2=4; now this latter is much less than the former. So is still too great a value for x; that is to say, 10 is greater than 2. For the cube of 10 is 10, and that of 2 is only 8. But, on the contrary, 4 we give it too small a value, because the fourth power of 10 being 10, and that of 2 being 16, it is evident that 10 is less than 2. So that x, or the L.2, is less than, and yet greater than . We may in the same manner determine, with respect to every fraction contained between and, whether it be too great or too small. 6 Making trial, for example, with, which is a fraction less than and greater than, 10, or 103, must be 2: or the seventh power of 10, that is to say, 102, or 100, must be equal to the seventh power of 2; now the latter is 128, and consequently greater than the former. We therefore infer that 103 is also less than 2, and that therefore is less than L.2, and that L.2, which was found less than, is however greater than 3. Let us try another fraction, which, in consequence of what we have already found, must be contained between and; is a fraction of this value, and it is therefore required to find whether 102; if this be the case, the tenth powers of those numbers are also equal; now the tenth power of 10 is 1031000, and the tenth power of 2 is 1024; we conclude, therefore, that 10,3 is not 2, that is too small a fraction to produce that equality, and that the L.2, though less than, is yet greater than 236. This discussion serves to prove, that L.2 has a determinate value, since we know that this logarithm is certainly greater than Te, and less than . We shall not proceed any farther at present, but since we are still ignorant of its true value, we will represent it by x, so that L.2=x; and endeavour to shew how, if it were known, we could deduce from it the logarithms of an infinity of other numbers. For this purpose we shall make use of the equation already mentioned L.cd=L.c+L.d, which comprehends the pro perty, that the logarithm of a product is found by adding the logarithms of the factors. 237. First, as L.2=x, and L.10=1, we shall have L.20=x+1; L.200x+2; L.2000=x+3; L.20000=x+4; and L.200000=x+5, &c. 238. Farther, as L.c2L., and L.c3-3L.c, and L.c+=4L.c, &c. we have L.4=2x; L.8=3x; L.16=4x; L.32=5x; L.64=6x, &c. Hence we find also that, L.4000=2x+3; L.40000=2x+4, &c, L.80co 3x+3; L.80000=3x+4, &c. 239. $ 239. Let us resume also the other fundamental equation, L.-L.-L.d, and let us suppose c=10, and d=2; since L.10=1, and L.2=x, we shall have L., or L.5=1-x, and 2 shall deduce from hence the following equations: L.50-2-x; L.500=3-x; L.5000—4—x, &c. L.25=2-2x; L.125=3-3x; L.6254—4x, &c. L.250-3-2x; L.2500=4-2x; L.25000-5-2x, &c. L.1250 4-3x; L.12500-5-3x; L.1250006—3x, &c. L.6250-5-4x; L.62500=6-4; L.625000=7-4%, &c. and so on. 240. If we knew the logarithm of 3, this would be the means of farther determining a prodigious number of other logarithms. We shall subjoin a few examples. Suppose the L.3 expressed by the lettery. Then, L.30y+1; L.300 y+2; L.3000=y+3, &c. L.9=2y; L.27=3y; L.81=43; L.243=5y; &c. we shall have also, L.6=x+y; L.12=2x+y; L.18=x+2y; and L 15=L.3+L.5=y+1−x. 241, We have already seen that all numbers arise from the mul tiplication of prime numbers. If therefore we only knew the lo garithms of all the prime numbers, we could find the logarithms of all the other numbers by simple additions. The number 210, for example, being formed by the factors 2, 3, 5, 7, its logarithm will be =L.2+L.3+L.5+L.7. In the same manner, since 360=2×2 × 2 × 3 × 3 × 5 = 23×3×5, we have L.360-3L.2+2L.3+L.5. It is evident, therefore, that by means of the logarithms of the prime numbers, we may determine those of all others; and that we must first apply to the determination of the former, if we would construct tables of logarithms.' The second volume treats of the Indeterminate Analysis, and is, according to an able mathematician (Condorcet) " une theorie presque complete de cette partie de l'algebré." The first who considered the nature of indeterminate problems was Diophantus, of the school of Alexandria. Little progress, however, was made in this branch of the mathematics till the beginning of the 17th century, when Backet de Mezeriac published a learned commentary on Diophantus. The subject was soon enriched by the labours of Fermat, Descartes, Frenicle, and Wallis. Yet investigations of this kind began to be neglected, when the curiosity of the learned was revived by the publications of Euler. By these, and by the accurate and comprehensive methods of M. de la Grange, the indeterminate analysis was rapidly advanced to a high degree of perfection. In regard to the notes, we wish that the English translator had placed them at the bottom of the several pages, in imitation of the French editor. It must also be remarked that these notes, notes, though good, are few, and are given (in our opinion) on those parts which least required elucidation: so that, had we not bowed to the respectable name of Bernouilli, we should have applied to the authors of the notes the words of a great philosopher: In istius modi autem laboribus, pessimus ille criticorum nonnullos quasi morbus invasit, ut multa ex obscurioribus transiliant, in satis vero perspicuis ad fastidium usque immorentur et expatientur." In the preface to the present translation, an attack is wantonly made on the French