Prof. De Morgan on the Invention of the Signs -f and — . 135 



scheme begins, and the interpreter having (so far as may be 

 in his works,) " found out the Almighty to perfection," will 

 be left to the Word and his own free-will to form his judge- 

 ment of the past, present and future, and " He who reads the 

 secrets of the heart," will know to what end he applies his 

 knowledge, and whether his faith be or be not without virtue. 



I have now, Sirs, to apologize to yourselves and readers 

 for this lengthy and 1 fear tedious detail, but I felt I owed a 

 duty to what I believed to be the truth to make it. I have an 

 utter aversion to polemical discussions of any kind ; circum- 

 stances of a local nature have unfortunately involved me in 

 the present controversy with gentlemen to whom I owe the 

 deep obligation of a great addition to my amount of know- 

 ledge; it was their valuable labours elsewhere which stirred 

 my ambition to elucidate the structure of a region, some of 

 whose headlands and hills and valleys bound my daily land- 

 scape. I unaffectedly feel gratitude, and something more, for 

 what they have taught me. I accord my full confideiice in 

 their observations when they do not write and travel too last ; 

 but for this, 1 am well assured, I should not now feel con- 

 strained to adopt the mortifying alternative of proclaiming 

 how essentially I differ from the results of their Devon and 

 Cornwall labours, and in this spirit I venture to conclude, 



"Amicus Sedgwick, amicus Murchison, sed magis arnica 

 Veritas." 



I have the honour to remain. Gentlemen, 



Your obliged and obedient Servant, 

 Bleadon, near Cross, Jan. 1, 1842. D. WiLLIAMS. 



XXI. On the Invention of the Signs + and — ; and on the 



sense in "which the former was used hij Leonardo da Vinci. 



By A. De Morgan, Esq. 



To the Editors of the Philosophical Magasine and Journal. 

 Gentlemen, 

 I N M. Libri's History of Mathematics, vol. ii. p. 46, lie 

 *- attributes the invention of the signs -f and — to Leonardo 

 da Vinci, for which he cites, without quotation, a manuscript 

 of that extraordinary man in the library at I'aris. 1 tiiink it 

 most likely that M. Libri, when called upon, will be able to 

 substantiate his own assertion by an express quotation which 

 shall put the matter beyond doubt: but it will be necessary 

 that he should do so, on account of the occurrence of -f and 

 — in one manuscript at least of" Da Vinci's, in a sense alto- 

 gether different from that of algebraic addition and subtraction. 



Being lately at the British Museum, I was informed by my 



