COMPOSITION OR AGGREGATION OF FORCES IS A CONSEQUENCE. 297 



3. Successive and simultaneous rotation about axes passing through a point. Motion 

 of rotation is brought under the case of tendency having magnitude and direction, by having 

 for its determinants an axis and an angular magnitude. Let two axes remain fixed ; and let 

 a point revolve first round one axis and then round the other. All the postulates are satisfied 

 except the third : the result is not indifferent to conversion. But when the angles of rotation 

 are infinitely small this third postulate is satisfied, the difference between the results before 

 and after conversion being infinitely small compared with the wholes. And hence demon- 

 stration of the diagonal law in composition of successive infinitely small rotations. Of simul- 

 taneous rotations, as of simultaneous translations, it must be said that they cannot coexist. 

 But revolution of a point about one axis may be aggregated with revolution of that axis and 

 surrounding spaces about itself or another axis; and in infinitely small rotations the disturb- 

 ance of the second axis produces an effect which is but an infinitely small part of the whole. 

 It need hardly be stated that, in these cases of translation and rotation, all the postulates 

 are pure results of thought. 



Before proceeding further, I must request permission to digress into some remarks on the 

 foundation of geometry. Rectilinear translation and rotation are the two simple elements into 

 which all motions are resolved : the lines in which a point moves by simple translation and by 

 simple rotation, the straight line and the circle, are all which elementary geometry has ever 

 taken into account. The straight line, considered as rotation about a point at an infinite 

 distance, may occur to a mind much accustomed to generalisation as unduly separated from 

 the other circles. And Mascheroni, by performing the constructions of geometry with the 

 compasses only, has shown the rejection of the straight line to be as possible as would be 

 the rejection of a circle of any one given radius. But it could have been made out, long 

 before Mascheroni, that such a principle as appears in the rejection of the straight line must 

 be carried further, if acted on at all. For Benedetti (1553) and others had shown that the 

 constructions of geometry require only the straight line and one circle, of some one given 

 radium. And this opens reasonable ground of suspicion that all the power of construction 

 which is given by the straight line and circle lies in any two circles of different radii, pro- 

 vided one of them be large enough to pass through any two points which can be wanted. 

 As no such limitation of means could be listened to, and as any limitation short of the 

 utmost which leaves sufficient means would be idle, the straight line must be allowed to 

 retain its place, and of course its importance. 



The straight line and circle are self-repeating curves : any arc of one of them is a 

 facsimile of any equal arc of the same. And they are the only self-repeating plane curves. 

 It is then the limiting law of plane geometry to stop after the admission of self-repeating 

 curves, and to allow no others. When we come to solid geometry, we admit the self- 

 repeating surface, the sphere : why do we exclude the self-repeating curve of double cur- 

 vature, the screw, the most simple aggregate of one translation and one rocation .'' I believe 

 we have only to answer that Euclid did not do it. If Euclid had possessed that fulness of 

 system in solid which he had in plane geometry, we may strongly suspect that the screw would 



38—2 



