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THE

ELEMENTS OF PLANE GEOMETRY

IN 48 PROPOSITIONS ;

Promising to serve for the First Sie Books of Euclid with all the deductions.

FROM THE

SANSCRIT TEXT OF ARYA BHATTA.

EDITED ON THE PRINCIPLE OF EUCLID

BY

JASODA NANDAN SIRCAR.

CALCUTTA.

PRINTED BY P. N. SHAHA AT THE CARISTIAN

HERALD Press.

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PREFACE. The Hindoo Geometry, as contained in the Dasa Gitica Parisistacam of Arya Bhatta, which is the subject of the present edition, differs from that of Euclid in several important particulars. The propositions in the Hindoo Geometry are arranged from right to left. That which ought to have been, according to modern taste, the last in the series is given first and is proved by the next following. The latter is proved by the next following it and so on.

This inverse arrangement has rendered the subject unnecessarily abstruse and definitely points out to the learner the necessity of attending Guruphadeshas or lectures of a competent Geometer with a view to his being initiated into the secrets of Geometry. One reason that the editor of the present work would venture to give for the light of Geometry having been extinguished from the land in an early age, is the necessity, as alluded to, of having recourse to the lectures of a .competent Guru, under which the student of Geometry was unavoidably placed. The excellent diction adopted by Euclid in his demonstrations, as would appear through its English version, is not only free from this hieroglyphic character of the Hindoo Goometry, but is what from its inimitable care and arrangement as well as from its easy simplicity, and masterly grasp, has become alike acceptable to the philosopher, the orator, the young and the advanced. The Hindoo Geometry, on the other hand, contains mere enunciations, without attempting at demonstrations, giving here and there cursory hints which serve as

dubious landmarks in a boundless exploration. Yet, in a few lines it has given all the useful matter, that is contained in the first six books of Euclid : besides it seems to give the trisection and, perhaps, polysection of an angle, and to allude as having found in a previous work, a straight line equal to a given arc. Moreover, when demonstrated on the principles of Euclid, the Hindoo Geometry becomes, as it were, an abridgment of Euclid containing all the useful propositions in his first six books and leaving out the useless ones.

There are, it must be said, propositions in Euclid which might be left out as useless but which are retained as subsidiary propositions that lead to the demonstration of the useful ones.

The Hindoo Geometry contains no such subsidiary propositions and is consequently much less complicated than Euclid in this respect. The whole appears as a continued demonstration, as if, of a single proposition. The chain is never lost and if lost for a time is resuined soon after, while in Euclid a complication arises from the second, third, fourth and sixth books being not progressively founded upon each other, but being independent books all founded upon the first and having no intermediate connections. Glimpses of connection appear now and then (for instance between the second book and the last three propositions in the 3rd book or the tenth in the fourth), but they rather contribute to puzzle the memory of the student and add to the complication in question.

In the manuscript of Dasa Gitica Parisistacam or the Hindoo Geometry in hand, the enunciations are stated in the left part of the page and the figures are given in the

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right. Hints or notes, if any, are given below the enunciations which they are intended to illustrate. Some of the figures agree with those of Euclid perfectly and there is internal evidence to suspect that either Euclid has borrowed from the Hindoo Geometry or that the Hindoo Geometry is an abridgment of Euclid. In twelve cases where no figures are given in the manuscript, the Elitor has adopted Euclid's figures. He has also made additions from Euclid, where the proofs in the original have appeared to be defective. It must also be said that without the light of Euclid to guide him, it would have been simply impossible for him to walk through the hazy maze of the Hindoo Geometry and he has been utterly at a loss to do where Euclid has failed to assist him. Elence it is that he has passed over the part in the manuscript where the trisection and, perhaps, polysection of an angle are given, and has merely contented himself with publishing the original in Sanscrit. One proposition, however, not contained in Euclid is deciphered and placed in this edition, which is to find two mean proportionals between two given straight lines by a straight line and circle and which, says Mr. Pott, if we understand him aright, is impossible. It appears from the context that the trisection of an angle is based on this proposition and the one following it i e. Similar triangles are to one another in the duplicate ratio of their homologous sides. The whole matter therefore deserves the best attention of the Geometers.

The student will find in this edition all the useful propositions of Euclid and also the inferior ones which are given in the corollaries. Thus in a compendium, it contains both Euclid and the Hindoo Geometry,

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