Traité du Calcul différentiel et Intégral, 2nd edit. tome 1 et 11, in 4to. par S. F. La Croix.

Traité élémentaire du calcul des probabilités; par S. F. La Croix, 8vo. 300 pages.

Traitè de Physique expérimentale et mathématique; par J. B. Biot, 4 vol. 8vo. 2400 pages.

Développemens de Géometrie, avec des applications à la stabilité des vaisseaux, aux déblais et remblais, aux défilemens, &c.; par Ch. Dupin, 4to. 400 pages.

Physique Mécanique; par E. G. Fischer,traduite de l'allemand, avec des notes de M. Biot, 8vo. 500 pages.

Principes de Mathématiques de feu Joseph Anastase du Cunha, professeur à l'université de Coimbre, traduits littéralement du portugais; par J. M. d'Abreu ; nouvelle édition, 8vo. 300 pages. Elémens de Mécanique, par J. L. Boucharlat, 8vo. 350 pages. Elémens de calcul différentiel et de calcul intégral; par J. L. Boucharlet, 8vo. 250 pages.

Réflexions sur la métaphysique du calcul infinitésimal, 2nd edit. in 8vo. par Carnot.

Nouvelles Tables d'aberration et de nutation, pour quatorze cent quatre étoilles; avec une table générale d'aberration pour les Planètes et les Comètes, &c. par le Baron de Zach.

L'attraction des montagnes et ses effets sur les fils à plomb ou sur les niveaux des instrumens d'astronomie; par le Baron de Zach, 2 vol. 8vo.

Mémoire de M. le Baron de Zach, sur le degré du méridien mesuré en Piémont, par le P. Beccaria, 4to.

Mémoire sur diverses intégrales définies; par M. G. Bidone; 4to. 120 pages.

Mémoire sur les intégrales définies; par M. Plana; 4to. 45 pages. Cet intéressant mémoire forme un utile complément aux travaux de M. M. Lagrange, Legendre, Poisson et Bidone sur le mème sujet.

Mémoire sur divers problèmes de probabilité, lu à l'académie de Turin; par M. Plana.

Mémoire sur le mouvement de rotation d'un corps solide libre, autour son centre de masse, par J. J. Français.

Mémoire sur le cercle qui en touche trois autres sur un plan, et sur la sphere qui en touche quatre autres dans l'espace; par J. D. Gergonne, 410. pp. 20. Turin.

Théorie de la distance d'un point à un autre, sur la surface d'un solide de révolution; par M. B. Goudin, 4to.

Traités élémentaires de calcul différentiel et de calcul intégral, indépendans de toutes notions de quantités infinitésimales et de limites; par M. J. B. E. Du Bourguet, 2 vol. 8vo.

Elémens de Statique, par J. B. Labey, 8vo.

Table des diviseurs pour tous les nombres du troisième million, par J. C. Burckhardt, 4to.

ARTICLE III.

Solutions to Questions proposed in Number XIV.

I. QUESTION 371, by the Rev. Mr. W. WOOD.

Given to find x and y, the two equations

3x2 - y2 = a, and x3 —y3 — 2xy2 = b.

FIRST SOLUTION, by the Rev. Mr. W. WOOD, the Proposer.

Put y 32, xwgz, then our two equations become

=

Make now w + z =m, and wz = n, then our two last equations become

w and z will be the two roots of the quadratic equation w2-mw + n = 0; lastly y = 3z, and x = w — 22.

SECOND SOLUTION, by A.

- b.

Given 3x2-y2 = a, and x3 — y3 — 2xy2 =

Put x + y = 2u, and x - y = 2z; then x = u+z, and y = u-z; these values being substituted for x and y in the given equations, they become

Qu2 + 8uz + 222 = 0,

and - qu3 + 8u3z + 2uz2 = b.

Let the first be multiplied by u, and the second subtracted from the product; the result will be

Find the value of u from this cubic equation, and then the value of z may be found from the quadratic 2u2+8uz + 2z2 = a, Consequently the values of x and y will then become known.

II. QUESTION 372, by B. A.

Let C be the circumference of a circle, D the diameter, c the chord of any arch a; then

4aD (ca)

c2 (Caja

Required the investigation.

= c, nearly.

SOLUTION, by A. B, the Proposer.

