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blems of importance; and to shew that, as it furnishes a greater variety of ways for expressing antecedentals than the fluxionary caculus does for fluxions, so it will open new and extensive rules for finding antecedents, as yet altogether unknown in the inverse method of fluxions.

Observations on the Trigonometrical Tables of the Brahmins. By John Playfair, F. R.S. Ed. Professor of Mathematics, Edinb. The principles and rules of the trigonometry of the Brahmins are contained in the Surya Siddhanta, one of the inspired writings of the Hindoos: a book which, with sedate and strict science, interweaves many wild and extravagant fictions. The subject of the present memoir is the construction of the trigonometrical tables in that work.

In the beginning of his paper, the Professor offers a reason why the division of the circumference was made into 360 parts, both in Greece and Hindostan: he supposes that the inventors of the circular instruments for measuring the heavens were naturally led to a division of them, which should correspond to the space daily described by the sun in the ecliptic; and this was nearly to be effected by making each division one 360th part of the whole circumference. This reason appears more probable from the Chinese division of the circle into 365 parts and.

Ptolemy and the Greek mathematicians had two measures for arches and chords: the former were parts of the circumference, the latter of the radius: but the Brahmins, adopting, it would seem, the notions of an advanced science, used only one measure for arches, chords, and sines, &c. The circumference was divided into 21600 parts, of which the radius contained 3438.

The trigonometrical tables contained in the treatise previously mentioned are two, the one consisting of sines, the other of versed sines: the use of sines was unknown to the Greeks, in the 2d who calculated by chords. In the first table, are given the sines to every twenty-fourth part of the quadrant; table, the versed sines: the sines are expressed in minutes of the circumference, without any fraction of a minute: but there is a reason for thinking, as Mr. Playfair observes, that this want of exactness did not originate from any inability in the Hindoo mathematicians to advance their calculations nearer to the truth.

The following extract contains the rules for constructing their tables:

• Their rules for constructing their tables of sines, may be reduced to two, viz. the one for finding the sine of the least arch in the table,

that that of 3°. 45', and the other for finding the sines of the multiples of that arch, its triple, quadruple, &c. Both of these Mr. Davis has translated, judging very rightly, that it was impossible to give two more curious specimens of the geometrical knowledge of the Hindoo philosophers: the first is extracted from a commentary on the Surya Siddhanta; the other from the Surya Siddhanta itself.

• With respect to the first, the method proceeds by the continual bisection of the arch of 30°, and correspondent extractions of the square root, to find the sine and co-sine of the half, the fourth part, the eighth part, and so on, of that arch. The rule, when the sine of an arch is given, to find that of half the arch, is precisely the same with our own: "The sine of an arch being given, find the co-sine, and thence the versed sine, of the same arch: then multiply half the radius into the versed sine, and the square root of the product is the sine of half the given arch." Now, as the sine of 30o, was well known to those mathematicians to be half the radius, it was of consequence given: thence, by the rule just laid down, was found the sine of 15°, then of 7°. 30', and lastly of 3°. 45', which is the sine required. Thus the sine of 3°. 45', would be found equal to 224, 44", as above observed, and, the sine of 7° 30', equal to 448. 39", and, taking the nearest integers, the first was made equal to 225, and the second to 449 *.

When, by the bisections that have just been described, the sine of 30. 45', or of 225', was found equal to 225", the rest of the table was constructed by a rule, that, for its simplicity and elegance, as well as for some other reasons, is entitled to particular attention. It is as follows: "Divide the first jyapinda, 225 by 225; the quotient 1, deducted from the dividend, leaves 224', which added to the first jyapinda, or sine, gives the second, or the sine of 7°. 30, equal to 449. Divide the second jyapinda, which is thus found, by 225, and deduct 2, the nearest integer to the quotient, from the former remain. der 224, and this new remainder 222', added to the second jyapinda,

