Makch 3, 1916] 



SCIENCE 



313 



reference, by noting the distance it compresses 

 an (idealized) standard spring. 



2. To measure a force, we require a unit 

 force, and a scale of multiples and submultiples 

 of that unit. Such a scale can be readily con- 

 structed by simply opposing one or more 

 springs, in various combinations, against the 

 standard, or unit, spring. (This process of 

 calibrating a spring balance does not involve 

 any assumption in regard to Hooke's Law; it 

 requires merely that the elastic properties of 

 the spring, whatever they may be, do not, in 

 the ideal case, vary with the time.) 



3. A material body, or lump of matter, is a 

 familiar notion, in the sense that if any mate- 

 rial is added to or taken away from the body, 

 it ceases to be the same body. (In this discus- 

 sion, " body " is used in the sense of " particle," 

 that is, a body which may be supposed, for the 

 purpose in hand, to be concentrated at a 

 single point.) 



4. A motion of a body, with respect to a 

 given frame of reference, is also a familiar 

 idea. The scientific concepts of velocity and 

 acceleration serve merely to make quantita- 

 tively precise our qualitative notions of 

 " faster " and " slower." 



5. The ejfect of a force when applied to a 

 body free to move, is to change the velocity of 

 the body. As a matter of common observation, 

 the force required to produce a given change of 

 velocity in a given time is larger for some 

 bodies than for others. 



6. If a given body is acted on, at two differ- 

 ent times, by two forces, F and F', and if a 

 and a' are the corresponding accelerations, then 



F/F' = a/a'; 



that is, in the case of any given body, the 

 accelerations are proportional to the forces. 

 This statement is best regarded as a scientific 

 hypothesis, the consequences of which have 

 been abundantly verified by experiment. 



7. In order to predict the behavior of any 

 given body under the action of various forces 

 (and this is the central problem of dynamics), 

 it is sufficient, and necessary, to know from 

 some (direct or indirect) experiment, what 

 acceleration some one force would produce in 



that body. The acceleration, a, that would be 

 produced by any other force can then be com- 

 puted at once by the fundamental proportion. 



In the special case in which the acceleration 

 a is constant, and the body starts from rest, 

 v=^at and x = ^af^ ; in the general case, v 

 and X must be obtained from a by integration. 



The foregoing items 1-7 are quite general;^ 

 the following, 8-11, are suggested primarily by 

 observations on the earth's surface. 



8. The observed acceleration, g, of a freely 

 falling body, in any locality, is the same for 

 all bodies. By the "standard locality" is 

 meant any locality (for example, approxi- 

 mately 45° latitude, sea level) in which g = g, 

 = 980.665 cm./sec.2 = 32.1740 ft./sec.^, this 

 being the convention now generally adopted.* 



9. (a) The force required to support a body 

 at rest with respect to the earth in the standard 

 locality, and (b) the force which would give 

 that body, if free to move (in any locality) 

 the standard acceleration g„, are the same. 

 This force, which is characteristic of the given 

 body, is what I have called the standard weight, 

 W„, of the body.5 By the fundamental pro- 

 portion, if F is any other force, and a the 

 corresponding acceleration, then F/W„ = a/g^. 



10. The " standard weight " of a body can 

 always be found (in any locality) by the famil- 

 iar process of " weighing " the body on a beam 

 balance. 



For example, suppose a given body balances 



2 The principle of action and reaction, the prin- 

 ciple of the vector addition of forces and the prin- 

 ciple of the independence of two perpendicular 

 forces, together with the definitions of such terms 

 as work, kinetic energy, impulse, momentum, etc., 

 although necessary for the development of the 

 science, may be passed over without comment, aa 

 they are not now in dispute. 



< International Conference on Weights and 

 Measures, Procfes-Verbaux des Stances, page 172, 

 Paris, 1901; U. S. Bureau of Standards, Circular 

 No. 34, second edition, page 6, 1914. 



5 Science, July 30, 1915, page 161. A defect 

 in my earlier form of the definition (Bulletin of 

 the Society for the Promotion of Engineering Edu- 

 cation, June, 1913; compare U. S. Bureau of Stand- 

 ards, loc. cit., page 7) was called to my atten- 

 tion by Professor Hoskins 's criticism in Science, 

 April 23, 1915. 



