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for the cofine of PA, line of 169 3'35", Aldebaran's declination; and x the cof. of ZA, its zenith distance, we have 1~*2=5dx, when the fluxion of P is equal the flux. of 2: consequently, 1-s=x2-dx, and x={d+vi-s+d2=,62428, the nat. fine of 38° 371, the altitude of the star when the change in azimuth is 15 in one minute of time. Hence the star was S. 64° 7' E. or W.

But to determine what azimuth circle the star is on when the motion in azimuth bears the least ratio poflible to the diurnal motion; it is manifest that sin. 2 ZA muft bear the greatest ratio possible to fin. ZPXR-cos. ZA x cof: PA; that is, retaining

soda the preceding notation, must be a minimun ; and by making its fiuxion =

-N52-02 , we obtain x =

- =,182547, the nat, fine of 109 31's", the star's

d altitude when it changes its azimuth the flowest, and hence its azimuth is N. 77* 224 E. or W.

It is well known that all objects change their azimuth fastest when on the meridian.

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43. QUESTION (III. Feb.) answered by Tasso, of Bristol, the proposer: The second equation being the sum of xy and zv, and the fourth their product, the former is readily found to be = 24, and the latter 360; and these values being Substituted in the third equation, it becomes 242+360 y = 1944, or z=81-15Y; 360


818-360 but z = i consequently v=

and, as y =

we have za

27-50 and v=

These values being subftituted in the first equation give x + 9.140 24 818-360

40X +

+ = 57, or x3 +2472–4424x + 1493} = 0; which gives

9X-40 * = 8. Hence y=3, Z=36, and v=10.

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44. QUESTION (IV. Feb.) answered by Mr. Todd, the proposer. Put n = 30, the complement of a life of 36 years old, according to Mr. DeMOIVRE's hypothesis, r= 1,04,5=1721. and a the annual payment: then, because 2

is the probability that a life, the complement of which is n, will fail in any one year of its duration, the present value of s, payable at the failure of the said life, is + that to me yol = £.73,898715, = p. Again, the present value of an annuity of 11. for the said life is, according to the same author,



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=-X gry ; confequently a=4-4,9333, &c. that is, very near sl. as given by Dr. Price at p. 123 of his Treatise on Rever. fonary Payments.

To find x, the years this life should continue, so that the amount of p, and the amount of a, the annual payment, may each of them be equal to s (1721.) =pr*, =art tark-start-2 tar? tar, arx .; p* will be = 2,32751002, log. of 2.32751002 ,36689156

= 21,5396 years.
lug, of 1,04


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SCHOLI U M. This reverfionary annuity is worth more than the reversionary fum by the present

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the present worth of the firêt payment a; therefore, - *

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= the present worth

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of the reverfionary annuity.

In answer to what has been advanced against the note E, in Dr. Price's Rever. fionary Payments, it may be observed, that when n is put in the third line on pages 286, 287, 2d edit. the whole will be perfectly right. For 1-5 Х +


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1 +

+ + &c. ad

ph go"? +3 infinitum, +


+ &c. ad infinitum g2

+ + &e. (to n terms) equal a perpetuity of il, minus the present worth of sl. annuity for the given life, =

X += present worth of one pound per ann, for ever, after a given life, the complement of

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(s) :: 1 : 1, the same as Dr. Price makes it, when r=1,04.

Mr. Brand also, at p. 65 and 66 of his book on affurances and annuities on lives, has made very free with the Doctor, because at p. 123, 2d edit. of his Reverfionary Payments, he hath said, “ That an annual payment, beginning immediately, of sl. during a life, now at the age of 36, should entitle, at the failure of such a life, to 1721. interest at 4 per cent. and taking Mr. De Moivre's valuation of lives.” Mr. Brand, in his attempt to refute this, has taken 12,1 years purchase of an annuity of il. for a life of 36 years, at 4 per cent. as given by Mr. Simpson, from the London bills of mortality; and then asks, “ How is it possible that an annual payment of gl. with its compound interest, at 4 per cent. fhould in 12,1 years amount to 1721.?". It certainly cannot: it amounts to no more than 75l. 185. 7d. But another might say to Mr. B. Pray, Sir, how should it? Your 12,1 years is not the duration of a life of 36, according to Mr. De Moivre’s hypothesis, but a number of a very different kind. And every one must see that it is very wrong to give the annuity from equal decrements, and the supposed time from the London bills of mortality.