This Theorem is given in the Preface to the Bija Ganita, or Hindoo Algebra, translated by E. Strachey, Esq. It seems to be derived from the Hindoo method of computing the sines, which we find in a paper by S. Davis, Esq. in the 2nd vol. of the Asiatic Researches. Mr. Davis has detailed the process which is thus: The quadrant is divided into 24 equal parts of 225' each, and 225 is assumed for the length of its sine (the sine of 3° 45'):-then 225 divided by 225 gives 1 (the 2d difference) which taken from 225 leaves 224 (the 1st difference); and 224 +225 = 449 the sine of twice the arc 3° 45'. Next 449 divided by 225 gives 2 the integral quotient (the 2d diff.) which subtracted from 224 leaves 222 (the 1st diff.); and 222 +449

671 the sine of 3 times 3° 45'. Again, 671 divided by 225 gives the integral quotient 3, which taken from 222 leaves 219; and 219671890 the sine of 4 times 3° 45'. In this manner by adding the 1st and 2d integral differences, the 24 sines of the quadrant are completed.*

Professor Leslie in the notes to his Geometrical Analysis gives the following investigation: "The successive differences of the sines of the arcs AB, A, and A + B, are sin A — sin (A — B), and sin (A + B) — sin A; and consequently the differences between these again, or the second difference of the sines, is sin (ATB) + sin (A — B) — 2 sin ▲ —— ·2 vers в sin A. The second differences of the progressive sines are hence subtractive, and always proportional to the sines themselves. Wherefore the sines may be deduced from their second differences, by reversing the usual process, and recompounding their separate elements. Thus the sines of A- B, and A being already known, their second and descending difference, as it is thus derived from the sine of A, will combine to form the succeeding sine of AB, which is — 2 vers B sin A + (sin a — ·sin (A — B)) + sin A."Now the diameter being 6876, and taking 225 for the chord of 225', 2 × 2252 we have twice the versed sine, this is 6876

= 14.7 nearly

=

=

of the radius, and is the constant multiplier of the sine; but as 225 differs but little from 233, the fraction is assumed for the multiplier, or 225 for the divisor.

Mr. Leslie calls this "an elegant and very ingenious mode of forming the approximate sines:" but his antipathy to the Bramins appears so inveterate, that he will not on any account allow them to be the inventors; yet he is not able to discover any author from whom they could have obtained the information.

Now it appears from the process that each column of differences is a recurring series: and that any sine is equal to the sum of all the preceding differences: thus 224 +222 + 1 + 2 = 449224 + 222 + 219 + 1 + 2 + 3 = 671, &c. And when the table is extended to 24 arcs the sum is 3438 the sine of go° or half the diameter.

If that part of the quadrant whose sine is required be denoted by n (the quadrant being divided into 24 equal parts) then the whole sum of the 24 terms of the two series of differences being = D(D= the diam.) the sum of n terms of the series will be

(48-n) 4n 482 + (24 — n)2

XD nearly, the sine of the arc which is n

parts of the quadrant; and its double or

is the chord of twice that arc.

But if the circumference be denoted by 96 (c) then will represent the arc itself, and the expression for the chord

(c-n)n

4c2 + (4C —n)2

× 4D =

× 4D; and multiplying both numera

is the proposed theorem when a is put for the are an.

It may, however, be remarked that the theorem brings out the chords too great when the arcs are small; and too little when they are about of the circumference.

3

III. QUESTION 373, by Z.

Find what conditions must have place among the coefficients of x3 — px2 + qx — r = o, that the roots may be in harmonical progression, and find those roots.

L &

FIRST SOLUTION, by Mr. JOHN WALLACE, R. M. College.

Let a, b, c be the roots of the proposed equation. Then, by the theory of equations, we have

a+b+c=p, ab +ac + bc = q, abc = r.

But since by the conditions of the question a, b, c are in harmonical progression, so that a:c: a-bb-c; it is evident that ab+bc = 2ac, and therefore we have

(ab + bc + ac) × b = 3abc, that is bq 3r; wherefore

=

+ bc = 2ac, we obtain Зас But the equation ab + bc =

a+b+c= 26c+b=

2ac + b2
b

Substituting therefore for a+b+c, for b and for ac their respective values, we obtain

and this equation expresses the relation which must subsist among the coefficients of the given equation, in order that the roots may be in harmonical progression.

. One of the roots of the equation has already been found,

namely b = ; and from the equations a + c = p − b =

3r
9

?- 3 and ac = q we obtain the other two roots,

a =

= 1 ( p

c = 1

(p

SECOND SOLUTION, by A.

Let u, v and w be the roots required.

By the nature of harmonical progression, u: w :: u — v : v — w,

31