GEOMETRICAL STATICS. XXXV 



a special ingenuity is often needed to reach the goal. Just here analysis 

 comes to our aid with its rich treasures of developed methods, and here it 

 is most certainly not for geometry to " undervalue the advantage afforded 

 by a well-established routine, that in a certain degree may even outrun the 

 thought itself " (F. Klein : Verglewhende Betrachtungen uber neuere geome- 

 triscJie ForscJiuhgen. Erlangen, 1872, p. 41). Algebraic operations are 

 thus, however, not the chief thing, but only the instrument a most excel- 

 lent instrument indeed, which can be almost universally applied, and 

 which, by reason of its connection with an extensive and independent 

 mechanism, often needs only to be set in action in order to work of itself. 



Geometrical mechanics, moreover, can never entirely free itself from 

 analytical formulae and operations. For though it may be both interesting 

 and useful to follow, with Poinsot, the body during its entire rotation, yet 

 practically this is of minor interest, and the chief problem remains still, 

 " d determiner le lieu ou se trouvera le corps an bout d'un temps donne." 



In the present day all those familiar with both methods of treatment 

 hold fast the good in each ; they supplement each other. Often in the 

 course of the same investigation we must interrupt the general analytical 

 process with synthetic deductions, and inversely. Thus we may well close 

 these considerations with the sentence with which Schell begins his " TTieo- 

 rie der Bewegung und der Krdfte " both methods, the analytic and the 

 synthetic, can only, when united, give to mechanics that sharpness and 

 clearness which at the present day ought to characterize all the mathemati- 

 cal sciences. 



m. 



GEOMETRICAL STATICS. 



Statics is a special case of dynamics, though earlier treated as indepen- 

 dent of the latter. The principle of d'Alembert furnishes the means of 

 passing from one to the other. In technical mechanics the distinction is 

 still preserved, and indeed, in view of the distinct branches in which the 

 applications on either side, are found, not without propriety. 



After the mechanics of the ancients, as comprised in the mathematical 

 collections of Pappus, the first great step towards our present geometrical 

 statics was made by Simon Stevinus (1548-1603), when he represented the 

 intensity and direction of forces by straight lines. Stevinus himself gave 

 a proof of the importance of his method, in the principle deduced from it, 

 that three forces acting upon a point are in equilibrium when they are pro- 

 portional and parallel to the three sides of a right-angled triangle. 



A main discovery was the parallelogram of forces by Newton (1642- 

 1727). The composition of two velocities in special cases was long famil- 

 iar. Galileo made use of it for two velocities at right angles, and exam- 

 ples also occur in Descartes, Rdbereal, Mersenne, and Wallis, but the funda- 

 mental principle was first established when Newton replaced the theories 



