constant, namely M.) Thus, at least in the case where the net force on the body remains parallel to the direction of motion, we have established that the change in speed is proportional to that parallel force.

Now suppose that the net force on the body is not in the direction of motion. The body then moves along a curved trajectory rather than a straight line, but we can still apply the above argument using the tangent line PL to this trajectory and using as our parallel force the component of the net force which is parallel to the direction of motion at P. The only change in the argument is that Q now lies on the curved trajectory and not on the tangent line, but this just introduces a second negligible error term into our speed equation –

PQ h

PG

+ o(h) + o(h)

h = v0 ±

188

= v0 ±

PG

+ o(h) h

B. Pourciau

– and once again we conclude that the change in speed is proportional (exactly or in the limit) to the motive force. Thus the compound second law has the following simple corollary which explains every instance where Newton employs an increment in speed to measure a force: Corollary on Force Parallel to the Direction of Motion The increment in speed (generated in a given time) is proportional to the motive force (generated in the same given time) which is parallel to the direction of motion. The argument we have given for this corollary, at least in the initial case where the net (impulsive or uniform) force lies in the direction of motion, is hardly different from the argument in Horologium Oscillatorium that Christiaan Huygens gives for his Prop-osition 1: “In equal times equal amounts of velocity are added to a falling body. . ..” [22, 35] We should also point out that Newton uses the phrase “increment in velocity” to mean the increment in (straight-line) displacement in a given time, that is, what we might call an increment in average speed. In modern usage, however, “increment in speed” would probably mean the change in instantaneous speed, namely v(t0 + h) − v(t0), where v(t ) ≡ v(t ) is the 2 instantaneous speed at time t.

21. From the “Waste Book,” we move to the Principia and the famous scholium on “time, space, place, and motion.” [32, 408–415] The scholium begins with four defini-tions, the last a definition of “absolute [or true] motion”: “Absolute motion is the change in position of a body from one absolute place to another.” A few paragraphs later, we find an assertion very similar to the second law: “True motion,” he writes, “is neither generated nor changed except by forces impressed upon the moving body itself. . ..” Here by a generated “true motion” Newton manifestly means, not the change in “quantity of motion,” (which would be the change in the product of the quantity of matter and the speed), but rather the change in position. The scholium ends with the Principia’s reason for being: “In what follows, a fuller explanation will be given of how to determine the motions from their causes, effects, and apparent differences, and, conversely, of how to determine from motions, whether true or apparent, their causes and effects. For this was the purpose for which I composed the following treatise.” [32, 415] In our interpre-tation of the second law, a change in position is generated by an impressed force: the

2

Speaking of “speed,” this may be a reasonable place to make a comment about the translation of Newton’s Latin “velocitas.” According to [1], “velocitas” means “swiftness, fleetness, speed, rapidity, velocity,” and Cohen and Whitman [32] generally choose the translation “velocity.” But this choice has the potential to cause some confusion, for mathematicians and physicists today make a clear distinction between what thay call “velocity”, which is the speed together with a direction, that is, a “vector velocity,” and what they call the “speed,” which is just the speed part

of the vector velocity. Velocity is thus a vector, while speed is a scalar. In the notation used in this paper, velocity is the vector v(t ), while speed is the scalar ||v(t )|| (the length of the veloc-ity vector). On most of the occasions where Newton uses the Latin “velocitas,” he has in mind what we would call today the “speed” (although often he means an average speed, rather than an instantaneous speed). Perhaps then a more accurate translation of “velocitas” would be speed, rather than “velocity.”

Newton’s Interpretation of Newton’s Second Law

189

−→ moving deflection LQ, a change in position from L to Q, equals the resting deflection −→

P G, which measures the force. Nothing could be more in harmony with this scholium. On the other hand, if the “change in motion” were to mean a change in the “quantity of motion,” as some of the impulse-only advocates suggest, much of the harmony and consistency between the second law and the scholium which immediately precedes it is lost. 22. Let us stay on the page with the first two laws of motion. According to the first law, “Every body perseveres in its state of being at rest or of moving uniformly straight forward, except in so far as it is compelled to change its state by forces impressed.” [32, 416] Cohen [8, 145–148] asks a natural question: Why does Newton list the first law as a separate law, since “it may seem to us that the [first law] is simply a special case of the [second law] when the impressed force is zero.” Answering his own question, Cohen suggests four possible reasons, three historical and one technical, in a discussion that covers three pages. But under our interpretation of the second law, a completely compelling reason for a separate first law can be given in one line: in its compound interpretation, the second law predicts the force by measuring the deflection from L, the place the body would have been had there been no force, yet only through the first law do we know where to find the place L, and so without a separate and prior statement of the first law, the second law has no clear meaning! We may still argue historically, along with Cohen, that in separating the first two laws Newton is following the example of Huygens in Horologium Oscillatorium, but now we know, courtesy of the compound interpretation, why Newton could not conceivably have done anything else.

23. If we remain with Huygens for a moment, we can bring out more evidence for the compound interpretation. Here is what Cohen says about the example Huygens provided for Newton’s laws of motion [32, 110–111]: No doubt, another factor in Newton’s decision to have a separate law 1 and law 2 was the model he found in Huygens’s Horologium Oscillatorium of 1673, a work he knew well. In part 2 of that work, Huygens began the analysis of gravitational dynamics with three laws which he labeled “hypotheses,” in the same way that Newton – a decade later in De motu would set forth the laws of motion as “hypotheses.”

