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for the cofine of PA,

fine of 16o 3'35", Aldebaran's declination; and x the e of ZA, its zenith diftance, we have 1-x2-s-dx, when the fluxion of Pis equ the flux. of Z: confequently, 1-s-x2-dx, and x={d + √ 1−s+1 d2=,62428 the nat. fine of 38° 374, the altitude of the ftar when the change in azimuth is r in one minute of time. Hence the ftar was S. 64° 7' E. or W.

But to determine what azimuth circle the ftar is on when the motion in azim bears the leaft ratio poflible to the diurnal motion; it is manifeft that fin. ZA bear the greatest ratio poffible to fin. ZPXR-cof. ZAx cof PA; that is, retain 5-dx must be a minimum; and by making its fluxion 31x2

the preceding notation,

e, we obtain x =

,

,182547, the nat. fine of 10° 31′ 5′′, thef altitude when it changes its azimuth the floweft, and hence its azimuth is 224 E. or W.

It is well known that all objects change their azimuth fafteft when on the ridian.

43. QUESTION (IIL. Feb.) anfwered by Tasso, of Briftol, the propole The fecond equation being the fum of xy and zv, and the fourth their pr the former is readily found to be 24, and the latter 360; and these values fubftituted in the third equation, it becomes 242+360y=1944, or 2=81

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24 and, as y= x

we have x=

Thefe values being fubftituted in the firft equation gi

9x-40
81x-360 40x

and v

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x

x

x=8.

+

9x-40

·= 57, or x3 +24x2-4423x + 1493}=0; Hence ey=3, z=36, and v10.

which

44. QUESTION (IV. Feb.) anfwered by Mr. TODD, the propofer. Put 50, the complement of a life of 36 years old, according to M MOIVRE'S hypothefis, r= - 1,04,5=1727. and a the annual payment: then,

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is the probability that a life, the complement of which is #, will fail in year of its duration, the prefent value of s, payable at the failure of the fai

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prefent value of an annuity of 1. for the faid life is, according to the fame

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that is, very near 57. as given by Dr. Price at p. 123 of his Treatije on fionary Payments.

To find, the years this life should continue, fo that the amount of A, 31 amount of a, the annual payment, may each of them be equal to s (172)

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SCHOLIU M.

439

This reverfionary annuity is worth more than the reverfionary fum by the prefent

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the prefent worth of the first payment a; therefore,

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In infwer to what has been advanced against the note E, in Dr. Price's Reverry Payments, it may be observed, that when z is put in the third line on pages

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417, 2d edit. the whole will be perfectly right. For 1

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terms) equal a perpetuity of 17. minus the present worth of 17. annuity for the

r3

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at worth of one pound per ann. for ever, after a given life, the complement of

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Er: 1, the fame as Dr. Price makes it, when r1,04.

Brand alfo, at p. 65 and 66 of his book on affurances and annuities on has made very free with the Doctor, because at p. 123, 2d edit. of his Rerary Payments, he hath faid, "That an annual payment, beginning immeof 51. during a life, now at the age of 36, fhould entitle, at the failure of alife, to 172/. intereft at 4 per cent. and taking Mr. De Moivre's valuation es." Mr. Brand, in his attempt to refute this, has taken 12,1 years purchase annuity of 1. for a life of 36 years, at 4 per cent. as given by Mr. Simpson, the London bills of mortality; and then afks, "How is it poffible that an payment of 5. with its compound interest, at 4 per cent. fhould in 12,1 amount to 1721. ?" It certainly cannot: it amounts to no more than 75%. 4. But another might fay to Mr. B. Pray, Sir, how fhould it? Your 12,1 is not the duration of a life of 36, according to Mr. De Moivre's hypothefis, number of a very different kind. And every one muft fee that it is very g to give the annuity from equal decrements, and the fuppofed time from the don bills of mortality.

MATHEMATICAL QUESTIONS.

60. QUESTION I. by R. M.

Required a general method of drawing the reprefentation of a great circle on orthographic projection, to cut the reprefentation of a given great circle era given angle, and touch the reprefentation of a given lesser circle.

