336 THE POPULAR SCIENCE MONTHLY. 



place both lived and worked in our own century. When we consider 

 what astronomy would be without these three great men that is, 

 what it was only so few years ago we are better prepared to appre- 

 ciate the studies which laid the remote foundations of their triumphs. 



It would be impossible, within moderate limits, to determine the 

 value of Hindoo astronomy, however interesting the effort might be, 

 since we should enter at once into debateable ground, and come among 

 great authorities in conflict. 



Bailly, Delambre, Bentley, Davis, Hunter, Sir William Jones, 

 and others, have various, often contradictory, beliefs to maintain. 

 Some are partisans of the Greek, some of the Arab, others of the 

 Hindoo scientists of long ago. But, fortunately, some of the original 

 manuscript books of the Hindoos have come down to us : among others 

 various complete treatises on mathematics, and these are authentic and 

 of great age. Precisely of how great age it is difficult to ascertain. 

 Bailly, a Hindoo partisan, accepts the largest estimate ; Delambre, a 

 deti*actor of Hindoo science, and an advocate of the Greek, believes 

 the most important of them to have been written about a. d. 1114; 

 while the translator of this manuscript, Colebrooke, a distinguished 

 Sanscrit scholar, places the date of writing, in a. d. 1150. 



This treatise, the "Lilivati" of Bhascara Acharya, is supposed to 

 have been a compilation, and there are reasons for believing a portion 

 of it to have been written about a. d. 628. However this may be, it is 

 of the greatest interest, and its date is sufficiently remote to give to 

 Hindoo mathematics a respectable antiquity. 



The " Lilivati," according to Delambre, was written to console the 

 daughter of its author for her ill-success in obtaining a husband, and 

 it speaks well for the Hindoo gentlewoman that such a means could be 

 considered worth the attempting. It was called by her name, and 

 many of the questions are addressed to her, as we shall see. 



It opens most auspiciously with an invocation to Ganesa, as follows : 

 " Having bowed to the Deity whose head is like an elephant's ; whose 

 feet are adored by gods ; who, when called to mind, relieves his vota- 

 ries from embarrassment, and bestows happiness upon his worship- 

 pers ; I propound this easy process of computation, delightful by its 

 elegance, perspicuous with words concise, soft, and correct, and pleasing 

 to the learned." 



Thus fairly launched, the author gives various tables of Hindoo 

 moneys, weights, etc., and proceeds to business, not without another 

 invocation, however, shorter this time : " Salutation to Ganesa, re- 

 splendent as a blue and spotless lotus ; and delighting in the tremu- 

 lous motion of the dark serpent, which is perpetually twining within 

 his throat." 



The principles of numeration and addition are then stated con- 

 cisely, and he affably propounds his first question : " Dear, intelligent 

 Lilivati, if thou be skilled in addition and subtraction, tell me the sum 



