636 PHILOSOPHICAL TRANSACTIONS. [anNO 1(571. 



those five lectures, the two first are lemmatlcal, or preparatory to the demon- 

 stration of the propositions delivered in the other three, containing some curi- 

 ous theorems. In the tenth is delivered a general analytical method of deter- 

 mining tangents, extending to all sorts of curve lines, both geometrical and 

 mechanical, as M. Descartes distinguishes them. 



The llth lecture contains several general theorems, about the dimension of 

 magnitudes, or the comparison of them with one another. To which is sub- 

 joined an appendix concerning the dimension of circular and hyperbolical 

 segments, with divers theorems and rules serving to that purpose. 



The 12th lecture contains also several theorems concerning the dimension 

 of magnitudes, but chiefly respecting the dimension of surfaces produced by 

 the rotation of curve lines, and the dimension of curve lines themselves. To 

 this lecture there are also three additaments ; the first containing some theo- 

 rems about the dimensions of spaces constituted by the tangents and secants of 

 a circle. The second shows how the foregoing theorems may be demonstrated 

 by the apagogic way, or by reduction ad ahsurdum ; together with a way of 

 finding the dimension of the surfaces of conical bodies. In the third divers 

 problems and theorems are added, akin to those of the llth and 12th 

 lecture. 



The 13th lecture propounds an explication of the nature and constitution of 

 equations, with the variety of roots, their limits, &c. by construction and con- 

 sideration of certain curve lines appropriate to each equation : with some notes 

 respecting each particularly, and all in general. 



So much of these two excellent treatises : since the publication of which, 

 their worthy author has been pleased to communicate to a friend of his some 

 corollaries, belonging to the second problem of his third appendix to the 12th 

 lecture ; which because we conceive they will be very acceptable to the mathe- 

 matical reader, we shall here subjoin.* 



III. A Continuation of the Memoirs of M. Bernier, concerning the Empire 

 of the Great Mogul. From the French. London, 1671, in 8vo. 



The first volume of these memoirs, containing little but political affairs, was 

 left unmentioned in these books; but in this second, besides an accurate de- 

 scription of the two famous cities of Indostan, Delhi and Agra, and many things 

 showing the genius of the Moguls and Indians, also those which belong to their 

 militia, &c. is an account given. 



First, of the extravagant opinions of the Gentiles of Indostan ; of their odd 



* These may be seen introduced into Mr. Stone's translation of the work, towards the con- 

 clusion. 



