122 PETER GUTHRIE TAIT 



Fluxions (' methodi hujus generalis fundamentum continetur in lemmate praecedenti ') 

 Newton expressly says...' Neque enim spectator in hoc lemmate magnitude momen- 

 torum, sed prima nascentium proportio. Eodem recidit si loco momentorum 

 usurpentur vel velocitates incrementorum ac decrementorum (quas etiam motus, 

 mutationes et fluxiones quantitatum ^ominare licet) vel finitae quaevis quantitates 

 velocitatibus hisce proportionales.' The finite differentials of Cauchy & myself, & 

 doubtless of other moderns, are therefore really the fluxions of Newton in disguise ; 

 and I ought to talk, or at least might talk, of fluxions of quaternions, and of their 

 functions. 



"(30) Before I was 17 years old, I had diligently studied at least the three first 

 sections of the ist Book of the Principia..,.'B\A. I think it was about that age, that 

 I was carried away by the attractions of the French School, & specially by that of 

 Lagrange. The Calcul des Fonctions charmed me, & for several years I supposed 

 it to be, not merely an elegant and original production of a genius, whose mathematics 

 almost sublimed themselves into poetry, but a sound and sufficient basis for the 

 superstructure of the Differential Calculus.... But you may possibly be aware that 



it is now a long time since I pointed out a fatal defect in the foundation of 

 Lagrange's theory, as set forth in the Calcul des Fonctions. ...... I suppose that no 



one now contests the necessity of founding the differential calculus on the notion of 

 limits; at least, if it be desired that the structure should be a weather-proof and 

 habitable house: or, in short, good for anything. In that respect, at least, though 

 certainly not in the notation of fluxions, we are all glad to go back to Newton. 



"(31) To connect my definition more closely still with Newton's views, we have 

 only to conceive that, if r = dq = A^, the quaternion function, fq, of the quaternion 

 variable q, GROWS,... and passes, GRADUALLY, by such GROWTH, through the n I 

 intermediate stages (of state, rather than of quantity) 



where n is a large positive whole number, until it ATTAINS at last the state 



/(*+?) =/<* + r > =/ (* + **> =fl + A /<?." 

 Tait's reply to this long letter was as follows : 



Q. C. BELFAST, 



October 19^/58. 

 My dear Sir William Hamilton 



Plunged as I now am in the middle of the entrance & Scholarship 

 examinations for this session, I shall not have for some days the amount of time 

 requisite for a careful reading of your excellent No. V ____ 



I am tolerably familiar with the works of Moigno and I quite agree with you 

 in your estimate of him. Did you ever see his ' Repertoire d'Optique Moderne ' ? 

 It is the strangest mixture of valuable matter and utter trash I ever came across. 

 I should like very much to know your opinion of Cauchy's investigations in the 

 Undulatory Theory for I have found it possible by apparently legitimate uses of 



