373 



VIGA GANITA. 



VIGA GANITA. 



376 



writings : nor would it be unlikely that at the same time those notions 

 of algebra from which Diophantus wrote bis work were given in 

 exchange. It is exceedingly difficult to make any other conjecture 

 which will explain the existence of this solitary work on algebra 

 among the Greeks ; but that the Hindoos received at this time all 

 their astronomy is very unlikely. In several points it differs mate- 

 rially from the system of the Greeks, and in some it is more correct : 

 for instance, in the precession of the equinoxes, the length of the 

 tropical year, and the synodic period of the moon. 



It is worth noting that the disposition whioh existed among Greek 

 writers to send their old sages to India to learn the principles of 

 astronomy and other sciences does not commence till after the 

 Christian era. 



We may now leave the question of the antiquity of Hindoo science, 

 and proceed to give some account of its materials. The works in 

 which it is contained are usually written in verse, and in short and 

 obscure precepts, intended to be committed to memory : the commen- 

 tators take every verse, and almost every word, in succession. The 

 most peculiar feature of these books is the general absence of demon- 

 stration : results only are frequently announced. It cannot be denied 

 that there is, particularly in the algebraical part, a frequent succession 

 of steps, of which the connection is pointed out in a manner which 

 makes the last of those steps a necessary consequence of the first. 

 But though a Hindoo writer may fall into the road of demonstration 

 in any part of his journey, and remain there for a time, it is evident 

 that this is with him entirely a matter of convenience, and that he 

 does not feel himself at all bound to give proof. 



It seems to us by no means to be taken for granted that there ever 

 was any such thing among those writers, or their predecessors, as a 

 connected system of demonstration ; there are few propositions either 

 of their geometry or algebra which might not have been found by 

 trial, and verified numerically or graphically ; or else procured from 

 empirical propositions by the mode of occasional demonstration just 

 alluded to. But it must be allowed that here and there we have a 

 proposition for which it is difficult to suppose an origin without pre- 

 suming, not only power of demonstration, but methods of considerable 

 generality. Though the Greeks, after the time of Euclid, never pub- 

 lished anything of a mathematical nature without demonstration, it 

 does not follow that even they had demonstration from the beginning; 

 and the hints given by Proclus on the progress of geometry would 

 almost support the contrary notion. The idea of an undemonstrated 

 mathematical system may appear a strange one, but it must be remem- 

 bered that the nations of modern Europe are, in this matter, the 

 pupils of the Greeks, and never, till of late years, even so much as 

 heard of any science which was independent of their own masters, 

 except what has been added among themselves; and it is no wonder 

 that any different mode of proceeding may seem strange, when the 

 mere possibility of such a mode has never been made a matter of 

 discussion among us. 



The following is Colebrooke's comparison of the daily motions of 

 the several planets, according to the Hindoos, Ptolemy, and Lalande 

 (it is not worth while to substitute any astronomer more modern than 

 the latter). Degrees, minutes, and seconds are common to all : 



It appears then, that Ptolemy's daily motions are generally too 

 small, but that the Hindoos err still more in the same direction; 

 except only in the synodic motion of the moon, in which they are 

 much more correct than Ptolemy : the Surya Siddhanta in particular, 

 probably the later work of the two, and therefore the more likely to 

 be misled by Ptolemy's numbers if they were known, agrees entirely 

 with Lalande. This is what might have been expected : the Hindoos 

 were not, as far as appears, noted for good observations, nor very apt 

 to record them ; but they sedulously attended to eclipses, the pre- 

 diction of which was the most important duty of the astronomer, and 

 hence the goodness of their determination of the moon's synodic 

 motion. 



The length of the sidereal year is given 365 d 6 h 12 m 30 s , more than 

 three minutes too much ; the Hindoo astronomical year is sidereal, 

 and begins when the sun enters the sign of the Ram. But their 

 tropical year is 365 d 5 h 504 m , much nearer the truth than that of 

 Ptolemy and Hipparchus, which was 365 d 5 h 55 m . The meridian from 

 which they reckon is that of Lanka, which some take to be Ceylon, 

 others the name of a lake near the sources of the Ganges ; it passes 

 through Ujein. Their precession of the equinoxes is 54" iu each year, 

 which is much more correct than that of Hipparchus or of Ptolemy. 

 Most of the Hindoo writers do not suppose a permanent precession, 

 but imagine the oscillatory motion or trepidation, as it was called 



when it was afterwards introduced into Europe by the Arabs, who 

 seem to have borrowed this idea from India. Those who hold the 

 oscillatory motion fix it at from 24 to 27" on each side of a mean 

 position. The revolutions of the apsides and nodes of the moon are 

 given within a fraction of a day of what they are now known to be; 

 the obliquity of the ecliptic is 24, too large even for their time. The 

 inclination of the moon's orbit is made 4 30'; those of Mercury, 

 Venus, and Saturn, 2 each; of Mars, 1 30'; of Jupiter, 1. The 

 circumferences of the orbits (obtained, it is said, upon the purely 

 speculative idea that they all move with the same actual velocity) are 

 given in yojemas, a measure which appears to have been used in 

 different senses, and which cannot be very well settled. This yojana 

 contains four crosas, aud the modern crosa is 1*9 statute miles. 