Here are Huygens’s first two hypotheses of motion [22, 33]: Hypothesis 1 If there were no gravity, and if the air did not impede the motion of bodies, then any body will continue its given motion with uniform velocity in a straight line. Hypothesis 2 By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of uniform motion in one direction or another and of a motion downward due to gravity.

Newton’s Law 1 is manifestly a generalization (to any force, not just gravity and air resistance) of Huygens’s Hypothesis 1. Nothing could be more natural than that Newton

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would make his Law 2 a generalization (to any force, not just gravity) of Huygens’s Hypothesis 2, and using the compound interpretation, Law 2 is exactly that! Here is the compound second law written expressly in the language of compounding: Law 2 (Compound Interpretation) By the action of any force (impulsive or continuous), bodies are moved by a motion compounded from uniform straight line motion and from the motion generated by that same force on that same body at rest. The compound interpretation thus puts Newton and Huygens in harmony, a harmony lost under an impulseonly interpretation. 24. Besides the harmony with Huygens, the compound interpretation produces har-mony with our expectations as well: indeed the compound second law is what we would naturally expect of a second law, given the first law. To see this, imagine for a moment that we have seen only the first law, nothing more. Could we then anticipate the second law? At least its general form? Because each of the examples used to illustrate Law 1 – projectile motion, a “spinning hoop,” and planetary motion – involves a continuous force, we may safely assume (and indeed all commentators agree) that Newton intends the first law to apply, not only to impulses, but also to continuous forces, and because the central problem of Book 1, the analysis of orbital motion, involves a continuous force, we would naturally expect the second law to apply to continuous forces as well. Furthermore, since the first law tells us that a deviation from uniform straight line motion implies the existence of a force, we would naturally expect a second law to relate the magnitude and direction of that deviation to the magnitude and direction of the force. Under the compound interpretation, this is precisely what the second law does! In this sense, the compound second law is the natural sequel to Newton’s first law. The same

cannot be said for any impulse-only interpretation of the second law. 25. Consider now this claim which follows Newton’s statement of Law 2 in the Principia: “If some force generates any motion, twice the force will generate twice the motion, and three times the force will generate three times the motion. . ..” [32, 416] According to Cohen, this assertion holds for impulsive forces, but “has no meaning for continuous forces since the latter produces a net change of motion that depends on both the magnitude of the force and the time during which the force acts.” [10, 66] It follows, he argues, that Newton must intend the second law to apply to impulsive forces only. But −→

−→

LQ P G in fact with our interpretation of the second law, we have M h = M h , whether the force is impulsive or continuous, and clearly (given a fixed time interval h) three times −→

−→

P G LQ the force, 3M h , generates three times the change in motion, 3M h , just as Newton claims. (Of course these equalities are equalities in the limit if the force is continuously varying.) 26. We move now from the “three times the force” sentence immediately following the statement of the second law to the facing page in the Principia in order to take up Corollary 1 of the laws: “A body acted on by [two] forces acting jointly describes the diagonal of a parallelogram in the same time in which it would describe the sides if the forces were acting separately.” [32, 417] To illustrate this corollary, Newton draws the following figure (where we have replaced his A, B, C, D with our P , L, G, Q):

Newton’s Interpretation of Newton’s Second Law

191

Newton would never have labeled this assertion a corollary if he did not believe it to be a relatively simple consequence of the laws, in this case clearly of Law 1 and Law 2. Yet here is a startling fact: no Newtonian commentator (to my knowledge at least) has ever demonstrated how Corollary 1 follows from the first two laws (alone) in any way at all, much less in any simple way! (Of course from our point of view this difficulty in seeing Corollary 1 as a simple consequence of the first two laws derives from an incorrect interpretation of the second law.) In contrast it would appear that Corollary 1 is indeed a simple and obvious corollary of the first and second laws, provided we give the second law its compound interpreta-tion. To be more specific, suppose an impulse I applied to a body resting at P would move that body (in uniform straight line motion) from P to L in a given time h, and suppose an impulse J applied to the same resting body at P would move that body from

P to G in the same time h. Then Corollary 1 tells us what happens when the two impulses I and J are applied simultaneously: the claim is that the body moves (in uniform straight line motion) along the diagonal P Q of the completed parallelogram P LQG, arriving at Q in time h. At first glance, this result may seem to be just a statement of the impulse case of the compound second law. But this is not quite so. For the impulse case of the compound second law concerns an impulse applied to a body already in uniform straight line motion, while Corollary 1 concerns two impulses applied “jointly,” that is, simultaneously. Nevertheless, let us show that Corollary 1 is still a simple corollary of the compound second law, even if it is not a mere restatement of the impulse case of that law. We first must decide what it would mean, from the mathematical point of view, to say that a body is “acted on by two impulses acting simultaneously.” We choose the natural and obvious definition: We define the motion of a body acted on by two impulses simultaneously to be the limit of the motions generated when the impulses act separately on that body as the separating time tends toward zero. Using this definition, we can now show how Corollary 1 follows easily from the compound second law (together of course with the first law): Suppose in a given time h a given body at rest at P , by an impulse I applied alone at P , would be made to move in uniform straight line motion from P to L, and by an impulse J , applied alone at P , would be made to move in uniform straight line motion

from P to G. Let the point Q complete the parallelogram P LQG. Suppose the impulse I acts on this resting body at P , moving the body from P to P in a small time h . In an additional time h, this body if unimpeded would continue to move in uniform straight line motion from Pto L , say, where P L = P L.

