a II. Geognosy. I ought now to give a sketch of the recent improvements in this branch of science, which of late years has become a fashionable object of study in Great Britain ; but I have already extended this article to such an enormous, and I fear improper length, that I must, however reluctantly, stop short here. The great object of geognosts on the Continent at present seems to be to trace to their utmost extent the formations discovered in the neighbourhood of Paris; and to extend as much as possible the transition formations in those countries hitherto considered as primitive. In this country we have no fewer than three geological societies, the Wernerian, the London, and the Cornwall. The first two have recently published each' a volume of Transactions, I shall give an analysis of each of these books as speedily as possible. They contain almost all the important geological facts that have been lately ascertained in Great Britain. ARTICLE II. Solution of a Problem of Col. Silas Titus. (See Wallis's Algebra, Chap. 60.) By the Abbé Buée.* SIR, HAVING for many years considered the different algebraical methods for the solution of arithmetical problems by approximation to be deficient in their fundamental principles, I have been led to mistrust the whole science of algebra as generally taught, and am convinced that if we place an implicit faith in it we shall be involved in the most "revolting absurdities. Pell's problem (see Wallis's Algebra, chap. 60, 62, &c.), and all those which can only be resolved by approximation, are examples of this kind. The absurdity belonging to the solution of these sort of problems is to represent numbers of which we know not the fundamental unity. In speculation this absurdity is not felt; but we easily perceive it when we quit speculation, and are engaged in questions respecting real beings, such as men. In this case the solution gives for units fractions as much smaller as the approximation is farther extended. If, then, the real unit be a man, the solution gives for unit a fraction of a man, which goes on always diminishing, and by that means becomes more and more absurd. In general the speculative a * The following curious solution of a well-known problem was sent by the Abbé Buée to a mathematical gentleman in London, who declines communicating his name to the public. Though I do not participate in the Abhe's objections to algebraic approximation, yet i conceive the solution of the problem itself to be so curious as to be well entitled to the attention of mathematicians ; and on that account I agreed without hesitation to insert it in the Annuls of Philosophy. The letter which serves as an introduction to the problein is written by the Abbé Bube. --T. 1 n 12 a unit is the QUOTIENT of a number divided by itself, while the real unit is the product of a number multiplied by the inverse of that number. Let n be any number, n divided by n is a speculative unit, and n x is a real unit, as a rectangle would be. If, then, I be the supposed unit, it is necessary, in order to obtain a real unit, to multiply by n; consequently, the nearer we approach one side, the farther we recede on the other. The following solution, which for the first time is given of Pell's problem, is the only one exempt from this absurdity : Problem. The following 'equations being proposed, viz. a + b c = 16 .. (1) 12 + ac = 17 (2) C2 + ab = 18 (3) To find a, b, c, (See Wallis's Algebra, chap. 60, 62, &c.) let there be a series of concentric circles, (Plate. XXVII. fig. 1) 1l 1, 11 l, 1 1 1 11, .11 1.. . 1 l, and let C 1 1 1 2 2 2 13 13 13 17 17 17 21 21 21 them be so described that we have 0 1 = 11 = 11 = 11 = 21 21 21 C = the sector 0 1 1 1 (6) Then if the areas of these sectors be substituted for a, b, c, in the equation (1), (2), (3), according to the following method, their differences will reduce these equations to identical ones. Demonstration. This demonstration is founded upon a remarkable property of the concentric circles of this figure. This property is, that the areas of each of the rings intercepted between two consecutive circumferences are equal to the area of the central circle. If we take the area of the central circle for an unit, the areas of each of the rings will be = 1. To prove this, let O 1 be the radius of the central circle, we may easily perceive that the radii of the successive circles are the hypotheneuses of right angled triangles, whose sides are, 1. The radius of the preceding circle. 2. A constant tangent equal to the radius of the central circle. The series of radii will be then expressed by 01(Vī, 2, V3, V4, &c.)....(7) Now the circumferences of circles are constantly proportional to their radii : if, then, we designate by 2 the ratio of the circumference to its radius, the series of circumferences will be expressed f that e unit, by the progression ( 1 )* * 2 [Vī, v2, 13, vã, &c.].. (8) 2 hit , to e side , But the areas of circles are as the squares of their radii ; so that the series of circles will be expressed by the progression (O 1). 2 Pell's d let 1 = 3 [1, 2, 3, 4, &c.] .(9) In this series (by taking away the common factor (O 1). 2 ) 1 expresses the area of the central circle; 2, 3, 4, &c. express those of the successive circles. If we take the difference of each of the areas of these consecutive circles, we shall have the areas of the rings. Now these differences are constantly equal to l; consequently the areas of the rings are = 1. It follows from the above conclusions that if we reckon the central circle for the first ring, the series of rings will be expressed by the common urdinals, 1st, 2d, 3d, 4th, &c. (10) while the series of circles are expressed by the absolute numbers ), 2, 3, 4, &c. ... .(11) These ordinal numbers follow the direct or inverse order: when they follow the direct order, the areas of the rings are positive ; when they follow the inverse order, they are negative. These areas are constantly = 1, and are represented by the equation l (12) ·(15) These equations, which are fundamental, I thus demonstrate : in every system of logarithms the logs. are exponents, and these exponents are ordinal numbers, because they are the indexes of the terms of a geometrical progression whose first term is 1. Now from the principles demonstrated by Euler (Introductio in Analysin Infinitorum, Cap. VIII. No. 39,) we may prove the truth of the two following equations : Log. (+ 1) = = 2 kon-T = (14) .(16) .(18) From the above we easily derive the equations (12), (13). The following is an explanation of these equations : k can only be an cs , in their the as of um the Engs may log. e... : |