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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. Hence 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 augle 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,

The original in Sanskrit is at present reproduced so far as ends in the finding of two mean proportionals between two given straight lines, which will serve as a specimen for the rest. There ought to have been three figures for the second proposition, but as there is only one figure in the original, it has been preferred. The 18th and 21st propositions appear to involve two propositions, each. Perhaps they ought to have been four instead of two. But the editor did not like to depart from the original in any way whatever for the sake of mere niceties. Each proposition has been demonstrated agreeably to its figure or figures as given in the original and the reader can realize well the difficulties in which the editor was placed in explaining a figure (for instance the figure : Prop. 47) when it presented itself to his eyes as altogether different from that which is in Euclid. Meanwhile if the present edition will meet with encouragement, the condition of the print and figures will be improved and the original published in full.

It is not unlikely, to quote the opinion of Colbroke, that Arya Bhatta flourished several centuries before Christ. According to reliable authorities, Euclid fourished 300 years B. C. This is suggestive of a good deal of reflections which antiquarians will do well to indulge in regarding the Geometry of the Aryans.

EDITOR.

ॐ नमो गणेशाय।

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पथाष्टिशतके गोलाध्याये परिधिखण्डरेखयोः साम्यं विश्त्य समाप्य च सम्प्रत्यारम्भे म्लेच्छान्तेवासिनामववोधाय गोलमेवानुसरति। गौतिकायान्वखाणामेव परिगणनं । अथ प्रयोजनाभिधेयमङ्गलादीनि। मूढ़ानां वोधाय च विनोदाय च विशेषज्ञानां। नत्वा क्षेत्रपतिं ध्र वं निगद्यते क्षेत्रोपपत्तिः॥

मूढ़ानामिति म्लेच्छानामन्येषान्तु गोलाध्यायेनैव पर्याप्तिरिति प्रसक्तिः । क्षेत्राणामुपपत्तये क्षेत्राणां पत्ये इत्यादि वृहस्पतिवचनम् । मितिरित्येक प्रवदन्ति न्यायगणितमित्यपरे प्राडः। व्याविधिरिति वा योगविधिर्वा विश्वगणितमिति चापि वदन्ति ॥

मितिरिति नाषे, न्यायगणितमिति भाष्ये, व्याविधिरिति पाराशरे, योगविधिरिति गर्गः,विश्व गणितमित्याचार्याणां । तथाहि,

पराशरादधिगतं गर्गेण विशदौछातं ।
मार्थाचार्येण रचितंमितिशास्त्रं प्रचक्ष्यते ॥

व्यतिरेकान्वयज्ञानं व्याप्तिज्ञानञ्च जायते ।
यतोस्थानवायवीजन्तदित्युवाच रहस्पति:।।अपिच
मानिन्या नयनसलिलमिव तरला बुद्धिः
तस्यापि स्यात् सिवियदि सिध्यति मित्यां ।।
कारिकालोचनं कृत्वा क्षेत्रालोचनमेव च ।

गुरुपदेशं संस्मृत्य मितिचिन्तां समाप्नुयात् ॥ अथ समत्वे प्रयोजकत्वानि

आकृतिः प्रकृतिस्तत्वं समत्वे प्रभवन्ति च ।
विषमत्वेपि तद्भावः समस्य सन्निवर्तते ।।

तिरित्याकार एव समत्व प्रमाणं । प्रकृति-
रिति मानं (area)। तत्त्वमिति इयोरेकस्यापि च
समाभिधानत्वे (see def. 26)। विषमत्वमिति सति
तद्भूतेवैपरीत्यमपि प्रमाणम् (if it is that, its
converse also proves that ) । अनुपाते चतुणां
समाने तु इयोस्तस्यापि च दूति गर्गः। अपिच
कर्णीयौ गणयेत् समौ विषमयोः साम्यं प्रमाणे पि च
साकल्यं सममेव संहितवतां साम्यं भवेत् तहतां ।
वृत्ताई सममेव योगयुगलं व्यावतमानं समं
प्रायेण त्रिभुजाः समाः सममथ प्राक्तन्य कोणहयं ।

कर्णीयौ लम्बसंसृष्टौ योगावित्यर्थः । योगयुगलमिति कोणवयं लक्ष्यते, कोणानां रेखायोगोत्प- ।

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