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But let the impulse J act at P , moving the body instead from P

to Q , say, in the

−→

additional time h. Let P G be the resting deflection of the impulse J at P . By the −→

impulse case of the compound second law, this resting deflection P G equals the mov−→

−→

ing deflection L Q . But the resting deflection P G of the impulse J at P must equal −→ the resting deflection P G of the impulse J at P , and (because Q completes the paral−→

−→

−→

−→

lelogram P LQG) P G equals LQ. Hence LQ equals L Q . It follows that LL Q Q is a parallelogram. Yet clearly L tends to L as the small time h tends toward zero. Hence Q tends toward Q as well. Using our definition concerning impulses applied simulta-neously, we conclude that the body if acted on by the impulses I and J simultaneously would be made to move in uniform straight line motion along the diagonal from P to Q in the time h, just as Newton claims in Corollary 1. 27. Corollary 2 of the laws of motion, dealing with the composition and resolution of forces, is not an agenda item here, but a comment might be in order before we pass on to discuss Corollary 3: The point of Corollary 2 is to show how the familiar composition and resolution of forces in mechanics, especially static forces in equilibrium, follow from the laws of motion. Under the compound interpretation, with its subjunctive mea-surement of motive force (involving the displacement that would have been observed had the body been unconstrained and at rest), the second law makes Newtonian forces which produce motion one with forces in statics. Since Newton clearly does intend his motive forces to represent static forces as well as forces that produce motion – see for example the treatment of constrained motion in Section 10 of Book 1, where Corollary 2 is cited repeatedly – we have a strong argument for the compound interpretation of the second law. Now, as we continue to explore the simplicity and harmony in the Principia gener-ated by the compound interpretation of the second law, let us move on to Corollary 3: “The quantity of motion, which is determined by adding the motions made in one direc-tion and subtracting the motions made in the opposite direction, is not changed by the action of bodies on one another.” [32, 420] We shall analyze this corollary using the compound second law and Newton’s third law. The simplicity of the argument will add to our case for the compound interpretation. Newton says the net “quantity of motion” is “determined by adding the motions made in one

direction and subtracting the motions made in the opposite direction.” We can use the directed line segment notation to keep a simple record of this directional adding and subtracting. Imagine a body M in uniform straight line motion which would, if unhindered, move from P to L in a given time h and a body m in uniform straight line motion which would, if unhindered, move from

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p to l in the same given time. (By an efficient abuse of notation generally adopted in the Principia, we use M and m to stand both for the bodies themselves and for their respective “quantities of matter.”) Suppose these two (hard) bodies collide at the place P = p.

Then the total “quantity of [directional] motion” before the collision would be −→ −→ V pl PL M 0 . If the body h M h , which we could write more compactly as mv0 M had been resting at P , suppose a collision with the body m would have moved M from P to G in time h, and if the body m had been resting at p, suppose a collision with the body M would have moved m from p to g in time h. We would then say the m

+

+

−→

“action” (i.e. the motive force) produced by M on m is m −→ PG

force) produced by m on M is M

pg h and the action (motive

. At the collision, according to Law 3, the action

h

and the reaction have the same magnitude but opposite directions: m

−→

−→

pg

PG

h

= −M

h

.

But according to the compound second law, the moving deflection equals the resting −→ −→ −→−→ deflection, that is, lq = pg and LQ = P G, and together with Law 3 we infer that −→

lq

−→

LQ

m h = −M h . (Newton puts this simple equation into words in his argument for Corollary 4: we have “equal changes in opposite directions in the motions of these bodies.” [32, 422]) It follows that + mv

V 0

−→ pl

=

h

−→ PL

+

h

m

M

M 0 −→

−→

−→

−→

= m pl + M P L + m lq + M LQ h

h

−→

−→

pq PQ =m h +M h =

+ mv1 MV1.

h

h

In other words, the total “quantity of [directional] motion” before the collision equals the total “quantity of [directional] motion” after the collision, as predicted by Corollary 3.

28. Let us turn our attention to Corollary 4 –

The common center of gravity of two or more bodies does not change its state whether of motion or of rest as a result of the actions of the bodies upon one another; and therefore

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the common center of gravity of all bodies acting upon one another (excluding external actions and impediments) either is at rest or moves uniformly straight forward. [32, 421]

– and show how it follows easily from the three laws of motion, provided we give the second law its compound interpretation. For simplicity we concentrate on a two-body collision, using the same notation (and referring to the same figure) that we used in our discussion above of Corollary 3. When the body m is at p and the body M is at P , we call the point

C ≡ mp + M P m + M the center of mass. (We will use the bold italic letters C, L, and Q when we are referring to locations of the center of mass.) If there were no collision when p = P , then in the given time h, p would move to l, P would move to L, and so (by algebra) C would move to

L = ml + M L . m + M When in fact the bodies do collide at p = P , then in the same given time h, p moves to q and P moves to Q, and so (by algebra) C moves to

