426 Mr DE morgan, ON THE THEORY OF 



and denominator into numer and denomer, which the arithmeticians dare not do. That 

 nothing shorter than ' the differential coefficient of y with respect to tv,' sixteen syllables of 

 sound and forty-three letters of writing, can be found to express the ultimate element of the 

 differential calculus, is a misfortune and a discredit. And more especially when it is remem- 

 bered that this conglomerate of letters does not express the modern meaning of the symbol. 

 This meaning is 'the limiting ratio of the increment of y to the increment of a?;' and, 

 when first introduced, must be preceded by explanations which would allow ' the limiting 

 ratio of Ay to Aa? ' to be sufficient. It is a grand absurdity that the common name of the 

 most common symbol, the least amount of phraseology which gives a complete designation, 

 should be longer than its definition need be. 



The reform which I should propose, if it were possible to create a discussion, would 

 consist in expressing dy : dx as ' the rate of y to a? ' and ' the oc rate of y,' in abbreviation of 

 ' the ratio of the rate of variation of y to that of a?.' This is a most useful notion, and 

 gives all the simplification of expression which can be imagined to be practicable. 



A. DE MORGAN. 



University College, London, 

 July 31, 1861. 



ADDITION. 



In the last sentence a nomenclature is recommended which is simply fluxional. It 

 is very much to be regretted that the notion of fluxions disappeared with the notation. 

 Though satisfied that the doctrine of limits must be the basis of sound demonstration, 

 I advocate the early introduction and use both of the infinitesimal and of the fluxional 

 principles in aid of conception : and I observe that the fluxional principle begins to gain 

 some currency in works published on the continent. It is not correct to make Newton 

 the first proposer of the notion of magnitude as generated by flux : the intension and 

 remission of the schoolmen were really positive and negative fluxions. I had made up 

 my mind that Newton was more conversant with the schoolmen than is supposed, long 

 before it was made known that the very scholastic Logic of Sanderson was a study of 

 his early youth. It is impossible here to give any sufficient account of the old doctrine: 

 I will content myself with one quotation. Nicolas D'Oresme (Horem, Oresmius) who died 

 Bishop of Lisieux in 1382, wrote a tract De Latitudinibus Formarum, which was printed 

 in 1482, I486, 1515, and perhaps oftener. Though consisting of definitions and statements, 

 without any calculus, there is in the work a certain prselibation of co-ordinates. The • 

 latitude being constant, we have a rectangle : the variation, therefore, of the latitude makes 

 the difficulty of finding areas. Among other statements, we find that in every semicircle 

 the intension of the breadth (which is nothing but dy : dx positive) begins from the utmost 