• By such continual bisections, the Hindoo mathematicians, like those of Europe before the invention of infinite series, may have approximated to the ratio of the diameter to the circumference, and found it to be nearly that of 1 to 3.1416 as above observed. A much less degree of geometrical knowledge than they possessed, would inform them, that small arches are nearly equal to their sines, and that the smaller they are, the nearer is this equality to the truth. If, therefore, they assumed the radius, equal to 1, or any number at pleasure, after carrying the bisection of the arch of 30, two steps farther than in the above construction, they would find the sine of the 384th part of the circle, which, therefore, multiplied by 334, would nearly be equal to the circumferenee itself, and would actually give the proportion of 1 to 3.14159, as somewhat greater than that of the diameter to the circumference. By carrying the bisections farther, they might verify this calculation, or estimate the degree of its exactness, and might assume the ratio of 1 to 3.1416 as more simple than that just mentioned, and sufficiently near to the truth,'

will give the third jyapinda equal to 671. Divide this last by 225, and subtract 3, the nearest integer to the quotient, from the former remainder 222′, and there will be left 219', which, added to the third jyapinda, gives the fourth; and so on unto the twenty-fourth

or last."

Making a slight change in the enunciation of the rule, the Professor demonstrates the principle on which it is founded; which may be thus generally stated:- If there be three arches in arithmetical progression, the sine of the middle arch is to the sum of the sines of the two extreme arches, as the sine of the difference of the arches to the sine of twice that difference. Mr. P. remarks that

This theorem is well known in Europe; it is justly reckoned a very remarkable property of the circle; and it serves to shew, that the numbers in a table of sines constitute a series, in which every term is formed exactly in the same way, from the two preceding terms, viz. by multiplying the last by a certain, constant number, and subtracting the last but one from the product.

Now, it is worth remarking, that this property of the table of sines, which has been so long known in the East, was not observed by the mathematicians of Europe till about two hundred years ago. The theorem, indeed, concerning the circle, from which it is deduced, under one shape or another, has been known to them from an early period, and may be traced up to the writings of Euclid, where a proposition nearly related to it forms the 97th of the Data: "If a straight line be drawn within a circle given in magnitude, cutting off a segment containing a given angle, and if the angle in the segment be bisected by a straight line produced till it meet the circumference; the straight lines, which contain the given angle, shall both of them together have a given ratio to the straight line which bisects the angle." This is not precisely the same with the theorem which has been shewn to be the foundation of the Hindoo rule, but differs from it only by affirming a certain relation to hold among the chords of arches, which the other affirms to hold of their sines. It is given by Euclid as useful for the construction of geometrical problems; and trigonometry being then unknown, he probably did not think of any other application of it. But what may seem extraordinary is, that when, about 400 years afterwards, Ptolemy, the astronomer, constructed a set of trigonometrical tables, he never considered Euclid's theorem, though he was probably not ignorant of it, as having any connection with the matter he had in hand. He, therefore, founded his calculations on another proposition, containing a property of quadrilateral figures inscribed in a circle, which he seems to have investigated on purpose, and which is still distinguished by his name. This proposition comprehends in fact Euclid's, and of course the Hindoo theorem, as a particular case; and though this case would have been the most useful to Ptolemy, of all others, it appears to have escaped his observation, on which account he did not perceive that every number in his tables might be calculated from the two preceding numbers, numbers, by an operation extremely simple, and every where the same; and therefore his method of constructing them is infinitely more operose and complicated than it needed to have been.

• Not only did this escape Ptolemy, but it remained unnoticed by the mathematicians, both Europeans and Arabians, who came after him, though they applied the force of their minds to nothing more than to trigonometry, and actually enriched that science by a great number of valuable discoveries. They continued to construct their tables by the same methods which Ptolemy had employed, till about the end of the sixteenth century, when the theorem in question, or that on which the Hindoo rule is founded, was discovered by Vieta. We are however ignorant by what train of reasoning that excellent geometer discovered it; for though it is published in his Treatise on Angular Sections, it appears there not with his own demonstration, but with one given by an ingenious mathematician of our own country, Alexander Anderson of Aberdeen. It was then regarded as a theorem entirely new, and I know not that any of the geometers of that age remarked its affinity to the propositions of Euclid and Ptolemy. It was soon after applied in Europe, as it had been so many ages before in Hindostan, and quickly gave to the construction of the trigonometrical canon all the simplicity which it seems capable of attaining. From all this, I think it might fairly be concluded, even if we had no knowledge of the antiquity of the Surya Siddhanta, that the trigonometry contained in it is not borrowed from Greece or Arabia, as its fundamental rule was unknown to the geometers of both those countries, and is greatly preferable to that which they employed.'

