XX. On the Influence of Signs in Mathematical 
_ Reasoning. 
By CHARLES BABBAGE, Esq. M. A. Trin. Cott. 
FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, MEMBER OF THE ROYAL 
IRISH ACADEMY, FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, MEMBER OF 
THE ASTRONOMICAL SOCIETY, MEMBER OF THE ROYAL ACADEMY OF DIJON, 
CORRESPONDING MEMBER OF THE PHILOMATH. SOCIETY, PARIS, 
AND OF THE ACADEMY OF MARSEILLES, &e. 
[Read Dec. 16, 1821.] 
Ir can scarcely excite our surprise that the earlier geometers, 
engaged in successfully employing the most powerful instru- 
ment of discovery which human thought has yet contrived, and 
seduced by the splendour of the view their science had opened 
to them, should press with earnestness to enlarge its boundaries 
by new applications, rather than exert their genius in explaining 
the causes which have combined to advance it to such unrivalled 
eminence. On the discovery of those branches which have so 
completely altered the face of the science, the use of the new 
acquisitions was too inviting to allow time for any very scrupu- 
lous enquiry into the principles on which they were founded : 
satisfied with the accuracy of the results at which they arrived, 
the desire of multiplying them naturally prevented any return 
on their steps for the purpose of applying themselves to the less 
promising task of establishing on secure foundations, principles 
of whose truth they felt confident. 
These efforts to extend the reach rather than fix the basis of 
the new calculus, were undoubtedly to be admired at the period 
to which we refer: an acquaintance with its extensive bearings 
ought justly to have no inconsiderable influence on the form in 
which its elements should be delivered; hence the lapse of 
TT 2 
