= 1: the truth of which propositions is well known; and hence the definition itself, is universally correct. Indeed, when powers are thus defined, these propositions are self-evidently true, independent of the demonstrations here given: but on the contrary, when powers are considered, as generated according to the common definition; the same propositions, (notwithstanding the demonstrations of modern analysts,) must inevitably appear, as incomprehensible paradoxes. For, if powers be the successive products, resulting from the continual multiplication of any number into itself; such multiplication, must be essential in the production of a power; and where there has been no such multiplication, there can be no power. So long, therefore, as the common definition is retained, it will be absurd to consider a number as the first power of itself. For by that definition any number x multiplied by itself, gives the first power of x = x2 = x x x; and when this multiplication by x, is abstracted we have xx, which is a manifest absurdity, and utterly repugnant to the well known nature of logarithms. Further, since in adhereing to the same definition it has been shewn, that, xo = x; it would therefore be impossible for x, to be universally equal to a unit. But we know, from the nature of logarithms, and also, from other principles, that x, is actually and universally equal to a unit; and hence we infer, that, the inexplicable mysteries, which have long been attributed, to the nothingth powers of numbers and quantities; originated, in a wrong definition of the word power.

From what has here been said on the subject, it is plain, that, the powers of any number, form a geometrical series from unity, whose ratio is the number itself; and that the exponents of such a

Beries of powers, form an aritmetical series, from whose common difference is a unit or in other words, that, the exponents of the powers of any number, are a certain system of the logarithms of their corresponding powers. And hence it follows, that, all the operations of numbers, performed by logarithms, may in a like manner, be done of powers, by means of their exponents. Also, since x° = 1, whatever be the value of x; of consequence, in every system of logarithms, the logarithm of 1 = 0.

ARTICLE X.

NEW QUESTIONS to be answered in the next Number

I. QUEST. 19. by Niel Gray, New-York.

Out of an annuity of 1000 dollars per annum, for 10 years, the first payment being due one year hence, the owner desires to know how much he may spend a year; so that his annual savings, with the simple interest arising therefrom, at 7 per cent. per ann. may, at the expiration of this annuity, amount to a sum whose interest at 7 per cent. per ann. shall be equal to the yearly expenditure required.

H. QUEST. 20. by Alexander Walsh, New-York.. It is required to determine 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.

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

Given x = a + b", and y = ✅✔✅aTM — b”, supposing r = 5, m = 9, n = 7, a = 684 588 and b=24.5632; required a general rule, for finding the values of x and y, by a table of common logarithms.

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.

gentleman a garden has, In area, just an acre;

In form a circle, on whose verge,
Was sat John Bulla quaker;
Whose head oft-times, it must be
owned,

Was teeming with strange fangles
And horticulture studied less,
Than circles, lines, and angles.

The third I'd plant outside the
fente,

For this thou seest grows in it ;
Just 90 feet from that whose boughs
So much frequents the linnet;
But so thatwhence soe'er thou view'st
Them from the garden s ring,
hey Il equi-distant seem--if not---
May I ne'er be a king.!

Here, master, look, thee, on ex-John', master knowing what he

claim'd,

Three trees thou seest--- blunder Two trees I mean.I would have

three;

Pray what excites thy wonder?

mean't

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

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.

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

he courses and distances from the same point 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 a-x=the expenditure required. Then, by the common formula, and the nature of the t-1xr+2 x trx = a X. Whence x= 2

question;

2 a

t-1xtr2+2 tr +2

-520-6977 dollars, and a-x=

479.3023 dollars, the annual expenditure sought.

II. QUEST. 20. Answered by James M'Ginness, 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+f+1+g+1=37, an odd number. Now, by the question, each of the seven letters a, b, c, d, e, f and g, is either a unit or an odd number: consequently a+1, b+1, c+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.