60. QUESTION I. by R. M. Required a general method of drawing the representation of a great circle on the orthographic projection, to cut the reprefentation of a given great circle under a given angle, and touch the representation of a given leffer circle.

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

Keeping Keeping still the same course, at the same rate, till four in the afternoon, I then found that the tide had set me as far, within ten leagues, to the E. S. E. of my reckoning, as I was distant at noon from the place of my departure : required the drift of the current.

62. QUESTION III. by Mr. WILLIAM RICHARDS. Given AC, the hypothenuse of a right-angled triangle ABC; if the bafe BA be produced to D, so that AD = the perpendicular BC; and if C and D be joined, and AE drawn perpendicular to AB, meeting CD in E, the areas of the triangles ABC, ADC, so formed, will be equal: it is required to construct the triangles.

63. QUESTION IV. by SENEX. Mr. Emerson, p. 177 of his Fluxions, 2d edit. proposes to find the weight y, to be raised by the descent of w, so that y may receive the greatest motion possible in a given time; the weight w, and the radii of the wheel and axle being given: it is proposed to examine whether liis solution to that problem be true or false; and if false, to point out the error.

The answers to these questions may be directed (post-paid) to Mr. Baldwin, in Paternoster-row, London, before the ift of September.

THAT gentleman, in his lemma, finds by division,



15 =j-iti-1, &c. ad infinitum. By the same method,

4= is found = 1-2+3–4, &c. ad infinitum.

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Here, it is plain, no regard is had to the remainders which result in performing the operation, but they are at length rejected without assigning any reason, though in all fuchi operations they ought to be retained, unleis in the end they become indefinitely mall. Now, it may easily be shewn, that is in general = 1- x + x2–43 (n) {

I+x and

= 1-2x+3x2-444 (n) + +- let n and x be what I+x ital


Itx they will; and where the upper or lower of the double signs takes place, according as n, the number of terms in the series, is even or odd; and where, how great foever n may be, the terms with the double ugns can never be rejected on account of their smallnels unless x be less than 1.

Is it not then obviously wrong to say, that 1-1+1-1, &c. ad infinitum is = the fraction.d; and that 1–2+3_-4, &c. ad infinitum is the fraction 4, seeing that the terms of the series are all integers; that the fum of the series 1-1+1-1, &c. is manifestly equal o or 1; that the sum of the series 1-2+3–4 (») is in general =F!F; that the more terms you take of the last written series, the more will the aggregate of those terms differ from the fraction l; and that, by increasing the number of terms, ihe difference between their aggregate and that fraction (1) may be greater than any given number, how great foever it be?

I must confess that I cannot help thinking mathematics will never be improved by the admission of such principles as these. And surely, that the fum of a series of integers may be equal to a proper fra Tion, is a proposition too paradoxical to be admitted as a mathematical axiom!

In applying the lemma, we are told that the series 1-1+1-, &c, ad infinitum,

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is neither =

&c. ad infinitum, nor = 1+

2.3 4.5

3.4 &c. ad infinitum, though each of these last two series appears to consist of all The terms (or parıs) of the preceding series collected two into one, in a very obvious manner : but we are taught to correct these series (by adding or subtracting the fraction 1) to make each of them equal to the feries from which they are derived.

Now, I would ask, what terms of the first series (if any) are omitted in so coilecting the terms? If none be omitted, the series obtained by fo collecting the terms of the first series can want no correction: if any be onnitted, the corrector will do well to point them out, and prove that their aggregate correlponds with his correction.

3 4 5

6 To argue, that of these two series



7 (m), upon fuppofing m infinite, the former is continued after the latter 45 6.7 terminates, is fallacious. The latter properly never terminates : its terms may indeed be conceived to become indefinitely Imail but not absolutely nothing; and so far from terminating whilst the former is continued, the number of its terms depends on the number of terms in that former series; the number of terms in the one teries being manifestly equal to half the number of terms in the other series. Can the number denoted by 2m increase after the number denoted by m ceases to increase?

I must say, I think it an improper problem to propose to compute the sum of the series {-}+1-4, &c. ad infinitum, without being more explicit with regard to the continuation of the series. To me the proper problem seems rather to be, to determine the limit of the sum of the series {-3+1-(2m) or of the series } +*-, (2m+I) supposing the integer m to increale ad infinitum.

