VI PREFACE. 



trusting mainly to independent knowledge and judgment, I 

 derived assistance from the labours of well known authors on 

 the subject, especially Cournot, De Morgan, Moigno, Navier, 

 and Schlb'milch. In the subsequent editions a considerable 

 amount of fresh matter has been introduced, and this rests 

 almost exclusively on my own authority ; increased experience 

 as a teacher naturally gave stronger confidence to the writer. 

 Thus the work now contains on the whole much that is 

 original in substance, and much that is ew in form. 



The present edition has been carefully revised and some- 

 what enlarged. I have examined with attention and interest 

 treatises on the Differential Calculus recently published by 

 eminent mathematicians, in order to discover if the methods 

 of explaining and developing the principles of the subject 

 had gained any real improvement during the last twenty 

 years. I have not however found reason for concluding that 

 I could with advantage make any essential change in this 

 elementary work. 



I have much reason to be grateful for the approbation 

 bestowed by teachers and students on this volume, the 

 first of a long series relating to various branches of mathe- 

 matics. My thanks are especially due to Professor Battaglini 

 of Naples for the honour which he has conferred on me by 

 translating my treatises on the Differential and the Integral 

 Calculus into Italian. 



I. TODHUNTER. 



ST JOHN'S COLLEGE, 

 April, 1871. 



