Imágenes de página
PDF
ePub
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

vanishing when y = 1, the fluent in that cafe will be

+1/3

I

3

[ocr errors][merged small][merged small]

X Arc tang. 3 rad. 1. and the true fum of the feries

[blocks in formation]

which is = X:

2.

2.3 1.5.7

6.7

[blocks in formation]

187

14

14

Arc tang. I rad. 1. +

L. 2 +

Arc

19.39

13.193

247

[blocks in formation]

tang. 3, radius 1. And the fame will readily appear by transforming the numerators of the feries to 5 7-39-4, 9 - 3. 11- 4, &c. for by this easy artifice the feries will be refolved into others wherein the numerators are all unity. In fome cafes this is more readily done by an artifice fomewhat different; thus taking the feries

[blocks in formation]

4 7 + &c. at p. 85, the fum of which Mr. Lorgna 2.6.9 3.8.11 makes less than the first term itfelf, though all the figns are affirmative; here the numerators are refpectively, 5-42 2x5-6, 3×5-8, &c. and the feries by divifion

[merged small][merged small][merged small][merged small][ocr errors][subsumed][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

the true fum of the feries. Mr. Lorgna's error seems to arife here from the perplexity of the method, it being difficult to avoid errors, in feeking out fo many different quantities, affected with various figns, from general formula that are so much compounded. At Art. 92, p. 100, he has committed one error upon another; the feries he there proposes, is evidently equal to one half of the following one; namely, 7 1.23 9-1.3+5

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors]

530081 &c. the true fum of the feries, which Mr. Lorgna makes less than the four firft terms connected under their proper figns. At p. 110, Examp. V. he has fallen into an error of another nature, by miftaking the form of his own feries; for that, the fum of which he has there given, is very different from the proposed one: but it is high time to draw towards a conclufion, we shall therefore finifh our exemplifications with Mr. Lorgna's finishing example at p. 111, where the great perplexity of his general formula has again led him into errors, and made him give the fum of the feries much greater than it ought to be.

1.8

The feries is manifeftly =

+

3.18
5.28
+
1.2.5.6 2.3.7.8 34.9.10

-&c.

the numerators of which are refpectively 65. 25-17,

[blocks in formation]

5. 35-17, 10 5. 45-17, &c. and then the feries 25-17 35-17 45-17 &c.

1.2.5

I

+

2.3.7

+

X: + + &c. +85 X:

divides at fight into

25

I

[ocr errors]

2.

[blocks in formation]

3.4.9.10

1.2.3 2.3.4 3.4.5

[blocks in formation]

-&c. But the fecond of these feries is =

found above, and

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

third feries is evidently

REV. May 1781.

I

I

+ + + 1.2.5 2.3.7 3.4.9

[ocr errors]

I

I

I

+

+
2.3.7 3.4.9

[blocks in formation]
[blocks in formation]

I

+

1.2 3

[blocks in formation]

68 X:

I

2

1.2.5

+ + 4.6.7

X

2.3.4

I

+ &c. and the latter of 3.4.5

hence the original feries = 5

&c. = 2 + 272 ×

[blocks in formation]

272

70

[blocks in formation]

&c.

5

[blocks in formation]

8

+

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

3

X

3

1.206457, &c. the true fum of the feries, which Mr. Lorgna makes more than fix times as great as it should be.

Mr. Clarke feems to have been ftrangely prepoffeffed in favour of his author, and his method, for he has filled many pages of close quarto letter-prefs, in commenting on feveral of the feries, whofe fums we have given above; and that, without dif covering any one of the errors here specified.

It may be proper to observe, that the method by which we have transformed the numerators and denominators above, is fubject to some certain rules, and reafons; which to give here would be taking up too much room.

Mr. C. has indeed greatly extolled the method made ufe of by Mr. Lorgna, in his 9th fection; but, for the even powers of the reciprocals of the natural numbers, it is certainly much inferior to that made ufe of by Mr. John Bernoulli, in the fourth volume of his Works, and by Mr. L. Euler, in the 1ft volume of his Introduction to the Analyfis of Infinites; and as to the odd powers, it is far from being the thing that is wanted. The method is taken from Mr. Cotes's account of the Newtonian differential method, publifhed at the end of his Harmonia Menfurarum. On the whole, Mr. Lorgna's is a very unequal performance, fome things in it being very curious and accurate, and others as much the contrary.

To his Author's Differtation, Mr. C. has added a long Appendix, intituled, the Summation of Series, exhibiting general formulæ, for the fumming of both a finite and infinite number of terms, of all poffible numeral or literal feries whatsoever, &c. Surely this is faying a great deal, and more than Mr. C. or the greatest mathematician living, could be certain of the truth of.

