car ied through the press by this author: we avail ourselves of the present opportunity to lay before our readers a few more specimens. In the fourth chapter of the first part, a chapter which relates to the possible and impossible forms of square numbers, there are several very interesting propositions, and some comprehensive and useful tables of formulæ, by which the possible and impossible cases may be easily detected and separated. These however, we cannot transcribe; but we give the following simple proposition, for the sake of its curious corollary, relative to two series, which, though it has been long known, we have never seen publicly demonstrated before. The area of a rational right angled triangle cannot be equal to a square. For if it were possible, and x, y, and s, were made to represent the two sides and the hypothenuse of such a triangle, we should have. [x2+ y2 = x2, 'Or Zxy = 2 x2+2xy + y2=x2+4w2, and x2-2xy+y2=x2—4w2 ̧; that is, Sx2+4w2=(x+y)2, 2 4 w2 = (x—y)2. But these expressions cannot be both squares at the same time (art. 55); and, consequently, the area of a rational right angled triangle, cannot be equal to a square.-Q. E. D. Cor. 1. Since, in order to have a rational right angled triangle, we must have x2+y=x2; it follows (from art. 54), that Sx=r2—s2, and denominator be taken for the sides of a right angled triangle, it will be a rational one; and in these expressions we may give any values at pleasure 2rs each of which expressions, reduced to an improper fraction, gives the sides of a rational right angled triangle. And if in the fraction. 1, and r=2n+2, our expression becomes and here, making n = 1, 4n+3 7 +- - 4n2+8n+3 4n+4 2rs 2, 3, 4, &c. we have this other series., 11 =1-, 2-, 3-, 4-, 5-, &c., 4n+ 4 8 12 16 20 24 which has the same property as the former." Pp. 121, 3. 2 we make The chapter on the different scales of notation, contains much original matter, and some useful observations., We shall quote two or three of the author's remarks on the comparative advantages and disadvantages of different scales. On this head, simplicitly is evidently the first consideration to be attended to, for in that alone consists the superiority of one system over another; but this ought to be estimated on two principles, viz. simplicity in arithmetical operations, and in arithmetical expressions: Leibnitz, by considering only the former, recommended the binary scale, which has certainly the advantage in all arithmetical operations, in point of ease; but this is more than counterbalanced by the intricacy of expression, on account of the multiplicity of figures necessary for representing a number of any considerable extent; thus we have seen (prop. ii. of this chapter,) that 1000 in the binary scale would require ten places of figures, and to express 100000 we must have twenty places, which would necessarily be very embarrasing, at the same time that all calculations would proceed very slow, on account of the number of figures that must be made to enter into them. • The next scale that has been recommended is the senary, which certainly possesses some important advantages: first, the operation with this system would be carried on with facility; the number of places of figures for expressing a number would not be very great; beside, that those quan tities equivalent to our decimals, would be more frequently finite than they are in our system: for example, every fraction whose denominator is not some power of one of the factors of 10 is indefinite, and those only are finite that contain the powers of these factors: and it is exactly the same in every other scale of notation: namely, those fractions only are finite, that have denominators compounded of the powers of the factors of the radix of that system; therefore, in the decimal scale only fractions of the a form are finite, but in the senary scale the finite fractions are of the 2o 5m form a 23m and as there are necessarily more numbers of the form 2" 3m, within any finite limit, than there are of the form 25m, it follows, that in a system of senary arithmetic, we should have more finite expressions for fractions than we have in the denary; and, consequently, on this head the preference must be given to the senary system: And, indeed, the only possible objection that can be made to it is, that the operations would proceed a little slower than in the decimal scale, because in large numbers a greater number of figures must be employed to express them. This leads us to the consideration of the duodenary system of arithmetic, which, while it possesses all the advantages of the senary, in point of finite fractions, is superior even to the decimal system for simplicity of expression; and the only additional burden to the memory is two cha racters for representing 10 and 11, for the multiplication table in our common arithmetic is generally carried as far as 12 times 12, although its natural limit is only 9 times 9, which is a clear proof that the mind is capable of working with the duodenary system, without any inconvenience or embarrassment; and hence, I think, we may conclude, that the choice of the denary arithmetic did not proceed from reflection and deliberation, but 'was the result of some cause operating unseen and unknown on the inventor of our system; and it may, therefore be considered as a fortunate circumstance, that for this accidental radix, that particular one should have been selected, which may be said to hold the second place in the scale of general utility. All nations, both