The latter part of this excellent memoir is employed in ascertaining the date of the Surya Siddhanta, and the origin of the mathematical sciences in India. Trigonometry cannot be supposed to have been introduced till considerable advancements had been made in geometry and astronomy. The first step in astronomy was made in Greece, 1140 years before the Christian æra; and Hipparchus invented trigometry 130 years before the same æra. If, therefore, the age of the Surya Siddhanta be taken at 2000 years before Christ, we must add 1000 more to arrive at the origin of the sciences in India : which will thus appear to be placed near the celebrated period of the Caly Yug, the year 3:02 before our æra. It is to this æra that M. Bailly, in his antient Astronomy, refers the construction of the tables in Hindostan.

Some Geometrical Porisms, with Examples of their Application to the Solution of Problems. By Mr. William Wallace, Assistant Teacher of the Mathematies in the Academy of Perth. Communicated by Mr. Playfair. -The subject of porisms has been amply, ingeniously, and learnedly discussed in a former volume by the communicator of this paper*. The present memoir

* Sce Rev. vol. xix. N. S. p. 243.

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will not be unacceptable to these mathematicians who have perused that of Mr. Playfair.

Determination of the Latitude and Longitude of the Observatory at Aberdeen. By John Andrew Mackay, LL. D. F. R.S. Ed. These papers display considerable labour, but do not admit of our making any abstract from them.

An Account of certain Motions qubich small lighted Wicks acquire, when swimming in a Bason of Oil; together with Observations upon the Phenomena tending to explain the Principles upon which such Motions depend : communicated in a Letter from Patrick Wilson, F. R. S. Edin. and Professor of Practical Astronomy in the University of Glasgow, to John Playfair, F. R. S. Edin. and Professor of Mathematics in the University of Edinburgh.

It would be a just reproach of philosophy, if it should confine itself to the contemplation of the grand operations of nature, and disdain to attend to the causes of common appearances; nevertheless, there is still such a thing as trifling in philosophy. We heartily wish Professor Wilson much amusement with his floating lights;-his minikin lamps, we observe, are treated in rather lofty language.

Account of a singular Halo of the Moon. By Wm. Hall, Esq. of White Hall near Berwick, F.R. S. Edinburgh.

After a mild day, and in a pretty clear evening, the moon shining bright, on the 18th February 1796, a large and a small halo were observed from the author's residence, about the moon; the diameter of the larger one subtending an angle of a hun dred and twelve degrees, that of the small one being under 129 and more than 8°. - Mr. Hall concludes by remarking:

This halo appears to be of the kind called by the learned a Corona; and as it somewhat resembles the famous one of the sun, observed at Rome in the year 1629, and described by Scheiner *, it deserves the more attention, especially as the great halo, on the pre. sent occasion, having its south-western limb elevated to the height of 54°, and its north-eastern depressed to within 14° of the hori. zon, was in an oblique position, not easily reconciled with the theory of Huygens, which seems to require that such circles should be equally elevated above the horizon all round. It also shews, that Scheiner's original plan of the halo at Rome, which represented it as oblique, may have been right, and that Huygens's correction, which makes it parallel to the horizon, was probably an erroneous conjecture.'

A new Series of the Rectification of the Ellipsis; with some Observations on the Evolution of the Formula (a2+b2-2ab cos. 4)". By James Ivory, A. Μ.

* Smith's Optics, vol. i. § 534.'

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