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Editorial Items.

We have received the following solutions of the Prize Problems in the May number of the Monthly :

HARRIET S. HAZELTINE, Worcester, Mass., Probs. I., II.

J. G. WEINBERGER, Pennsylvania State Normal School, Millersville, Prob. I.

GEORGE B. HICKS, Cleveland, Ohio, Probs. III., IV., V.

ASHER B. EVANS, Madison University, N. Y., Probs. I., II., III., IV., V.

DAVID TROWBRIDGE, Perry City, N. Y., Probs. I., II., IV., V.

D. G. BINGHAM, Ellicotteville, N. Y., Prob. II.

GUSTAVUS FRANKENSTEIN, Springfield, Ohio, Prob. IV.

Mathematical Monthly Notices.

Translation of the Sûrya-Siddhânta, A Text-Book of Hindu Astronomy, with Notes and Tables, and an Appendix containing additional Notes and Tables, Calculations of Eclipses, a Stellar Map, and Indexes. From the Journal of the American Oriental Society, Vol. VI. New Haven, 1860. 8vo. pp. 364. Sold by the Society's Agent, John Wiley, 56 Walker Street, New York.

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The astronomy of the Hindus, though discussed at length by BENTLEY, Delambre, BIOT, and various others, has never before been presented to us in the only satisfactory form, by a translation of one of their own treatises. We have had no means of knowing the real extent and character of Hindu science, or of comparing it with that of other nations to learn its origin. The present work furnishes that information, and is a very important contribution to the history of the exact sciences, by far the most important ever published in this country. The first rough draft of the translation, as we learn from the preface, was made in India by Rev. E. BURGESS, formerly Missionary of the A. B. C. F. M., aided by a native astronomer. The whole collected material was however placed in the hands of the committee of publication of the Oriental Society for revision, expansion, and reduction to a suitable form. It is not difficult to see in every line the careful, conscientious scholarship of Professor WHITNEY, and the lovers of the exact sciences owe to him the greater debt, that he, a philologist, has made so rich a contribution to their favorite department of knowledge.

The Sûrya-Siddhânta is one of the earliest and most esteemed of the many text-books of Hindu Astronomy. We are by no means, however, compelled to believe in the Hindu date, which makes it a revelation from the Sun given about 2,163,101 B. C. There are, however, indications of its existence as early as the sixth century of our era, and the astronomical data correspond best with even an earlier epoch.

The treatise is poetical, and consists of very brief elliptical rules for calculating the places of the heavenly bodies, solar and lunar eclipses, and other celestial phenomena. These rules would be wholly unintelligible without a teacher or commentary, or some knowledge beforehand of what should be said. The whole system presents a strange mixture of very exact knowledge and the most arbitrary and fanciful assumptions. The planetary theory is essentially that of the Greeks, given by PTOLEMY in the Syntaxis, differing from it only in some minor particulars. In each the system of epicycles is fundamental. A full and careful comparison with the Syntaxis

is presented in the notes. It is only by such means that we can decide how far the Hindus were indebted to others for their astronomy. To show the accuracy of some of their determinations we quote (from p. 24) the following table of the times of sidereal revolutions, as given by dif ferent authorities.

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As a contrast to this exact knowledge, it may be mentioned that the positions of twenty-eight fixed stars are given, but eight of them have errors of over three degrees either in latitude or in longitude. We mention again, as characteristic of the system, that the nodes and apsides of all the planets are made to revolve, but with so slow a motion withal that even with the best modern instruments, it could not in some cases be detected in hundreds of years. We commend the note on pp. 26-28 to the attention of the curious.

This mixture of exact knowledge, gross error, and arbitrary assumption suggests the conclusion, which the arguments of Prof. WHITNEY prove, that the science of the Hindus was derived from the Greeks, being worked over by the borrowers to suit their fanciful mythology and chronology, and to serve as a book of rules for the astrologers and for the calculation of eclipses.

The origin of the Hindu division of the zodiac into twenty-seven or twenty-eight parts, and its connection with the Chinese zodiac and the Arabic lunar mansions is very fully discussed, and illustrated by a stellar map, and much light is thrown on this obscure subject.