ancient and modern, with a very few exceptions, divide their numbers into periods of 10s, which singular coincidence of different people, entirely unconnected and unknown to each other, can only be attributed to some general physical cause, that operated equally on all, and which there is little doubt is connected with the formation of man ; namely, his having ten fingers, by the assistance of which, in all probability, calculation, or at least numbering, was first effected. Our present scale of notation, however, though founded on this principle, was not the immediate consequence of this division, but was an improvement introduced a long time afterwards, as is evident from the arithmetic of the Greeks, who, notwithstanding they divided their numbers into periods of tens, had no idea of the present system of notation, the great and important advantage of which is, the giving to every digit a local, as well as its original or natural value, by means of which we are enabled to express any number, however large, with the different combinations of ten numerical symbols; whereas the Greeks, for want of this method, were under the necessity of employing thirty-six different characters, and with which, for a long time, they were not able to express a number greater than 10000; it was, however, afterwards indefinitely extended by the improvements of Archimedes, Apollonius, Pappus, &c.' The last chapter in this work, which relates to the solution of the equation a"-10, n being a prime number, and exhibits the application of that solution to the analytical and geometrical division of the circle, is one which we have read with a high degree of pleasure. Most of our mathemnatical readers will recollect the scepticism with which the news was first received in England, that M. Gauss of Brunswick had found, by means of quadratics only, the side of a seventeen sided polygon, inscribed in a circle. That scepticism, however, is now removed, and it is well known, that this is only a particular case of a far more general inquiry, which has been conducted with great success by M. Gauss, and which, indeed, gives the principal value to his "Disquisitiones." Mr. Barlow has here entered upon the same inquiry, for the first time, we believe, it has been attended to in England; and, although he proceeds upon the same general principles as M. Gauss, he has gone through the investigation in much smaller compass, and, we think with more perspicuity. We cannot exhibit the whole of his process; but we will endeavour to present our readers with the spirit of it. Since the discovery of the Cotesian theorem, it has been generally known, that the division of the circle into any number of parts n, depends upon the salution of the binomial equation -1=0; an equation, however, which was considered as beyond the reach of analysis, till Gauss conquered the difficulty, in the work just mentioned, by resolving the equation into others of inferior dimensions. The number of these equations, as well as the degree to which they arise, depends upon the factors of the quantity n-1: that is, if οι βγ α n be a prime number, and n-1-a b c, then the solution. will be affected by equations of the degree u, B equations. of the degree b, y equations of the degree c, &c. and consequently, if n-1-2m, the solution will involve m quadratie equations only; whence, in such cases, the problem is susceptible of geometrical construction. The principle of solution, then, employed by our author, consists in dividing the series of imaginary roots of the equation 1-0 (which roots, it is well known are all 2 k ̧ T comprehended in a general formulæ such as 2-2cos. n +1=0) into periods, and finding the sums of the roots of each period; then subdividing those periods into others, and those again into others, till the whole series is finally divided into periods of single roots. For this purpose, it is first demonstrated that the imaginary roots of the equation a"-10, (n being a prime number) are powers of the same imaginary quantity; so that if R be one of those roots, the whole series will be R, R2, R3, R4, &c. to R1, the real root being unity. And, since from the theory of equations the sum of all the roots is equal to the co-efficient of the second term, we have I+R+R2+R3+ &c.=0, or so that the sum of all the imaginary roots is known; and conse-. 1 quently, if this series of roots be distributed into two periods A 3 R= It will be seen, then, that the only difficulty in the solution of these equations consists in so selecting the periods that their product may become a known quantity: the means for accomplishing this are not always obvious; but Mr. Barlow furnishes the reader with a variety of remarks and examples, tending to facilitate this part of the inquiry. We shall select the most curious of these, which we hope may be tolerably well comprehended, after the preceeding observations have been duly considered. Find the cosine of 7 360° and the roots of the equation Since 17 is a prime number of the form 2m+1, or 17=2+1, the roots of the above equation may be obtained by four quadratic equations, and the 360° cosine of by three quadratic equations; in order to which, the imaginary 17 roots of the equation must be decomposed, first, into, two periods of eight terms each, then these into two periods of four, and these again into two periods of two terms each. Now 3 being a primitive root of the equation 16 the whole period of powers will be =M(17,) 1, 3, 32, 33, 34, 35, 36, 37, 38, 39, 310, 311, 31, 313, 34, 315; or |