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Qu E TI on the Tenth by J. T.
THEN tir’d with Bufiness, or perplex'd with Care,

Or minded am to breathe in purer Air ;
I to my Garden straightway then retire
To footh my Cares, and Nature's Works admire :
Here the Carnation and the ballaful Rose,
Their Virgin Blushes to the Sun disclose :
There the chaft Lilly rises to the Light,
Unveils her snowy Breast, and charms the Sight.
Here a Mxandring. Rivulet gently flows,
And a new Heaven in its fair Bosom fhows,
Inviting by its Murmurs soft Repose.
Two lofty Trees within the Garden stand
(Upright and freight) which o'er the rest command :
Between each Tree when measur'd on the Ground
In a streight Line, just forty Yards are found,
The higher Tree does fixty Yards contain,
Thirty the lower, ( both stand on a Plain :)
Betwixt the Trees a Fountain plac'd must be
But fo ; that when from it to th' Apex of each Treo
Two Lines are drawn, the Angle they contain,
Must be the greatest possible it * can.
He who by Fluxions resolves this Question true,
And gives a Geometrick Construction too,

To him I'll own that Thanks are justly due. 00000000050001000009099900

QUESTION the Eleventh, by J. T.
R. EDMUND HALLE Y in his Cometography,

says, That the Comet which appeared in the Year 1686, (which he supposes to move in a Parabolic Trac jettory,) its Perihelium Distance from the Sun was 32,5, fuch Parts as the mean Distance of the Earth from the

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* i e. The greatest Angle that can be formed by two right Lines drawn from any point in the line of Diffure, so the Vertices of the said Tree,

28

Arithmetical Questions to be Answered.

Sun is 1000. Suppose now the COMET to be in such a Part of its Orbit G, that the Area GPS is equal to 492916 za fuch Square Parts ; and letting fall the Semiordinate G B upon the Axis, its required to draw the Line B M to such a Point of the Curve, so that the Angle contained by the Line B M, and a Tangent drawn to the Point M, ('wiz, the Angle B M R) may be a maximum.

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To the above I will add, two more Questions which were some Years ago proposed in the LADIES DIARY ; the firit was there given unlimited, and the latter never had a true Algebraic Solution given to it hitherto ; for though we commonly reckon a Question folu'd when it is brought to an Equation, wherein there is only one unknown Quantity, yet in fome Cafes it is otherwise, viz. when it happens to be an Exponential Equation as

3C = b. we then are obliged to have recourse to other Methods, the common Rules of Algebra here failing us. The first of the Questions I have limited, by fixing ir to a particular Latitude, and the latter of 'em will, I doubt not (now Algebra and the higher Geometry are better known and understood) be truly answer'd by seve ral directly, without guessing it by repeated Trials,

QUESTION

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QUESTION the Twelfth.

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This is d.gg. L.O.

IN
N the North Latitude of Fifty Three,
As I with

some Ladies was drinking of Tears
The Room it was pleasant, and the First Day

[ of May.
I happen'd to applaud their bright Genius and Wit,
Yet thought Mathematicks. for them not fo fit

3
But nettld at this was a learned brisk Lass,
And the Sun shining plain on a Specular Glass,
Totho Top o'th' Eaft Corner of the Room came the Rays,
Let's see by your Learning ? The answering prays ;
This Glass is i'th' Middle o'th' Wall I do find,
Horizontally plac'd, which stands on th' South Wind.

The Area o'th'End of the Room I here show, [80 Sq. Feet ]
The whole Room's Dimensions from hence I wou'd know
With the Time of the Day? But I'll tell you below,
That the Height of the Sun is exactly the same
As his Azimuth, I truly do find hy the Beam.
A Dish of the Best Tall then be your due,
If you to this Quere a Solution give true.

2000

2.21. L. Diary

QU E'S T ion the Thirteenth.
A

Gentleman as he did ride,

Near to a pleasant Common Side,
Some Shepherdeifes chanc?d to meet
Driving their Flocks, whom he did greet :
God speed you well, and may you be
As happy as you're fair, quoth he!
Prosper your Flocks, and may they thrive,
Tell me how many Sheep you drive.

ONE of the Damsels straight reply'd,
Sir
you

shall soon be fatisty'd ;
For if the Flock we should divide
Amongst us equally, each share
Is twice the Number we Maids are ;

E

}

But,

But, if for One of us you do
Count one Sheep, for the next count Two;
For the Third, Four ; for the Fourth, Eight';
So doubling at each Maid aright :
At the last Maid, the Sum would be
As many as the Sheep you fee
From what I've faid, I make no doubt,
You foot will find the Numbers out.

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QUESTION the Fourteenth, by J. T. IT is required to find a Curve, the Sub-Normal of

which is equal to an invariable Line. a.

I Mould here have concluded, had not a small Treatise wrote by one T. Baxter, accidentally fallen into my Hands; in which he pretends to sày (not indeed to prove, for there is not a Demonstration in the whole Book) that he has found out the Quadrature of the Circle in finite Terms, viz. that if the Diameter be equal to 1; the Circumference is 3.0625. the chief Reason, if I may so call jt which he gives for his Affertion is, that there may be several Figures different in Form, yet equal with Regard to their Areas; and if so, why may not a Circle and * Square be also. This though it be very true in fome particular Cafes, cannot be affirmed of Universals ; we grant that a Square whose Side is 12. is exactly equal to à Rectangle, one of whose Sides is 16, and the other 9. Or that a Parabola, the Absciffa of which is 60, and the corresponding Ordinate 40, is equal to a Rectangle, one of whofe Sides is 80, and the other 20, because they are demonstrable ; but we cannot say the fame of a Circle and a Square. Again, fupposing it were poffible to find the Ratio of a Circle's Diameter to its Circumference in finite Terms, yet furely Mr. Baxter, has not found it ; who says they are in Proportion as I. to 3.0625 this will appear falie to any one who con sults Page 348, of Ward's, Young Mathematicians Guide, where he proves,

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That if a regular Polygon of 258280326 Sides be inscribed in a Circle, whose Radius is Unity, its Periphery will be equal to 6.28318530717958 : But the Circumference of the Circle must neceffarily be greater than that of its infcribed Polygon, confequently the Diameter of a Circle is to its Circumference as I to 3.14159265358979 nearly. I shall annex to this another Demonstration, which is deduced from the Principles of Fluxions. Let AD= x; and CD be the Sine of 30 Degrees = g (=.5) Radius equal to Unity, (for í fuppose that every one who knows what a Circle is, knows also that the side of a regular Hexagon infcrib’d in a Circle is equal to the Radius ; but the right Sine C D is cqual to half the Side of the Hexagon confequently = .5) now by the Property of the Circle

yy which in Fluxions is 2.4 -- 2xx = 2y; and x =gy

: This Squared will give

XXC

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نو و و

=?

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• Equal to

3

(because 230 + 1

уу by the Equation of the Curve 2x* -- ** = 99.). The Fluxion of the Arch AC is =

oc? t you?

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