He

He has, however, wrote an advertisement, by way of preface, to qualify this boafting title; and then given 53 forms of the most ufual feries, and added some examples, which may be useful and entertaining to all lovers of thefe fpeculations. But there are many kinds of feries whofe fums are attainable both in finite terms, and by means of the conic fections, of very different forms from any contained in Mr. C.'s table.

Mr. Landen's Obfervations are an excellent elucidation of the method of fumming feries given in Mr. Simpfon's Differtations, which, of itself, is obfcure and intricate enough; and, like most other general forms, much wants that perfpicuity, which, next to truth, is, in our opinion, the chief beauty of the mathematics.

ART. III.

Chemical Obfervations and Experiments on Air and Fire: By Charles William Scheele, Member of the Royal Academy at Stockholm, &c. Tranflated from the German by J. R. Forter, L.L. D. F. R. S. and S. A. &c. 8vo. 3 s. 6 d. fewed. Johnfon. 1780.

TH

THE Public is much obliged to Dr. Prieftley, to whom this valuable treatise is very properly dedicated by the Tranflator, Dr. Forster, for the pains which, we learn, he has taken to procure a tranflation of it from the German into our language. It is the production of a philofophical chemift, to whofe genius and induftry we are indebted for several excellent analyses of mineral fubftances; particularly for the difcovery of the fparry or fluor acid, or the analysis of the fparry fluor, or Derbyshire fpar. A tranflation of his curious memoir on that fubject was formerly given to the Public, by the Tranflator of the prefent performance. [See M. Review, Vol. xlvii. December 1772, pag. 460.]

One of the most striking difcoveries contained in this work, and which the Author appears to have made before he could have received any intimation of Dr. Priestley's prior discovery of the fame fubftance, is that of dephlogisticated, or, as Mr. Scheele has called it, Empyreal air. Alluding to this and other discoveries, the Tranflator, addreffing himfelf to Dr. Priestley, fays, the Northern philofopher has treated the fubject as a chemist, you as a philofopher: he came in many refpects to the fame conclufions, grounded however on different premifes. You knew nothing of his experiments, and he was ignorant of your great and numerous discoveries when he made his experiments: and you both, independent of one another, contributed by very different ways and methods to enlarge the field of fcience.'-It appears further, from the Author's preface, that he had already elaborated the chief part of his effay, when he first got fight of the excellent experiments of Mr. Priestley.”

The Author's intention being to inquire into the true nature of fire, he commenced this arduous undertaking by first attempting to investigate the nature of air; without the presence of which, or its agency in fome manner or another, no fire will burn. From a feries of experiments here related, in which air was diminished no less than one third (by including hepar fulphuris, burning phofphorus, &c. in a given quantity of it), he concluded, that atmospherical air confifted of two kinds of fluids; -one, refpirable and wholesome, and which he fuppofes to be the one third part that disappeared in the abovementioned experiments; and the other portion, or the two thirds remaining in the veffels, perfectly noxious. To the first of these fluids, or that peculiar part of common air by means of which fire burns, he gives the name of Empyreal air [Fire-air]; and to the remaining two thirds he gives the appellation of foul air. We fcarce need to add, that the first of thefe is the dephlogisticated, and the latter the phlogisticated, air of Dr. Priestley; whofe experiments, however, of a fimilar kind to those above mentioned, do not indicate the fame proportions as thofe here affigned by Mr. Scheele.

The Author does not follow the laudable method employed by Dr. Priestley, of relating his proceedings in an historical manner, and indicating the motives which actuated him in each process. A reader who is really anxious to receive information, particularly refpecting matters in which the minuteft circumftance is frequently of the greatest importance, cannot furely grudge the trouble of perufing a few pages, or even sheets, extraordinary, employed in this narrative ftyle; when the additional trouble thus occafioned is fo amply repaid by the advantages it procures towards understanding the views and motives of his author; whom he accordingly accompanies with equal fatisfaction and intelligence, or, as a Frenchman would fay, connoissance de caufe, through every step of the process.-We speak with doubt; but, from the context, we are inclined to infer, that the Author discovered, or firft became poffeffed of, empyreal air on the following occafion :

On diftilling fuming spirit of nitre, after the process had continued fome time, he applied to the end of the retort a flaccid bladder, moiftened on its infide with Lac Calcis, or lime-water, containing more quicklime than water can diffolve; in order to prevent the bladder's being corroded by the acid which came over. The bladder becoming inflated by an elaftic substance, or air, he transferred the air to a glafs veffel: into which, he says, I put a small burning candle: when immediately the candle burnt with a large flame, of so vivid a light that it dazzled the eyes. I mixed one part of this air with three parts of air in which fire would not burn; and this mixture afforded air, in

« AnteriorContinuar »