Q = mq + M Q . m + M For the “center of mass system,” where we imagine the total mass concentrated at the center of mass C, the change in motion would be −→

−→

−→

lq LQ LQ (m + M ) h = m h + M h −→

−→

pg PG =m h +M h where we have used the compound second law to get us from the first line to the second line. In words, this tells us that the change in motion for the center of mass system equals the combined motive forces of M on m and m on M. It follows that the center of mass C moves as if all the masses were concentrated at it and all the forces were applied to −→

−→

it. But by the third law, m pg / h = −M P G / h, so the right hand side of the second line above is zero. Hence the center of mass C moves as if no force at all were applied to it. From the first law, the center of mass must therefore remain at rest or in uniform straight line motion, just as Newton claims in Corollary 4. Newton’s own argument for Corollary 4 does not mention any one of the three laws of motion, but he uses all three nonetheless. At one point, for example, he assumes there must be “equal changes in opposite directions in the motions of these bodies,” [32, 422] and this assumption follows only from the compound second law together with the third law.

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29. Consider now Corollary 6 [32, 423]: If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces.

Advocates for an impulse-only interpretation of the second law must surely begin to doubt themselves when they read this corollary. “Accelerative forces”? Impulses do not accelerate. How could a second law which applies only to impulses have as a (presum-ably simple) consequence an assertion concerning “accelerative forces”? Yet Corollary 6 follows with ease from the compound second law: Bodies “urged by equal accelerative forces along parallel lines” are bodies which, if they were at rest, would all experience −→

the same (that is, equal in magnitude and direction) resting deflection, say P G, in the same given time h. The compound second law applies to such a uniform force and guar-antees that these same bodies in motion, no matter their speed or direction when the −→

−→

force is applied, will all experience the same moving deflection, namely LQ = P G, in that same given time h. Consequently, in the “relative motions,” which are the (vector) −→

difference motions, the common deflection LQ cancels out, leaving the relative motions unchanged, exactly as Newton claims in Corollary 6. 30. The “falling body passage.” The case for the compound interpretation of the sec-ond law builds as we continue to see examples, such as Corollary 6, where this interpre-tation restores harmony and simplicity to what had appeared unnatural or complicated under an impulse-only interpretation. Directly following Corollary 6 we find another example, and a truly striking one, namely the infamous “falling body passage,” inserted by Newton into the third edition of the Principia [32, 424]. (In the passage and its accom-panying figure, we have replaced Newton’s A, B, C, D, E with our P , L, G, Q, R.): When a body falls, uniform gravity, by acting equally in individual equal particles of time, impresses equal forces upon that body and generates equal velocities; and in the total time it impresses a total force and generates a total velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times jointly, that is, in the squared ratio of the times. . . . And when a body is projected along any straight line, its motion arising from the projection is compounded with the motion arising from gravity.

For example, let the body P by the motion of projection alone describe the straight line P L in a given time, and by the motion of falling alone describe the vertical distance P G in the same time;

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then complete the parallelogram PLQG and by the compounded motion the body will be found in the place Q at the end of the time; and the curved line PRQ which the body will describe will be a parabola which the straight 2 line PL touches at P and whose ordinate LQ is as PL

Why would we call this passage “infamous”? Because both the placement and context of this insertion indicate that Newton viewed it as a simple illustration of the second law applied to a truly elementary problem of mechanics – to predict the parabolic trajectory of a projected body under the sole influence of “uniform gravity” – and yet Newtonian scholars have found it anything but simple to explain this passage using an impulseonly interpretation of the second law. The passage “beginning ‘corpore cadente’ [‘when a body falls’],” writes Pierson [35, 654], “has exercised . . .historians of science . . . at considerable, controversial, and inconclusive length.” Indeed, as far as I am aware, no commentary on this “falling body passage” has explained (in any way at all, much less a simple way) how the parabolic motion arises from the first two laws of motion alone, with the second law given an impulse-only interpretation.

Let us now present, however, the entirely simple and direct way in which the par-abolic trajectory arises from the first two laws alone, with the second law given the compound interpretation: Suppose a body at P , urged by uniform gravity, falls from rest along a line from P to

G in a given time t . Divide the interval of time t into n very small and equal subintervals of time h. As a consequence of the Corollary on Force Parallel to the Direction of Motion (§20), in each of these equal subintervals gravity will generate an increment in (average) speed proportional to the motive force of gravity. But Newton assumes uniform gravity, so the motive force of gravity (generated in the same given time h) is the same at each point all along the line of motion. Hence the increments in speed are all equal. This gives us an approximation for the total speed v(t ) generated in the time t :

v(t ) ≈

v1 + · · · +

vn = h

t

v=

v

t.

h

(This is only an approximation, rather than an equality, because these increments in aver-age speed are not quite equal to the corresponding increments in instantaneous speed.) In the limit as the small time h tends toward zero, the ratio v/ h approaches a finite value (that we call g), and the approximation becomes exact: v(t ) = gt . But we quote Newton (replacing his N I with our P G): “In a body falling [with the velocity therefore proportional to the time] and describing in its fall the space [P G], gravity generates a [speed] by which twice that space could have been described in the same time, as Galileo proved, that is, the 2P G [speed] t .” [32, 657] In other words, 2P G = v(t )t , and Galileo’s “law of fall” is the consequence: PG=

1

2

v(t )t =

1

2 gt .

