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 appreciate 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 detractor 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 "Lílívatí" of Bháscara Achárya, 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 "Lílívatí," 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 votaries from embarrassment, and bestows happiness upon his worshippers; 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, resplendent as a blue and spotless lotus; and delighting in the tremulous motion of the dark serpent, which is perpetually twining within his throat."
The principles of numeration and addition are then stated concisely, and he affably propounds his first question: "Dear, intelligent Lîlîvatî, if thou be skilled in addition and subtraction, tell me the sum
