of the pulăbhů, or shadow, projected from a perpendicular gnomon when the sun is in

the equator."

minions.

"The longitude is directed to be found by observation of lunar eclipses calculated for the first meridian, which the Sooryu-Siddhantŭ describes as passing over Lŭnka, Rohitǎků, Ŭvǎntee, and Sunghita-sară. Ŭvùntee is said by the commentator to be "now called Oojjǎyinee," or Ougein, a place well known to the English in the Marhatta doThe distance of Benares from this meridian is said to be sixty-four yojŭnŭ eastward; and as 4,565 yojŭnă, a circle of longitude at Benares, is to sixty dundŭs, the natural day, so is sixty-four yojănăs to 0 dăndă, 50 pălă, the difference of longitude in time, which marks the time after midnight, when, strictly speaking, the astronomical day begins at Benares.* A total lunar eclipse was observed to happen at Benares fiftyone půlŭs later than a calculation gave it for Lunka, and 51+4565 4 sixty-four yojůně, the difference of longitude on the earth's surface."

60

=

"For the dimensions of the moon's kõkshŭ (orbit) the rule in the Sungskrită text is more particular than is necessary to be explained to any person, who has informed himself of the methods used by European astronomers to determine the moon's horizontal parallax. In general terms, it is to 'observe the moon's altitude, and thence, with other requisites, to compute the time of her ascension from the sensible kshitijă, or horizon, and her distance from the sun when upon the rational horizon, by which to find the time of her passage from the one point to the other; or, in other words, to find the difference ' in time between the meridian to which the eye referred her at rising, and the meridi'an she was actually upon;' in which difference of time she will have passed through a space equal to the earth's semidiameter or SOO yojănă: and by proportion, as that time is to her periodical month, so is 800 yojănŭ to the circumference of her kŭksha, 324,000 yojǎnǎ. The errors arising from refraction, and their taking the moon's motion as along the sine instead of its arc may here be remarked; but it does not seem

"This day (astronomical day) is accounted to begin at midnight under the rékha (meridian) of Lůnka; "and at all places east or west of that meridian, as much sooner or later as is their dė hantŭrŭ (longitude) re“duced to time, according to the Sōōryŭ-Siddhantů, Brümkŭ-Sıddkantŭ, Vŭshishthŭ-Siddhantŭ, Somŭ-Sid“ dhantă, Půrashŭrů-Siddhantů, and Uryŭbhuttů. According to Brŭmhu-gooptŭ and others, it begins at sun❝rise; according to the Romŭkŭ and others, it begins at noon; and according to the Arsbů-Siddhantŭ, at sun"set." (Comment on the Sōōryu-Siddhantă).

that they had any idea of the first, and the latter they perhaps thought too inconsiderable to be noticed. European astronomers compute the mean distance of the moon about 240,000, which is something above a fifteenth part more than the Hindoos found it so long ago as the time of Muyu, who acquired his knowledge from the author of the Sooryu-Siddhantă.

"By the Hindoo system, the planets are supposed to move in their respective orbits at the same rate; the dimensions therefore of the moon's orbit being known, those of the other planets are determined, according to their periodical revolutions, by proportion. As the sun's revolutions in a mŭha yoogů 4,320,000 are to the moon's revolutions in the same cycle 5,753,336, so is her orbit 324,000 yojunů to the sun's orbit 4,331,500 yojănă; and in the same manner for the kakshus, or orbits of the other planets. All true distance and magnitude derivable from parallax, is here out of the question; but the Hindoo hypothesis will be found to answer their purpose in determining the duration of eclipses, &c.

"For the diameters of the sun and moon, it is directed to observe the time between the appearance of the limb upon the horizon, and the instant of the whole disk being risen, when their apparent motion is at a mean rate, or when in three signs of anomaly; then, by proportion, as that time is to a natural day, so are their orbits to their diameters respectively; which, of the sun is 6,500 yojună; of the moon, 480 yojŭnă.”

"The diameter of the moon's disk, of the earth's shadow, and the place of the node being found, for the instant of opposition or full moon, the remaining part of the operation differs in no respect that I know of from the method of European astronomers to compute a lunar eclipse."