61. QUESTION II. by DISCIPULUS, of Greenwich Academy. After failing from fix o'clock in the morning till noon, S. S. E. at the rate tight knots, I found the port to which I was bound bore W. N. W.

Keeping

Keeping ftill the fame courfe, at the fame rate, till four in the afternoon, then found that the tide had fet me as far, within ten leagues, to the E. S. E of my reckoning, as I was diftant at noon from the place of my departure: required the drift of the current.

62. QUESTION III. by Mr. WILLIAM RICHARDS.

Given AC, the hypothenufe of a right-angled triangle ABC; if the baf be produced to D, fo that AD the perpendicular BC; and if C and D joined, and AE drawn perpendicular to AB, meeting CD in E, the areas the triangles ABC, ADC, fo formed, will be equal: it is required to cont the triangles.

63. QUESTION IV. by SENEX.

Mr. Emerfon, p. 177 of his Fluxions, 2d edit. propofes to find the we y, to be raised by the defcent of w, fo that y may receive the greateft mo poffible in a given time; the weight w, and the radii of the wheel and being given: it is propofed to examine whether his folution to that prob be true or falfe; and if falfe, to point out the error.

The anfwers to thefe queftions may be directed (poft-paid) to Baldwin, in Paternofter-row, London, before the 1ft of September.

ANIMADVERSIONS ON THE THIRD PART OF THE REV.
VINCE'S PAPER ON SERIES, IN THE PHILOSOPHICAL TRANSACTI
FOR 1782.

THAT gentleman, in his lemma, finds by divifion,

I

=1-1+1−1, &c. ad infinitum. By the fame method,

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Here, it is plain, no regard is had to the remainders which refult in perfor the operation, but they are at length rejected without affigning any reafon, the in all fuch operations they ought to be retained, unless in the end they be indefinitely fall.

I

Now, it may cafily be fhewn, that is in general 1−x + x2−x3

1+x

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1

they will; and where the upper or lower of the double figns takes place, accordin n, the number of terms in the feries, is even or odd; and where, how great i n may be, the terms with the double figns can never be rejected on account of fmailnefs unlefs x be lefs than 1.

Is it not then obviously wrong to say, that 1—1+1-1, &c. ad infinitum is fraction; and that 1-2+3-4, &c. ad infinitum is the fraction, feeing:

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the terms of the feries are all integers; that the fum of the feries 1-1+1−1, is manifeftly equal o or 1; that the fum of the feries 1-2+3—4 (#) is in get.

72

2

; that the more terms you take of the laft written feries, the more

the aggregate of thofe terms differ from the fraction ; and that, by increafing number of terms, the difference between their aggregate and that fraction (1) 14 be greater than any given number, how great foever it be?

I must confefs that I cannot help thinking mathematics will never improved by the admiffion of fuch principles as thefe. And furely, tast fum of a fenes of integers may be equal to a proper fraction, is a propofition paradoxical to be admitted as a mathematical axiom!

In applying the lemma, we are told that the feries 1-3+i—†, &c. ad infin.l.

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&c. ad infinitum, though each of these last two feries appears to confift of all Erms (or parts) of the preceding feries collected two into one, in a very obvious ter: but we are taught to correct thefe feries (by adding or fubtracting the jon) to make each of them equal to the feries from which they are derived. w, I would afk, what terms of the first feries (if any) are omitted in fo coilectre terms? If none be omitted, the feries obtained by fo collecting the terms e first feries can want no correction: if any be omitted, the corrector will do to point them out, and prove that their aggregate correfponds with his cor

aargue, that of thefe two feries.

6.7

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(m), upon fuppofing m infinite, the former is continued after the latter nates, is fallacious. The latter properly never terminates its terms may eed be conceived to become indefinitely mall but not abfolutely nothing; and fo taterminating whilft the former is continued, the number of its terms depends te tumber of terms in that former feries; the number of terms in the one feries gmiteftly equal to half the number of terms in the other feries. Can the denoted by 2m increase after the number denoted by m ceases to increase? day, I think it an improper problem to propofe to compute the fum of the * j − } + 4 − }, &c. ad infinitum, without being more explicit with regard Be continuation of the feries. To me the proper problem feems rather to be, to rmine the limit of the fum of the feries 1 − 3 + 1 − † (21) or of the feries ¿− † (2m+1) suppofing the integer m to increale ad infinitum.

wch limit may be ealily found by various methods; and the finding it may serve hew the fallacy of Mr. VINCE's imaginary correction.

