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j should be only ; and secondly is not 741 1+ y3
ý as he makes it; but it is =
+ Х 1-y+y?
+ 1-y+y 3' 1+ 3 Iy to go? 2 I-y+y^ and the fluent of the second term or L. 1-y + gy?
6 vanishing when y=1, the fluent in that case will be
3 X Arc tang. N3 rad. 1. and the true sum of the series 3 2.3 4.5
452 which is X
+ 1.5.7 4.7.9
14 Arc tang. I rad. 1. +
L. 2 +
13.1973 tang. 2, radius 1,
And the same will readily appear by transforming the numerators of the series to 5 - 3. 7-4, 7- 3. 9 --- 4, 9 - 3. II &c. for by this easy artifice The series will be resolved into others wherein the numerators are all. unity. In fome cases this is more readily done by an artifice fomewhat different; thus taking the series to 4
7 + &c. at p. 85, the sum of which Mr. Lorgna 2.6.9 3.8.11 makes less than the first term itself, though all the figns are affirmative; bere the numerators are respectively, 5
5 — 42 2 x 5-6, 3x5-8, &c. and the series by division = 5*:
5x: 4.7 6.9 8.11. 1.7 2.9 3.11
7 4 9 5 + &c. - X:I
499 19 +
L. 3 5 3 5
450 15 2 = the true fun of the series. Mr. Lorgna’s error seems to arife here from the perplexity of the method, it being difficule to avoid errors, in seeking out so many different quantities, affected with various figns, from general formulæ that are so much compounded. At Art. 92, p. 100,
he has committed one error upon another, the series be there proposes, is evidently equal to one half of the following one ; namely,
5 11 7 -- 1.2*3 9-1.3 + 5 of
3 3 5 4 I I
I &c. - X: +
+ + &c. 7 2 5 5 2.7. 3.7
&c. ut X:1
&c. 2.5 4.7 6.9
I VI - 11
2. + Arc 45° rad. 1 – tó 11
3 2 5
5 3 7 5 9
3 5 &c.
L. 2 + Arc 45° rad. 1. 8
3 3 530081 &c. the true sum of the series, which Mr. Lorgna makes less than the four first terms connected under their proper figns. At p. 110, Examp. V. he has fallen into an error of another naturé, by mistaking the form of his own series; for that, the sum of which he has there given, is very different from the proposed one: but it is high time to draw towards a conclusion, we shall therefore finish 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 sum of the series much greater than it ought to be.
1.8 The series is manifestly
-&c. 188.8.131.52 184.108.40.206
3 4.9.10 the numerators of which are respectively 6 5. 25-172 8 — 5. 35 – 17, 10 --- 5. 45 -- 17, &c. and then the series 25-17
35 – 17 divides at fight into +
1.2.5 2.3.7. 3.4.9 25
220.127.116.11 18.104.22.168 + &c. But the second of these series is =
4 found above, and + &c. = 5X1. moreover the
1.2.5 2.3.7 third series is evidently = 85 X :
+ + + 1.2.5 2.3.7
3.4.9 Rev. May 1781.
I &c. + +
&c. and the latter of 1.2 3
2.3.4 3.4.5 85
25 85 there = Х hence the original series = 5
8 8 4
70 68 X: + +
+ 272 X
8 1.2.5 2.3.7 3.4.9
2.4.5 70 272
2 + + t &c. 5
+ &c. 4.6.7 6.8.9
2 2.4 272
&c. + X: 1
272 272 &c.
X It 3 4 3
4 3 4 272 272
2219 + XL, 2.5 X hyp. log. of 2. 3 3 3
36 1.206457, &c. the true sum of the series, which Mr. Lorgna makes more than fix times as great as it should be.
Mr. Clarke feems to have been strangely prepoffeffed in favour of his author, and his method, for he has filled many pages of close quarto letter-press, in commenting on several of ine series, whose fums we have given above; and that, without discovering 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 subject to some certain rules, and reasons; which to give here would be taking up too much room.
Mr. C. has indeed greatly extolled the method made use of by Mr. Lorgna, in his oth section; but, for the even powers of the reciprocals of the natural numbers, it is certainly much inferior to that made use of by Mr. John Bernoulli, in the fourth volume of his Works, and by Mr. L. Euler, in the ist volume of his Introduction to the Analysis 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, some 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 summing of both a finite and infinite number of terms, of all possible numeral or literal series whatsoever, &c. Surely this is saying a great deal, and more than Mr. C. or the greatest mathematician living, could be certain of the truth of.
He has, however, wrote an advertisement, by way of preface, to qualify this boasting title ; and then given 53 forms of the most usual series, and added some examples, which may be useful and entertaining to all lovers of these speculations. But there are many kinds of series whose sums are attainable both in finite terms, and by means of the conic sections, of very
different forms from any contained in Mr. C.'s table.
Mr. Landen's Observations are an excellent elucidation of the method of summing series given in Mr. Simpson's Dissertations, which, of itself, is obscure and intricate enough ; and, like most other general forms, much wants that perspicuity, 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 ac
valuable treatise is very properly dedicated by the Translator, Dr. Forster, for the pains which, we learn, he has taken to procure a translation of it from the German into our language. It is the production of a philosophical chemist, to whose genius and industry we are indebted for several excellent analyses of mineral fubstances ; particularly for the discovery of the sparry or fluor acid, or the analysis of the sparry fluor, or Derbyshire fpar. A translation of his curious memoir on that subject was formerly given to the Public, by the Translator of the present performance.—[See M. Review, Vol. xlvii. December 1772, pag. 460.]
One of the most striking discoveries 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 same substance, is that of dephlogisticated, or, as Mr. Scheele has called it, Empyreal air. Alluding to this and other discoveries, the Translator, addressing himfelf to Dr. Priestley, fays, the Northern philosopher has treated the subject as a chemist, you as a philosopher : he came in many respects to the fame conclusions, grounded however on different premises. 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 science.'-It appears further, from the Author's preface, that he had already elaborated the chief part of his essay, 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 some inanner 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 phosphorus, &c. in a given quantity of it), he concluded, that atmospherical air consisted of two kinds of fluids;
-one, respirable and wholesome, and which he supposes to be the one third part that disappeared in the abovementioned experiments ;-and the other portion, or the two thirds remaining in the vessels, 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 scarce need to add, that the first of these is the dephlogisticated, and the latter the phlogisticated, air of Dr. Priestley; whose experiments, however, of a similar kind to those above mentioned, do not indicate the same proportions as those here assigned 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 respecting matters in which the minuteft circumstance is frequently of the greatest importance, cannot surely grudge the trouble of perusing a few pages, or even sheets, extraordinary, employed in this narrative style ; when the additional trouble thus occafioned is so amply repaid by the advantages it procures towards understanding the views and motives of his author ; whom he accordingly accompanies with equal satisfaction and intelligence, or, as a Frenchman would say, connaissance de cause, through every step of the process.-We speak with doubt; but, from the context, we are inclined to infer, that the Author discovered, or first became poffessed of, empyreal air on the following occasion :
On distilling fuming spirit of nitre, after the process had continued some time, he applied to the end of the retort a flaccid bladder, moistened on its inside with Lac Calcis, or lime-water, containing more quicklime than water can dissolve; in order to prevent the bladder's being corroded by the acid which came over. The bladder becoming inflated by an elastic substance, or air, he transferred the air to a glass 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