 According to Colebrooke, Aryabhatta gave 3300 yojauas for the cir- 

 cumference of the earth, which, if the crosa were the modern one, 

 would be 25,080 statute miles, or 69'7 miles to a degree : this degree 

 of accuracy must be accidental. With regard to the motions of the 

 nodes and apsides of the planets, which the Hindoo writers profess to 

 give, Colebrooke thinks they are inventions constructed from analogy 

 with those of the moon. As to the more theoretical parts of astro- 

 nomy, the Hindoos knew the inequality of the planetary motions 

 which is called the equations of the centre, though their values of 

 these equations are not very correct. They had about as much of 

 that which was afterwards called the Ptolemaic system as is reported 

 to have been invented by Hipparchus ; the principal variation being 

 that their epicycles are made (by several of their astronomers) oval, 

 instead of circular. This is enough of the actual details of the 

 astronomy for our present purpose ; those who would know more of 

 it must search the tedious and disjointed pages of the authors whom 

 we have cited. No one of them would trouble himself to collect into 

 one page the actual numerical elements, of the astronomy on which 

 they were all writing; and it is consequently so difficult to understand 

 their several accounts (since, in case of apparent contradiction, we 

 cannot know whether they speak of the same or of different values of 

 the elements), that we have not felt ourselves able to supply the 

 deficiency. It is not however of much consequence, for the elements 

 of the Hindoo astronomy are only interesting as connected with its 

 date and the discussions upon it. We have not at all entered upon 

 the refutations which it is still customary to give to Bailly on points 

 connected with the theory of gravitation. That writer imagined that 

 by correcting the various elements of the planets, as they now are, so 

 as to reduce them to what, according to the Newtonian theory, they 

 should have been at the beginning of the Cali Yug, a remarkable 

 agreement was found between the results and the recorded elements 

 of Hindoo astronomy. There is such agreement in one or two cases, 

 but the result of the whole is, that there is no reason to suppose the 

 few accordances to be due to anything but accident. 



The mixture of the mythological, which some of the Hindoo astro- 

 nomers [the author of the Surya Siddhanta aud also Bhascara ; the 

 latter, with apparent reluctance, not in the text, and only briefly iu the 

 notes] allow to appear in their works, and which seems to have be- 

 longed to the vulgar Creed, presents a very strange appearance. Both 

 in Hindoo and Burman systems eclipses are caused by a distinct planet, 

 Rahu, of a dark essence, which at times takes both the sun and moon 

 under its influence. The irregularities of the planetary motions, their 

 stations, retrogradations, and departures from the ecliptic, are caused 

 by deities provided for the purpose, who reside at the nodes and points 

 of conjunction. Aryabhatta, according to Colebrooke, not only gave 

 the true solution of the phenomena of eclipses, but asserted the 

 diurnal motion of the earth, which he affirmed to be carried round an 

 axis by a strong wind. Brahmegupta attributes this opinion to him 

 with reproach, and asks why, in such case, lofty bodies do not fall 

 (that is, off the earth). A commentator of Brahmegupta, who lived 

 before the 12th century (since he is mentioned by Bhascara), and whose 

 name (Prithudaca Swami) deserves to be mentioned, in spite of our 

 wish to keep as clear of these unretainable appellatives as we can, 

 says "The objection that lofty things would fall is contradicted; 

 for every way the under part of the earth is also the upper, since 

 wherever the spectator stands on the earth's surface, even that point is 

 the uppermost point." But the same commentator adds a very 

 scholastic reason for the earth's motion causing the diurnal changes. 

 He says a planet cannot have two motions : meaning that the orbital 

 motion is the only one it can have, and that the diurnal motion is 

 therefore to be attributed to the earth. 



The great point of contest seems to have been whether the earth is 

 stable in space or perpetually falling; if the former, whether it stands 

 by itself or upon a support We do not find that any astronomers 

 cited by our authorities support the notion which our books attribute 

 to the Hindoos, namely, that the earth stands upon an elephant, which 

 itself stands upon a tortoise, which tortoise swims in a sea of milk ; 

 but there is an allusion to this succession of supports in a passage of 

 Bhascara cited by Colebrooke, which is on other accounts worth 

 the quoting. The Jains, a species of Buddhist eect, affirmed the falling 

 motion of the earth ; on which Bhascara remarks " The earth stands 

 firm, by its own power, without other support, in space. If there be 

 a material support to the earth, and another upholder of that, and 

 again another of this, and so on, there is no limit. If finally self- 

 support must be assumed, why not assume it in the first instance ? 