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from above (m

gfuppofed any pofitive integer and 2m+1=n); we, from thence, by multi

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, that when x (fuppofed pofitive) is equal to, or less than 1, the value of fl.

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hall be less than any affignable quantity how small foever it be; we right

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1+x

4

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being rejected on account of

malinefs. But, if we would reafon farther from the equation between the hyp.

of 1+x and its value, we should do wrong to reject the quantity fl.

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rout enquiring whether the procefs may not produce there from another quantity at fhall be finite (and therefore of confiderable value) in the refult. The retainthe expreffion fl.

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; the rejecting it may.

it is evident cannot lead to an erroneous conclu

From the equation between the hyp. log. of 1+x and its general value, we have

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+ A.

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; and hence, by fuppofing m infinite, and taking x=1, we find

limit o + & − † (2m) = 1 + the hyp. log. of 2.

Thus by incontrovertible reafoning a value (or limit) of the feries -} + &c. ad infinitum is found; and in the fame manner may another value (or Eins of that feries be found, after writing 2m+2 for`n in the equation

1

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+x3-x1 (x) + : which fecond value (or limit) will be the hyp. kg

--

I+x 2. And these two values of the feries 4 − + − 4, &c. ad infinitum correl to the two obvious modes of fummation, or ways of collecting the values

terms.

To infer that the feries -+-, &c. ad infinitum, is not equal to eithe those values (or limi ́s) but that it is —— the hyp. log, of 2 (=R) the

tity that refults from fuch a procefs as the above when the expreffion fl.

is difregarded, is furely fuch a conclufion as ought not to have place in mather tics! for the more terms we take of the feries, the more will the aggregate of terms differ from that imaginary fum R!

We have feen above that the feries 3+ 4 , &c. ad infinitum, has two or limits and it is obfervable, that whenever the fummation of a feries is pro whofe terms are fome of them pofitive and fome negative, and they do not con fo as to become indefinitely fmall, if fuppofed to be continued ad infixum, propofition will be fo vague, that it may perhaps admit of various folutions, a the law of the continuation of the terms be indicated by a proper fymbol. For stance, the feries 4-3-3+4+39 — 3+, &c. ad infinitum, or + } } + 1 + 9 — 3 — 8, &c. ad infinitum, has three fums or limits: which are the -9—3+3 (n) upon taking nequal to 4m−1, 4m, and 4m+1; 3 + 3 + 9 - 3 - 8 (#) upon taking n equal to 4, 484 and 4+2 m being always an integer: and thofe limits are refpectively equal

2

6

2 — 3 — + 1 + 1 the limits of +3.

2

2

6

7

3

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and&+1;

G being circ. are, rad. 1, tang. 1, + ‡ hyp. log. of 2 ;

G

circ. are, rad. 1, tang. 1, — i̟ hyp. log, of 2.

It does not appear that it can be any way conducive to the improvement of doctrine of feries, to attempt to assign a certain fom to any fuch feries as 1-3 +3 −3+8—, &c. ad infinitum,

— — 3 — 4 + 3 + 38 −9−3+§, &c. ad infinitum,

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without any regard to the law of continuation on the contrary, it is (at lea appearance) an abfurdity, to affign, as the fum of fuch a feries, a quantity fr which the aggregate of the terms of the feries would differ more and more, increafing their number; as is always dene in pursuing the method which is fubj &t of these animadverfions.Indeed, the principle, that the fum of a feat of integers may be equal tó a proper fraction (upon which that method is founde is fuch, that no other than an abfurd conclufion can well be expected to fol from it!

T

AEROS TA TIC S.

J. LANDEN.

HE deferiptions of aeroftatie experiments, as well as an account of the principles on which they are performed, which have been prefented to tr readers of this work, form, as it were, an hiftory of this, difcovery, and of progrefs. To thefe narratives are now added a tranflation of large ext

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