treatise; which is stated to contain a needless multiplication of words, a redundancy of colloquial idiom, and unnecessary verbiage,'-together with fifty errors at least, which have been discovered and detected. We confess that we felt rather hurt, not to say provoked, by this act of hostility; as we had long been admirers of the French edition, for the excellence of its style and for its typographical correctOur judgment, however, might not be accurate: we therefore compared the two translations together; and, though we allow that, in the English, the sense is strictly preserved and rendered even with elegance, yet in the French there are an easiness and a familiarity of expression, which our language, from its genius, seems unable to attain. In respect of errata, the first volume of the present publication is tolerably free. In the perusal of the second volume, we corrected the errors as they appeared to us, and were surprised to find their number increase very rapidly. We had deter mined to notice, generally, that the plea of correctness had been urged by the present translator rather incautiously: yet, on farther reflection, considering how desirable accuracy was, especially in algebraic operations, we resolved to adopt a mode novel to our Review, and to give a list of errata. To us these faults have established, beyond all controversy, the superior correctness of M. Bernoulli's edition, though we acknowlege it to be sometimes erroneous in its algebraic notation: Vol. 1. P. 28. 1. 12. 39 for 34. P. 32. l. 1. omitted in the denominator. P. 44. last line, mn instead of mm. P. 46. 1. 5 from bottom, bb+4ac instead of bb-4ac. [Same error in French.] P. 48. I. 17. 2an instead of 2nn. P. 50 line 19. we read xx+(x+1)x(x−1)=xx¥ 2mx(x+1) ̧mm(x+1)2. This equation, after having destroyed n nn the terms xx, and divided the other terms by x+1, gives #nx¬nn=2mnnx+mm' Now, had the translator attended to the the steps of the algebraic process, he could not have fallen into this mistake; for the last equation ought to be nnx-nn= 2mnx+mmx+mm: but the fact is that he followed the French edition, in which there is precisely the same error. mediately afterward, we find x= mm + nn P. 73. 1. 19. axxyy instead of axx. mm +nn P. 76. 1. 12. the mark of equality (=) omitted. Im instead of 2gpq agg-pp P. 111. 1. 6. c+2fq+pp instead of c=2fq+pp. P. 145. line last, the character instead of √ P. 161. 1. 2, 3, and 4, apa and eq/c instead of ap1√ and cq3/c. P. 276. 1. 21. the quantity 3×3×5×5×5 instead of 2X4×4×6×6 3×3×5×5×7×7. [Same mistake in the French.] 2X4X4×6×6 P. 277. 1. 13. a for a. P. 332. 1. 2. qp" for q'p" P. 334. 1. 17. we read, and consequently the number can only be the integer number, which will be immediately above the quantity,' &c. Now this is certainly a translation of the French," et le nombre p' ne pourra etre par consequent, que le nombre entier qui sera immediatement au dessus de la quantité," &c. We apprehend that there is an error in the French, au dessus being put for au dessous; which we could easily prove from the values of the quantities concerned in the calculation. P. 336. 1. 23. the sign is used instead of the sign [Same mistake in the French.] P. 341. 1. 11. qm instead of Vqm, an easy and trivial error; yet remarkable, as there is precisely the same in the French edition. P. 350. there are at least ten errors, Ap &c. being put for Ap &c. The French copy has only one error. P. 351. 1. 13. Cq' for Cq" [Same error in the French.] SH H P. 290. 1. 4. 231111 instead of 23 X11: P. 419. 1. 9. B for B P. 427, 428. faulty position of the marks (') above the letters *. We find considerable difficulty in reconciling these errors with the assertions of the editor, concerning the time and labour which he has bestowed on the present publication. He must either have been ignorant of the nature or inattentive to the process of the calculations, to have adopted the errors of the French edition. As the present work is on the whole a very valuable one, and as accuracy is most especially desirable in mathematical treatises, we hope that the editor will take advantage of our corrections. We wish nevertheless to repeat that, notwithstanding the defects which we have pointed out, he is entitled to the best thanks of the public for introducing so valuable a production to their notice, and for executing his task with so considerable a portion of fidelity, spirit, and correct ness. As we have given, in a former volume + of our work, a sketch of the life of Euler, and discussed his merits, we do not here purpose to speak of him characteristically; yet we could willingly expatiate on his learned labours, his unwearied researches, his accurate views, his comprehensive methods, the suavity of his manners, and the soundness of his virtues: we can repeat with pleasure the words of his eulogist‡,"son nom ne perira qu'avec les sciences memes. Transmis à la posterité avec les noms illustres de Descartes, Galileé, Leibnitz, Newton, et tous les grands hommes qui ont honoré l'humanité par leur génie, son nom vivra encore, lorsque ceux de bien des personages qui la frivolité de notre siècle a illustré, seront enstvelis dans la nuit éternelle de T'oubli" *The French edition is free from this scrics of errata, except where we particularly specify to the coutrary. Vol. lxxiii. p. 436. Monsieur Fuss. Wood.... ART. |