2

We see then how Galileo’s “law of fall” follows from Newton’s assumption of uniform gravity and the compound second law. To continue with the “falling body” argument, suppose now that this same body, rather than falling from P , is projected from P in any direction and with any initial speed v0. By this projection alone, the body would move with uniform straight line

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motion (by the first law) from P to L, say, in the given time t . According to the com-pound second law (expressed in the language of compounded motions), this “motion arising from the projection,” as Newton puts it, “is compounded with the motion arising from gravity.” In other words, if the body actually moves from P to Q in the time t , then −→

−→

the moving deflection LQ equals the resting deflection P G, and it follows that −→

−→

−→

P Q = P L + LQ −→

−→

= PL+PG 1

−→

= v0

→

2

g

t+

2

t ,

which clearly describes a parabola! (In the final line above, our notation became more −→

−→

→

−→

g

has the direction of P L but length v0 and has the direction of P G but length g.) Indeed the “curved line P RQ which the body describes will be a parabola which the straight 2 line P L touches at P and whose ordinate LQ is as P L ,” exactly as Newton asserts. modern: v0

Thus the infamous “falling body passage,” so hard to explain with an impulse-only interpretation of the second law, turns out to be a simple application of the first two laws of motion alone, provided we give the second law its compound interpretation. Of course this is what we should expect, for Newton offers this passage as an example to illustrate how the first two laws may be used to solve an elementary problem of mechanics. It would be unexpected and unnatural for this application to be confusing and complicated, as it is when we insist on an

impulse-only interpretation of the second law. This adds quite dramatically to the evidence that the compound interpretation is actually Newton’s own interpretation of the second law. By the way, notice that the compound second law is applied twice in the argument, once in the derivation of the law of fall (at the point where we called on a consequence of the compound second law – the Corollary on Force Parallel to the Direction of Motion) and then again when we compounded the downward fall with the projection. 31. Newton’s Galilean attribution. Now that we have studied his “falling body pas-sage,” we may be able to evaluate Newton’s claim that “by means of the first two laws and the first two corollaries Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola, as experiment confirms, except insofar as these motions are somewhat retarded by the resistance of the air.” [32, 424] This “Galilean attribution” is as infamous as the “falling body pas-sage.” As we pointed out in §4, Newton scholars generally regard this attribution as misleading and mistaken. Cohen, for example, calls this claim a “complete misrepre-sentation of Galileo’s procedure.” [8, 176] Many of these scholars are forced into this uncomfortable position – of having to believe that Newton did not know what he was talking about here – at least in part by their commitment to an impulse-only interpreta-tion of the second law. We have regarded their discomfort as evidence against such an impulse-only interpretation. But now that we have proposed the compound second law as an alternative interpretation, we can be more constructive. We ask: If we suppose the compound second law to be Newton’s own interpretation, do we get a different picture

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of the Galilean attribution? The answer, as we shall now show, is yes. In fact, using the compound interpretation of the second law, Newton’s bow to Galileo appears, while certainly not totally accurate, at least more reasonable and appropriate and less mistaken or misleading than previously thought. Not having to believe anymore that Newton was guilty of a “complete misrepresentation” – well, that is rather good evidence that the compound interpretation is actually Newton’s own interpretation. Bernard Cohen [7] constructs a persuasive case that Newton never read the Discorsi, that his knowledge of Galileo’s derivation of the parabolic trajectory must have come from secondary sources, including, for example, Huygens’s Horologium Oscillatorium. In what follows, we certainly do not argue against Cohen that Newton had direct knowl-edge of Galileo’s work. Rather we argue as follows: if, under a compound interpretation of the second law, Galileo’s and Newton’s arguments for the parabolic trajectory become more similar, then Newton’s Galilean attribution becomes more appropriate, no matter where Newton may have come across a description of Galileo’s argument. Of course the two arguments differ in significant ways, perhaps most notably that Newton sees gravitational fall as produced by a “force” whose variable efficacy may be measured at each point (by what happens to a body at rest), while Galileo invokes no such agency to explain or derive his “naturally accelerated” motion. Nevertheless, as we shall see, the arguments become strikingly similar in structure when Newton’s second law is given the compound interpretation. We proceed by reviewing Galileo’s derivation of the parabolic trajectory in the Dis-corsi, noting as we go along any differences between and similarities with Newton’s own derivation (as we have interpreted that derivation above in §30). Galileo’s argument, just like Newton’s, has two parts: first deduce the “t squared law of fall” and then from the law of fall deduce the parabolic path. Galileo’s argument for the “law of fall” begins by assuming (motivated in part by experiments) that the correct mathematical model for “naturally accelerated motion” (meaning the accelerated motion of a heavy body falling from a point near the surface of the earth) is “uniformly accelerated motion”: “I say that motion is equably or uni-formly accelerated which, abandoning rest, adds on to itself equal momenta of swiftness in equal times.” [17, 154] Here we see an immediate contrast with the start of New-ton’s argument, for Newton does not assume that bodies fall with what Galileo calls “uniformly accelerated motion”; rather he derives this from a prior assumption, that of “uniform gravity,” which, we recall, is an assumption about how bodies fall from rest at every point: If at a point P a body falls from rest to G in a given time h, Newton −→