"The beginning, middle, and end of the eclipse, may now be supposed found for the time in Hindoo hours, when it will happen after midnight; but, for the corresponding hour of the civil day, which begins at sunrise, it is further necessary to compute the length of the artificial day and night; and, for this purpose, must be known the ŏyŭnangshŭ or

"But they are not wholly ignorant of optics: they know the angles of incidence and reflection to be equal, and compute the place of a star or planet, as it would be seen reflected from water or a mirror.”

distance of the vernal equinox from the first of Méshů, the sun's right ascension and declination; which several requisites shall be mentioned in their order."-See the second volume of the Asiatic Researches.

The Hindoo astronomical works, not improperly embrace their system of the Mathematics, in which branch of science they were eminently conspicuous. Indeed, in those departments of learning which require the deepest reflection and the closest applicatiThere can hard.on, the Hindoo literati have been exceeded by none of the ancients.

ly be a doubt, that their mathematical writings originated amongst themselves, and were not borrowed either from Greece or Arabia.* The Veeju-Gănită, a Săngskritŭ treatise on Algebra, by Bhaskŭracharyŭ, and other similar works, sufficiently establish these facts. Mr. Davis says, "Almost any trouble and expence would be compensated by the possession of the three copious treatises on algebra from which Bhaskără declares he extracted his Vēēju-Gănită, and which in this part of India are supposed to be entirely lost." A Persian translation of the Veeju-Gunitů was made in India, says Mr. Strachey, in the year 1634, by Ata Oollah Rusidee. The same gentleman says, "Foizee, in 1587, translated the Leelavŭtee, a work on arithmetic, mensuration," &c. from which work it appears that "Bhaskŭra must have written about the end of the 12th century or beginning of the 13th.” Foizee in his preface to this work says, “By order of king Ŭkbŭr, Foizee translates into Persian, from the Indian language, the book Leelavõtee, so famous for the rare and wonderful arts of calculation and mensuration." "We must not,” adds Mr. Strachey, "be too fastidious in our belief, because we have not found the works of the teachers of Pythagoras; we have access to the wreck only of their ancient learning, but when we see such traces of a more perfect state of knowledge; when we see that the Hindoo algebra 600 years ago, had, in the most interesting parts, some of the most curious modern European discoveries, and when we see, that it was at that time ap

* See Mr. Strachey's preface to the Vēējŭ-Gŭnită. In this preface Mr. Strachey observes, “It appears They knew from Mr. Davis's paper that the Hindoos knew the distinctions of sines, cosines, and versed sines. that the difference of the radius and the cosine is equal to the versed sine ; that in a right-angled triangle if the hypothenuse be radius the sides are sines and cosines. They assumed a small arc of a circle as equal to its sine. They constructed on true principles a table of sines, by adding the first and second differences. From the Vēĕjŭ-Gŭnitŭ it will appear that they knew the chief properties of right-angled and similar triangles. They have also rules for finding the areas of triangles, and four-sided figures; among others the rules for the area of a triangle, without finding the perpendicular. For the circle there are these rules (given by Mr. Strachey]. Also formulæ for the sides of the regular polygons of 3, 4, 5, 6, 7, 8, 9 sides inscribed in a circle. There are also rules for finding the area of a circle, and the surface and solidity of a sphere."

plied to astronomy, we cannot reasonably doubt the originality and the antiquity of mathematical learning among the Hindoos."

The author begs leave to conclude this article, by subjoining a few paragraphs of what he translated, and inserted in the first edition, from the Jyotishă-Tutwă:—

The twelve signs of the zodiac, considered as rising above the horizon in the course of the day, are called lugnus. The duration of a lugnŭ is from the first appearance of By the fortunate and unfortunate signs,

any sign till the whole be above the horizon.

the time of celebrating marriages and religious ceremonies is regulated.

