A HALF-CENTURY OF SCIENCE. 209 



In order to remove any impression that these extensions of algebra 

 are merely barren speculations of ingenious intellects, I may add that 

 many of these derivative forms, at least in their elementary stages, 

 have already found their way into the text-books of mathematics ; and 

 one class in particular, known by the name of determinants, is now in- 

 troduced as a recognized method of algebra, greatly to the conven- 

 ience of all those who become masters of its use. 



In the extension of mathematics it has happened more than once 

 that laws have been established so simple in form, and so obvious 

 in their necessity, as scarcely to require proof. And yet their appli- 

 cation is often of the highest importance in checking conclusions 

 which have been drawn from other considerations, as well as in lead- 

 ing to conclusions which, without their aid, might have been difficult 

 of attainment. The same thing has occurred also in physics ; and 

 notably in the recognition of what has been termed the " law of the 

 conservation of energy." 



Energy has been defined to be " the capacity, or power, of any 

 body, or system of bodies, when in a given condition, to do a meas- 

 urable quantity of work." Such work may either change the con- 

 dition of the bodies in question, or it may affect other bodies ; but in 

 either case energy is expended by the agent upon the recipient in per- 

 formance of the work. The law then states that the total amount of 

 energy in the agents and recipients taken together remains unaltered 

 by the changes in question. 



Now, the principle on which the law depends is this " that every 

 kind of change among the bodies may be expressed numerically in one 

 standard unit of change," viz., work done, in such wise that the result 

 of the passage of any system from one condition to another may be 

 calculated by mere additions and subtractions, even when we do not 

 know how the change came about. This being so, all work done by a 

 system may be expressed as a diminution of energy of that system, 

 and all work done upon a system as an accession of energy. Conse- 

 quently, the energy lost by one system in performance of work Avill be 

 gained by another in having w T ork done upon it, and the total energy, 

 as between the two systems, will remain unchanged. 



There are two cases, or conditions, of energy which, although sub- 

 stantially the same, are for convenience regarded separately. These 

 may be illustrated by the following example : Work may be done 

 upon a body, and energy communicated to it, by setting it in motion, 

 e. g., by lifting it against gravity. Suppose this to be done by a spring 

 and detent ; and suppose, further, the body, on reaching its highest 

 point, to be caught so as to rest at that level on a support. Then, 

 whether we consider the body at the moment of starting, or when 

 resting on the support, it has equally received an accession of energy 

 from the spring, and is therefore equally capable of communicating 

 energy to a third body. But in the one case this is due to the motion 

 VOL. xx. 14 