P G says the “accelerative force of gravity” is h , and to assume “uniform gravity” is to assume the “accelerative force of gravity” (generated in any fixed time h) is the same (in magnitude and direction) at every place in the region. According to our reading of Newton’s derivation, as we saw in the previous section, the compound second law (in the guise of its consequence, the Corollary on Force Parallel to the Direction of Motion) is used to infer “uniformly accelerated motion” from “uniform gravity.” (This difference in how their projectile arguments begin – that Galileo assumes“naturally accelerated motion” is “uniformly accelerated motion,” while

Newton derives “uniformly accel-erated motion” from an assumed “uniform gravity” – may reflect a difference in their attitudes toward the “cause” of such motion. Neither finds it profitable to speculate on the

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cause or mechanism, but Newton imagines an agency whose effect varies with location, and he measures the magnitude and direction of that effect at a given point, for example −→

P G by calculating the accelerative force h . This suggests a rather modern concept: that of a “force field.” When Newton’s “program is laid out of trying to account for the phe-nomena of nature as the effect of forces of attraction and repulsion,” writes Stein, “what emerges is the view that the natural powers . . . may all take the form of [central] fields of force associated with the particles of matter. . ..” [41, 287])

Back to Galileo’s derivation. After he assumes “uniformly accelerated motion,” Gali-leo then argues, in his Proposition II (Third Day), that the “law of fall” is a consequence: “If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any given times whatever are . . . as the squares of the times.” [17, 166] And what about Newton? After deriving “uniformly accelerated motion” from “uniform gravity,” he too infers the law of fall [32, 424]: . . .uniform gravity, by acting equally in individual equal particles of time, impresses equal forces upon that body and generates equal velocities; and in the total time it impresses a total force and generates a total velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times jointly, that is, in the squared ratio of the times.

Galileo now restates (from an earlier discussion) what we shall call his ‘first law of motion’ [17, 217] – Galileo’s ‘First Law of Motion’ A “moveable projected on a horizontal plane, all impediments being put aside,” has an “equable motion on this plane [that] would be perpetual if the plane were of infinite extent.” – and then compounds this horizontal equable motion with the vertically downward “uniformly accelerated motion” of free fall: If this horizontal plane were not of infinite extent but ended, and [situated] on high, the moveable (which I conceive of as being endowed with heaviness), driven to the end of this plane and going on further, adds on to its previous equable and indelible motion that downward tendency which it has from its own heaviness. Thus emerges a certain motion, compounded from equable horizontal and from naturally accelerated downward [motion], which I call “projection.” [17, 268]

Here Galileo has applied an obvious ancestor of the compound second law, a fundamental assumption concerning the compounding of horizontal projection with “naturally accel-erated downward” motion. We call this assumption Galileo’s ‘Second Law of Motion.’

Galileo’s ‘Second Law of Motion’ A moveable projected horizontally has a “motion compounded from equable horizontal and from naturally accelerated downward [motion].” These horizontal and downward motions “in mixing together do not alter, disturb, or impede one another.” [17, 217, 222] Provided we give Newton’s second law its compound interpretation, Newton’s next step mimics Galileo’s application of compounding, except that Newton is more general,

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for he allows the body to be “projected along any straight line.” Just like Galileo, Newton applies his own second law in compounding the two motions: “When a body is projected along any straight line, its motion arising from the projection is compounded with the motion arising from gravity.” [32, 424] Both Galileo and Newton conclude that such a compounded motion must trace out a parabola. As Galileo puts it:

Proposition 1 [Fourth Day] “When a projectile is carried in motion compounded from equable horizontal and from naturally accelerated downwards [motions], it describes a semiparabolic line in its movement.” [17, 217] Recall again now Newton’s Galilean attribution: “By means of the first two laws . . . Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola. . ..” [32, 424] As we have noted, many scholars, at least in part because of their commitment to an impulse-only interpretation of the second law, have been forced to the very uncomfortable conclusion that Newton did not know what he was talking about in this attribution. On the other hand, as we have just witnessed, when we interpret Newton’s second law as the compound second law, then Newton’s and Galileo’s arguments for the parabolic trajectory exhibit such similar structure that Newton’s Galilean attribution becomes more appropriate than mistaken. This is rather potent evidence for the compound interpretation of the second law, no matter where Newton learned about Galileo’s work. 32. The harmony of natural progression. We know that Newton read and admired Ho-rologium Oscillatorium, where Huygens’s second and third hypotheses of motion man-date the independent compounding of “uniform motion in one direction or another and . . . a motion downward due to gravity.” So we might expect the second law in the Principia to generalize or in some natural way succeed these hypotheses in Horologium Oscillato-rium. And indeed, this is precisely what we find, under our compound interpretation of the second law. This we have already pointed out earlier. Cohen argues persuasively in [7] that Newton never read the Discorsi, where Galileo assumes the independent compound-ing of “equable horizontal [motion] and . . . naturally accelerated downward [motion].” But Huygens knew Galileo’s work, so we might expect the compounding assumption of Huygens to generalize the compounding assumption of Galileo, and in fact it does.