There are twenty-seven nõkshůtrus, viz. stellar mansions, two and a quarter of which make up each sign of the zodiac, viz. Ŭshwinee, Bhărănee, and a quarter of Krittika, form Méshū, or Aries; three parts of Krittika, the whole of Rohinee, and half of Mrigŭshira, make Vrishŏbhu, or Taurus; half of Mrigushira, the whole of Ardra, and three quarters of Poonŭrvůsoo, make Mithoonă, the Twins; a quarter of Poonŭrvůsoo, the whole of Pooshya, and Ŭshlésha, make Körkŭtů, the Crab; Mŭgha, Poorvüphŭlgoonēe, and a quarter of Ooturphülgoonee, make Singhŏ, or Leo; three parts of Ootărphulgoonee, the whole of Hăsta, and the half of Chitra, are included in Kănya, or Virgo; half of Chitra, the whole of Swatee, and three quarters of Vishakha, form Toola, or Libra; a quarter of Vishakha, the whole of Ŭnooradha and Jyéshť'ha, are included in Vrishchikŭ, or Scorpio; Moola, Pōōrvasharha, and a quarter of Ootărasharha, form Dhunoo, or Sagittarius; three quarters of Ootărasharha, the whole of Shruvuna, and half of Dhunisht'ha, form Mňkură; half of Dhunisht'ha, the whole of Shutŭbhisha, and three parts of Poor vŭbhadrŭpůda, make up Koombhů, or Koombhu, or Aquarius; one part of Pōōrvůbhadrõpůda, the This work dewhole of Ootǎrbhadrăpăda, and Révétee, form Meeno, or Pisces.

scribes the ceremonies to be performed, and the things to be avoided, at the time of each někshůtrů.

The moment when the sun passes into a new sign is called sunkrantee: the names of the sunkrantees are, Mühavishoovă, Vishnoo-pădee, Shurasheetee, Dakshinayǎnů, Jülůvishoovů, and Ootürayănů. The sunkrantee Mühavishoovů occurs in Voishakhă; Vishnoo

Hhh

pudēvē occurs in Joishthň, Bhadrỡ, Ügrůhayŭnů, and Phalgoonŭ; Shŭrŭshēctee occurs in Asharhu, Ashwinŭ, Poushŭ and Choitrů; Dükshinayănă in Shravănŭ; Jůlăvishoovă in Kartikŭ; and Ootŭrayŭnů in Maghů. By performing certain religious ceremonies at the moment of a sunkrantee, the shastră promises very great benefits to the worshipper; but this period is so small,* that no ceremony can be accomplished during its continuance; the sages have in consequence decreed, that sometimes a portion of time preceding the sunkrantee, and at other times a portion after it, is sacred.

The Hindoos divide the phases of the moon into sixteen parts, called külas. The light parts they fancifully describe as containing the water of life, or the nectar drank by the gods, who begin to drink at the full of the moon, and continue each day till, at the total wane of this orb, the divine beverage is exhausted. Others maintain, that the moon is divided into fifteen parts, which appear and recede, and thus make the difference in the phases of the moon. The first kŭla, is called protipůdă; the next dwiteeya, or

the second, and so to the end. Each day's increase and decrease is called a tit❜hee, that is, sixty dundus,† or as others say, fifty-four. The latter thus reason: sixty dundăs make one nŭkshŭtră; two nŭkshŭtrŭs and a quarter make one rashŭ, containing one hundred and thirty-five dŭndos; by dividing the rashŭ into thirty parts, each part will be four dŭndŭs and a half; twelve of these parts make one tit'thee, or fifty-four dŭndus.‡ Other pundits declare, that there are 1,800 dŭndis in the zodiac, which, subdivided into twelve parts, each portion forms a rashŭ of one hundred and fifty dundus; this rashă they divide into thirty parts, of five dĭndŭs, and twelve of these parts make a tiť’hee of sixty dundus.

The sun is in Méshǎ in the month Voishakhŭ; in Vrishubhŭ, in Joisht'hu; in Mithoonŭ, in Asharhă; in Kărkătă, in Shravŭnă; in Singhŭ, in Bhadrŭ; in Kũnya, in Ashwină; in Toola, in Kartikŭ; in Vrishchikŭ, in Ŭgruhayănă; in Dhŭnoo, in Poushă; in Măkără, in Maghu; in Koombhu, in Phalgoonă; and in Meenă, in Choitră. The sun passes through the signs in twelve months, and the moon through each sign in two days and a quarter.

* As long as a grain of mustard, in its fall, stays on a cow's horn, say the pundits.

+ Two půlus and a half make one English minute, and sixty of these půlŭs makes one dündă, or Hindoo hour, so that two and a half Hindoo hours make one English hour. The Hindoos have no clocks; but they have a clepsydra, or water clock, made of a vessel which fills and sinks in the course of an hour. The sand hour-glass has been lately introduced. The Tit❜hee-Tutwe maintains this position.