The amount of the knowledge of the Hindus in arithmetic, geometry, and trigonometry are indicated by this treatise. The decimal notation and the sine of an arc were known. Angles are not mentioned in the treatise. A love of calculation is quite prominent. Multiplications and divisions, where the numbers have ten or twelve figures each, are quite frequent. Almost the only geometrical knowledge involved in the rules is that of the forty-seventh problem of Euclid and that of the proportionality of the sides of similar right-angled triangles. Of these, however, they make constant and dexterous use, solving, in fact, all ordinary cases of plane and spherical trigonometry.

The notes are quite full, being designed to satisfy the wants of two classes, the philologists who are not astronomers, and astronomers who are not philologists. We heartily commend this work to the attention of those who are interested in the history of astronomy or mathematics.

There are a great many other treatises of Hindu astronomy, and some of them are of considerable age. A comparison of these treatises would doubtless serve to show what points are peculiar to the Hindu astronomy, and what were borrowed from other systems. We sincerely hope that Prof. WHITNEY, who has shown himself so admirably qualified for the task, will place science under renewed obligations by undertaking it.

THE

MATHEMATICAL MONTHLY.

Vol. II.... SEPTEMBER, 1860.... No. XII.

PRIZE PROBLEMS FOR STUDENTS.

I. IF AB, CD be chords of a circle at right angles to each other, prove that the sum of the arcs A C, B D is equal to half the circumference.

II. Prove that the cube of any number and the number itself, being divided by 6, leave the same remainder.

III. It is required to divide a given right cone into two parts, having the ratio of m to n, by a plane making an angle of p degrees with the axis.- Communicated by Prof. WRAY BEATTIE, Iowa Wesleyan University.

IV. In any spherical triangle,

sin s sin (sa) sin (s—b) sin (s — c) cos S cos (S—A) cos (S—B) cos (S—C)

=

cos a cos b cos c -1
cos a cos b cos c+1'

in which s= {(a+b+c), S= } (A+B+C).- Communicated by Prof. H. H. WHITE, Harrodsburg, Ky.

V. Describe an ellipse having its foci at any points on one of the asymptotes of an equilateral hyperbola. The ellipse will cut each branch of the hyperbola and its conjugate in two real or imaginary points. Prove that a chord joining one pair of these points will be perpendicular to one of the other similar chords. Prove it first for a circle and then for an ellipse.

Solutions of these problems must be received by November 1, 1860.

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NOTES AND QUERIES.

1. Commensurable Sides of Right-angled Triangles nearly Isosceles. — If

(2 an bn)2 + (a‚2 — b„2)2 = (a„2 + b„2)2,

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(4 a 2-1 + 2 an-1 bn−1)2 + (3 a2-1 + 4 ɑn-1 bn-1 + b3⁄4-1)2 = (5 a-1 + 4 a„-1 bn-1 + b2-1)2.

Also,

-

(2 a, b1) — (až —— b;) = (a;-1 — b2-1) — (2 an-1 bn-1) ·
= (2 a„-2 bn-2) — (aå—2 — bå—2) =

Additional values of a, b, give

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=±1.

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(a2 + b2)2 = (a2 — b2)2 + (2 a b)2.

The general problem is to find two square numbers whose sum or difference shall also be a square number; and it may be solved by the Diophantine Analysis.

(1)

Let x be the greater, and y the lesser number; then,

x2 — y2 = □ = (x — ny)2 = x2 — 2 n xy + y2,

in which n is any arbitrary entire or fractional number.

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We may now put y equal n, 2 n, or 3 n, &c.; but to make the value of x as simple as possible, let y 2n; then x = √(x2 — y2) = n2 — 1.

n2

n2 + 1 and

Now, by substituting any numbers for n in these equations, we shall obtain three numbers, satisfying the conditions of the problem. Suppose n = 4; then 17282+152; and if in the equation

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Hence, by assuming any two numbers for a and b, we shall find that a2+b2 is the hypothenuse, and a2 — b2, 2 ab, the sides of a rightangled triangle. - Prof. D. WOOD, University of Michigan, Ann Arbor.

3. On the Logarithmic Solution of Cubic Equations. In the December number of the Monthly, as in various other publications, attention has been called to the trigonometric or logarithmic solution of equations. The subject appears of sufficient interest to merit further attention; and it is here proposed to combine the methods in a more systematic form, with such additional precepts as will adapt the operation even to persons who have attended only to the first principles of the logarithmic Tables. It is plain that all equations of the third degree, or cubics, may be distributed into three general classes, represented as follows:

First class,
Second class,

Third class,

x3 + ax = b.

23 + d x2 = b'.

x2 + α x2 + α1 x = b.

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