Not because it adds anything new to the arguments we have already made, but for the pure aesthetic joy it produces, let us gather together Galileo’s ‘second law’ (as we have called it) from the Discorsi, Huygens’s second and third hypotheses from Horologium Oscillatorium, and Newton’s second law from the Principia, reveling in the harmony of natural progression that blossoms before us – provided we give Newton’s second law its compound interpretation: Galileo’s ‘Second Law’ A moveable projected horizontally has a “motion compounded from equable horizontal and from naturally accelerated downward [motion].” These horizontal and downward motions “in mixing together do not alter, disturb, or impede one another.” [17, 217, 222]

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Huygens’s Second (and Third) Hypotheses “By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another and of a motion downward due to gravity. These two motions can be considered separately, with neither being impeded by the other.” [22, 33] Newton’s Second Law (Compound Interpretation) “A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.” In other words, a body in uniform straight line motion, acted on by any force (whether impulsive or continuous), has a motion compounded from the uniform straight line motion and from the motion that would have been generated by that force on that same body at rest. [32, 416] Observe the natural progression: In Galileo’s ‘second law,’ the “equable motion” is (at least locally) “horizontal,” that is, along a line tangent to the surface of the earth, the second motion is vertically downward and is the “naturally accelerated motion” of heavy bodies at the surface of the earth. In Huygens’s second hypothesis, the “uniform motion” may be in any “direction or another.” In Newton’s second law the uniform straight line motion

may be in any direction, and the second motion may be in any direction and may be generated by any impulsive or continuous force, not just gravity. What could be more harmonious? Using the compound interpretation, Newton’s second law becomes a natu-ral progression from Huygens’s second hypothesis, just as Huygens’s second hypothesis is a natural progression from Galileo’s ‘second law.’

Of course there are significant differences, as the word “progression” suggests – most notably that Newton introduces a “force” whose strength and direction, measured by the effect on a body at rest, can vary with location – but even so the harmony among these compounding assumptions is striking. 33. The disappearing second law. According to the impulse-only interpretations of the second law, an impulse generates a “change in motion” proportional to the magni-tude of the impulse. Yet supporters of this interpretation never provide a way to measure the magnitude of the impulse other than by observing the magnitude of the “change in motion.” Under these conditions, the second law, as we have noted before, is not a law at all, certainly not a physical law, but rather a mere definition. “The French philoso-pher Malebranche began an attack on forces late in the seventeenth century,” comments Hankins [19, 52–53], and his criticisms were continued by Hume and Berkeley in England and by d’Alembert and Maupertuis in France. Forces were never directly observed, only the effects which they produced, therefore if the word “force” was to have any real meaning, it could only be identified with an observed change of motion. As d’Alembert and Maupertuis pointed out, this criticism reduced Newton’s second law to a tautology or to a definition at best, because the second law equated force with an observed change of motion.

What d’Alembert and Maupertuis failed to understand was that Newton’s “motive force” does in fact have “real meaning,” because it is “identified with an observed change of motion” independent of the “change in motion” for the body in motion – namely the “change in motion” for the same body at rest! But they did not understand this, and so

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the second law appeared to them as no more than a definition. Seen as a mere defini-tion and as applying to impulses only, Newton’s second law could well have seemed peripheral, even unnecessary, to those scientists studying mechanics in the decades fol-lowing Newton’s death. This may explain an otherwise baffling fact: Newton himself clearly regarded his second law of motion as a fundamental axiom of mechanics, and yet, although the f = ma “form of Newton’s second law” certainly did appear, at least in its component form, in the works for example of Euler and Lagrange, Newton’s laws of motion [as stated in the Principia] seldom appeared in the major mechanical treatises published on the Continent. . . .the major contributors to the sci-ence of mechanics in the eighteenth century – the Bernoullis, Clairaut, d’Alembert, Euler, and Lagrange – never stated the laws at all [as they were given in the Principia]. They certainly owed much to Newtonian mechanics; his influence is apparent in their works, but the three laws of motion as stated in the Principia are not to be found. [19, 45]

How could this be? Provided we take the compound second law to be Newton’s own interpretation of his Law 2, a simple explanation becomes available: Newton saw his second law as affirming a fundamental axiom of mechanics, what we call today Galilean invariance – the effect of a force (impulsive or continuous) on a body in motion is the same as the effect of that force on the same body at rest – while the scientists who followed him misread the statement of Law 2 in the Principia and saw the second law as just the definition of an impulse and thus peripheral to their investigations. If we are correct in our claim that Newton’s own interpretation of his second law is the compound interpretation, then although the “major contributors to the science of mechanics in the eighteenth century – the Bernoullis, Clairaut, d’Alembert, Euler, and Lagrange” may have not stated the second law (as it was given in the Principia), they surely used it, without realizing it, over and over again. For every time they compounded the effects of forces using Galilean invariance, they were using Newton’s second law.

