IV. QUEST. by the Rev. T. P. Irving, Newbern, North Carolina.

In inches, the head of a Pamtico fish,

Which makes, on the board, a delectable dish,
The thirty-five thousand two hundredth, will be,
Of
years circumvolving, (the learn'd all agree,)
'Twixt the century current, and when in amaze,
The world on a spring sempiternal, shall gaze.
It's body bisected, that's half cut asunder,

"You see I'm Hibernian, but smart's, I'll not blunder;
I mean, that, one half of its body's whole length,
Conjoin'd to those inches, by plus's whole strength,
Its tail will express; while its body if sought,
Is more the head, more the tail, eke more a naught,
Now tell me ye skill'd in symbolical art,

The length of this fish, and each specified part.

V. QUEST. 23, by Diarius Yankee, Bunker's Hill.

VI. QUEST. 24. by the Rev. J. Blackburn, Cambridge, England.

If a circle ABDC, circumscribe an equilateral triangle ABC; the straight line AD is equal to the sum of the two straight lines BD and DC: required a demonstration.

B

C

D

VII. QUEST. 25. by Robert Adrain, York-Town, Pennsylvania.

If any quadrilateral ABCD, be inscribed in a circle, and the diagonals AC, BD be drawn; it will be, as the sum of the rectangles BA. AD, and BC. CD, is to the sum of the rectangles AB. BC, and AD, DC, so is the diagonal AC, to the diagonal BD. Required a demonstration.

a

VIII. QUEST. 26. by G. Baron, New York.
A ray of light issues from A

given point A, and is

constantly reflected to another given point B, by a plane speculum LQM, which moves parallel to itself; required the locus of all the points Q. L

B

Q

M

IX. QUEST. 27, by Diarius Yankee, Bunker's Hill.

The cavity of our chimney, is an upright parallelopipedon, the diagonal of whose base is 60 inches; and the height of the lower side of the lintel, above the plane of the floor is 40 inches. What is the length of the longest inflexible stick that can be put up this chimney?

X. PRIZE QUEST. 28. by Wm, Green, New-York. (The author of the best solution, to this Question, shall receive a handsome silver Medal, value six Dollars.)

St. John's, in Newfoundland is in lat. 47° 32′ N. long. 52° 26′ W. Cape Finisterre, in Spain, is in lat. 42° 51′ 52" N. long. 9° 17' 10" W. and Cape Barbas, in Africa, is in lat. 22° 15' 30" N. long. 16° 40′ W. Now there is a certain point, (on the same hemisphere of the earth,) which, on the arcs of great circles, is equally distant, from each of these three places; and it is required to determine this distance, the bearings of the three places from the point, the latitude and longitude of the point, and the courses and distances from the same point to each of the same three places.

ARTICLE XI.

ANSWERS to the QUESTIONS proposed in ARTICLE X.

I. QUEST. 19. Answered by Thomas Whittaker, Harrisburg, Pennsylvania.

=

PUT a the annuity 1000 dollars, t-the time of continuance=10 years, r—the interest of 1 dollar for a year 07, and d-x-the expenditure required. Then, by the common formula, and the nature of the t-1xr+2 x trx = a-x. Whence x=

question;

2

2 a

t-1xtr242 tr +2

520-6977 dollars, and a-x=

479-3023 dollars, the annual expenditure sought.

II. QUEST. 20. Answered by James McGinness, Middletown, Pennsylvania.

The question is, "Whether 30 horses can be put into 7 stalls, so that in every stall there may be either a single horse or an odd number of horses." Let a+b+c+d+e+f+g=30; then will a+1+6+1+ c+1+d+1+e+1+ƒ+1+g+1=37, an odd number. Now, by the question, each of the seven letters a, b, c, d, e, ƒ and g, is either a unit or an odd number: consequently a+1, b+1, ċ+1, d+1, e+1, ƒ+1, and g+1, are even numbers; and, by Prop. 21, book 9, Euclid, their sum (37) is also an even number. But this last conclusion is evidently absurd; and therefore the horses cannot be put into the stalls so as to answer the conditions of the question.

No. 4.

The same answered by the Rev. Thomas P. Irving, Newbern, North-Carolina.

Sirs, Mr. Walsh, I think, must own,
One horse is useless lumber;

Unless he'll make three odds combin'd
Produce an even number.

OTHERWISE.

I think Mr. Walsh Alexander, must own
That Bucephalus sleeps out o' doors;
Unless he has skill to rectangle three threes,
That shall do what is done by four fours.

III. QUEST. 21. Answered by Ebenezer R. White, Danbury, Connecticut.

After correcting a typographical error, which, in this question, made n=7, instead of n=17; let the first of the given equations, x=a+b, be transformed to

(+1)x": from which, the first of

the general rules required is self-evident. Hence the value of x will be found as follows:

log. x=log. 127321=5.1048999

Hence x=127321 nearly; and by a similar process we obtain the second general rule; from which is found, y=126657 nearly. In the same manner, the value of a binomial surd may often be easily found, when the common methods of calculation would render the operation exceedingly troublesome.

IV. QUEST. 22. Answered by Diarius Yankee, Bunker's Hill.

The learned do not expect the commencement of a sempiternal spring: for, in nature, no cause, cacapable of producing such an effect, has hitherto been discovered.* It may therefore be said, that the learned all agree, that, should an eternal spring commence, at some future period, the distance of that period from the present century is now unknown. Hence the first six lines of this question may be considered as an ingenious enigma, signifying that the length of the fish's head is not given. Let, now, 2y=length of the body, and x-the length of the head; then by the question y+x=the length of the tail, and 3y+2x=the length of the fish. Also by the question 2y=y+2x+0; whence y=2x; and consequently if x=1, y=2, y+x=3, 2 y=4, and 3y+ 2x=8; or the length of the head, tail, body and fish are as 1, 3, 4 and 8. Therefore the question admits of an indefinite number of answers.

* See Prop. 34, Emerson's Centripetal Forces.