Such limit may be eally found by various methods; and the finding it may serve to thew the fallacy of Mr. VINCE's imaginary correction.

x21 +1 Taking the equation S=i- x + x2 – 33 (2m+1)

from above (m 1 + x

1+x being supposed any positive integer and 2m +1=n); we, from thence, by multia plying by *, and taking the fluents, find A.

== the hyp. log. of itx, in ge

Itx x2

x211 + 1x neral =*

(2m+1) - A.

Now, as m may be taken so

itx great, that when x (Tupposed positive) is equal to, or less than 1, the value of A. *21 +1;

Thall be less than any asignable quantity how small soever it be; we right, it*

* ly conclude, that (x being fo) the hyp. log. of 1** (or A.

is ニメー

fo 1+x

*2m +13 &c. ad infinitum, the quantity fl. being rejected on account of 3

1+x its smallness. But, if we would reason farther from the equation between the hyp.

xam+'* log. of itx and its value, we should do wrong to reject the quantity f.

I+* without enquiring whether the process may not produce therefrom another quantity that shall be finite (and therefore of considerable value) in the result. The retain

x2m +1c ing the expression A.

it is evident cannot lead to an erroneous conclu.

I+x fion ; the reje&ting it may. From the equation between the hyp. log. of +x and its general value, we have

x? (by transposition and division)

+ (20) ==- hyp. log. of 3

5 *2m +13

x2 itx A.

: whence, by taking the fluxions, and multiplying by itx

å 3х4

*2m + + (2m) = the hyp. log, of itx

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*29% +' + f.

and hence, by supposing m infinite, and taking x=1, we find the 1+x limit o į

}+(2m) - it the hyp. log. of 2. Thu: by incontrovertible reasoning a value (or limit) of the series {-} + §. &c. ad infinitum is found; and in the same manner may another value (or limit) of that series be found, after writing 2m +2 foron in the equation

1+x +*3-** (n) + : which second value (or limit) will be = the hyp. log. of

I+x 2. And these two values of the series ļ – } + $- $, &c. ad infinitum correspond to the two obvious modes of lummation, or ways of collecting the values of its terms.

To infer that the series (-}+- , &c. ad infinitum, is not equal to either of those values (or limi's) but that it is = 1 + the byp. log. of 2 (=R) the quan.

*20;-+' tity that results from such a process as the above when the expression A.

itx is disregarded, is surely such a conclusion as ought pot to have place in mathema. tics! for the more terms we take of the series, the more will the aggregate of those terms differ from that imaginary sum R!

We have seen above that ihe series 1 - {+ - $, &c. ad infinitum, has two sums or limits: and it is observable, that whenever the summation of a leries is propoted whole terms are some of them positive and some negative, and they do not converge so as to become indefinitely small, if fupposed to be continued ad infinitum, the proposition will be lo vague, that it may perhaps admit of various folutions, unless ihe law of the continuation of the terms be indicated by a proper symbol. For inItance, the series ---+*+---+, &c. ou infinitum, or 5 + } 1+2+2-1--&, &c. ad infriturn, has three lums or limits: which are the limits of --++--+ (n) pon taking n equal to 40--1,479, and 4m+1; and the limits of į+1

$*&+; - - (12) upon taking n equal to 4m, 4m+1, and 47.2 +2 ; m being always an integer; and those limits are respectively equal to -6, 1-6, and 2-6; 6-1, ő, and ő +1;

Ġ being = circ. arc, rad. z, tang. 1, + ! hyp. log. of 2;

= circ. are, rad. 1, tang. 1, - } hyp. log. of 2. It does not appear that it can be any way conducive to the improvement of the doctrine of leries, to attenipt to align a certain fum to any such series as

-+-+-, &c. ad infinitum, --+*+---+$, &c. ad infinitum,

*+*-*-*+&+ - - , &c. ad infinitum, without any regard to the law of continuation : on the contrary, it is (at least in appearance) an ablundisy, to align, as the sum of such a feries, a quantity from which the aggregate of the terms of the series would differ more and more, upon increaling iher number; as is always done in pursuing the inerhod which is the fubj &t of these animadversions.- Indeed, the principle, that the sum of a series of integers may be equal to a proper fraction (upon which that method is founded) is luch, that no other than an ablurd conclukon can well be expected to follow from it!


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3 4

A EROS T-A TICS. T! THE deferiptions of aerostatic experiments, as well as an account of the

principles on which they are performed, which have been presented to the readers of this work, form, as it were, an history of this discovery, and of its progress. "To there narratives are now added a tranllation of large extracts


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