34. The curved trajectory. Presumably not quite satisfied with the wording of the second law as it appears in the 1687 Principia, Newton experimented with various “resty-lings” of this law in the early 1690s. In §8 we considered a figure drawn to illustrate one of these tentative revisions. This figure, because it is a curved trajectory,

appears to be generated by a continuous force, and it therefore poses a very serious problem indeed for any supporter of an impulse-only interpretation for the second law. (Our reproduction of Newton’s figure has the letters P, L, Q, G in place of his A, a, b, B.) Even if we were somehow able to show that such a curved path could be generated by a limit of a series of impulses, the curved trajectory could never be seen as a simple illustration of an impulse-only second law. (Consult [39] for a study of the subtleties – subtleties that Newton certainly ignored, whether he was aware of them or not – involved in expressing a given motion generated by a continuous centripetal force as a limit of a

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series of motions generated by impulses.) Yet Newton clearly intends this figure to be just that: a simple illustration of his second law. This contradiction is clear and cogent evidence against an impulse-only interpretation. This much has been pointed out earlier, in §8. But in that earlier section we had not yet developed the compound interpretation of the second law. Now we can test the compound second law against an impulse-only interpretation of the second law: Which interpretation yields the simplest explanation of Newton’s curved trajectory? Recall Cohen’s explanation, based on an impulse-only view of the second law [8, 181]: The Curved Trajectory Using an Impulse-Only Interpretation “The parabola-like orbit is thus an infinitesimal orbital segment, produced by a first-order infinitesimal force-impulse which itself proves to be compounded of an infinite number of second-order infinitesimal force-impulses, each of which we may consider to be acting instantaneously in a time-interval which is not the whole interval d t but rather d t /n as n → ∞, itself infinitesimally small.” Compare this convoluted and complex “explanation” with the simple explanation which becomes available under the compound interpretation of the second law: The Curved Trajectory Using the Compound Interpretation The curved trajectory merely illustrates the continuous force case of the second law.

Newton’s Own Interpretation 35. Staring through the lens of an impulse-only view of the second law, many pas-sages in the Principia which invoke the second law appear opaque, overly complex, incongruous, or just plain wrong. About this there is no debate. But through the lens of our compound interpretation, the scene alters dramatically: what had been opaque becomes transparent, what had been complex becomes simple, what had been incongru-ous becomes harmonious. Recall Newton’s own rules of interpretation [26, 6], where he instructs us to choose that interpretation which reduces our “contemporary visions” to the greatest harmony and simplicity: ch

. . . reduce contemporary visions to y greatest har-mony of their parts.

ch

without straining reduce things to the greatest sim-plicity.

8. To choose those constructions w 9. To choose those constructions w

e

e Newton believes that “truth is ever found in simplicity, & not in y multiplicity

& confusion of things.” So do we, that is, so do I, and I am convinced by the arguments in the present study that the compound interpretation is Newton’s intended meaning for the second law of motion. Then again, I am naturally biased in favor of my own arguments. Others might still disagree and cling to an impulse-only interpretation.

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It is rare in the interpretation of text for there to be a final arbiter. Absent the author of the text, no interpretation has the authority to silence all debate, and the author is generally unknown, mute, or dead. In the case of imaginative text, the author’s own interpretation, when available, might be influential but not necessarily final, for imagi-native text may have an interpretive life all its own, independent of its creator. In the case of scientific texts, however, the author’s own interpretation must be seen as definitive.

Has the author of the Principia ever interpreted his own second law with sufficient clarity to end the debate about its intended meaning? The answer is yes. And to find this authoritative interpretation we return to our discussion (at the end of the previous section) of the “curved trajectory,” the figure Newton draws to illustrate the following planned rewording of the 1687 Principia’s second law (one of several from the early 1690s, as we know) [29, VI, 539]: Law II “All new motion by which the state of a body is changed is proportional to the motive force impressed and occurs from the place which the body would otherwise occupy towards the goal at which the impressed force aims.”

He follows this rewording of the second law with an explanation of the law, where we find the following words, written down not once but twice: “. . .in the meaning of this Law. . ..” Observe the careful wordsmith choosing the word “meaning” (“mente”) to announce his intention to give the meaning of his second law – not to give an application or consequence – but to give the very definition of the law. Here then (replacing Newton’s A, a, B, b with our P, L, G, Q) is Newton’s own interpretation of his second law If the body P should, at its place P where a force is impressed upon it, have a motion by which, when uniformly continued, it would describe the straight line P L, but shall by the impressed force be deflected from this line into another one P Q and, when it ought to be located at the place L, be found at the place Q, then, because the body, free of the impressed force, would have occupied the place L and is thrust out from this place by that force and transferred therefrom to the place Q, the translation of the body from the place L to the place Q will, in the meaning of this Law, be proportional to this force and directed to the same goal towards which this force is impressed.

Whence, if the same body deprived of all motion and impressed by the same force with the same direction, could in the same time be transported from the place P to the place G, the two straight lines P G and LQ will be parallel and equal. For the same force, by acting with the same direction and in the same time on the same body whether at rest or carried on with any motion whatever, will in the meaning of this Law achieve an identical

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translation towards the same goal; and in the present case the translation is P G where the body was at rest before the force was impressed, and LQ where it was there in a state of motion. [29, VI, 540–543]

As he lays out for us with great care the very meaning of his second law, Newton is man-ifestly describing the fundamental equality between the moving and resting deflection that we have been calling the Compound Second Law: Compound Second Law A change in motion equals the motive force: −→

−→

M LQ = M P G . h

h

Equivalently, a moving deflection equals the resting deflection: −→

−→

LQ = P G In the present study, we have been arguing that the compound second law is Newton’s own interpretation of his second law. With the authority of Newton’s own words in front of us, the debate ends, and we rest our case. Acknowledgments. This paper has benefited in many places from the wise and incisive commentary of the editor, George Smith. All remaining errors are mine.

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(Received November 15, 2005)

Published online February 4, 2006 – © Springer-Verlag 2006