if the Westermoft Ship's Difference of Latitude be 36 Miles more than that of the Eaftermoft; and the Eaftermoft Ship's Difference of Latitude be a mean proportional between their Departures; it is required to find cach Ship's Courfe, Departure, and Difference of Latitude.

(7.) QUESTION the 63d, by J. T.

Brewer of Note in the Town lately fent

For his Cooper, and told him it was his Intent, That he fhould make for him a Conical Tub : Being minded withall for to give him a Rub. Quoth he, Yesterday as I happen'd to be At my Neighbour's, a Tub I chanced to fee, And oth'fame Size and Form you must make one for me.. The Width at the Top is-On Paper I wrote it, But being in hafte, in good Truth I've forgot it; Yet thus much remember, That a Hoop just fo wide As the faid Diameter is, did the Tub circumfcribe, When upright 'twas plac'd, and fixt very tight With the four Ends of the Diameters right; Befides the Height of it I found juft to be Equal to the leffer Diameter exactly:

And it held, my Friend told me, neither lefs nor more
Than Ale-Gallons Two Thoufand One Hundred and Four.
The Cooper this hearing to Work ftraightway went,
Bug could'nt make the Dimenfions fit to the Content :
So your Help at this Juncture be pleafed to lend,
And you'll much oblige your Servant and Friend.

L

(8.) QUESTION the 64th, by I. N.

ET a Spheroid Cafk, (whofe Heads Diameter being the Latus Rectum, is in Proportion to the Bungs as 5 is to 8.) contain One Hundred and Thirty Gallons of Ale; then fuppofing a Cone whofe Bafe's Diameter is 8 Feet, and its Altitude 14. to be fo cut, as that the Plain of the Section by its Rotation, about its Axis, fhould generate the Spheroid, whereof this Cafk is the Middle Zone. The Distance from the Vertex where that Plain cuts the Axis, its Inclination thereto, and the Dimenfions of the Cafk are required.

(9.) QUES

(9.) QUESTION the 65th, by E. H-x-l-y. A Dmit there be a Field in Form of a Trapezia, two of its Sides being each in Length 36. Chains: The One being the Sine of an Arch of a Circle, the other the Chord of the Complement of that Arch; and one of the unknown Sides, being the Co-fine of the first Arch, and the other unknown Side the Radius of the Circle. Required the Length of each of the unknown Sides, and the Area of the Trapezia.

IN

(10.) QUESTION the 66th, by W. W.

Derwent Ings there lies a Piece of Ground, Its Form a right-angled Triangle found: Which once I was requested to furvey, But met with Difficulties in the Way ; The River, fwell'd by long continu'd Rains, Had almoft overflown the neighb'ring Plains : 1 jul could come at the right Angle's Place, And likewife at a fall Part of the Base In the Hypothenufe a Tree there food, Whofe lofty Branches triumph'd o'er the Flood At which, from the Right-Angle, Obfervation I took, and found, that if unto my Station A Line was drawn directly from the Tree, The certain Confequence thereof would be, That the Right-Angle would precifely feem Cut into mean Proportion and Extream; The Mean it lying (if I right define). Betwixt the Bafe and the Bifecting Line. From the Right-Angle then with nimble Speed, I did in meafuring the Bafe proceed;

Soon as I found the Menfuration Sum

Did to Nine Chains and Forty Nine Links come, halted, then obferv'd the Tree again,

And found, that if a Line from it was drawn,
To this, which I my Second Station call,
It perpendic'lar to the Bafe would fall:
The Owner alfo gave me an Account,
The Diff'rence did to Fourteen Chains amount,
Betwixt th' Hypothenufe (as had been try'd)
And th' Perpendicular or fhorteft Side.

From

From the abovefaid Data I'm confin'd The feveral Sides demonftrably to find; And therefore honeft Brethren of the Quill, Pray lend the Help of your fuperiour Skill. (IL) QUESTION the 67th, by J. T--ph--m. TH Here is given the Radius of the Circle ASQ = r. (Vid. Fig. 15.) and therein are Three Circles of equal Diameters infcribed touching each other, and the greater Circle Required the Diameters of the lefs Circles, and the Area of the little Spherical Triangle H M N made by thofe Circles about the Centre.

:

(12.) QUESTION the 68th, by I. T. 'WO Towers let's fuppofe whofe Height in Feet Are Seventy Four, and Ninety Nine compleat : Their Distance likewife meafur'd on the Ground, One Hundred Fifty Feet exact is found.

If now two Ladders Foot and Foot to each

Were fet, and just to th' Top o' th' Towers fhould reach
That Point i' th' Distance Line I fain would know,
And beg that you the fame to me will fhow,
When the two Ladders Lengths join'd in one Sum
Are equal to
a Minimum.

(13.) QUESTION the 69th, by R. F. Suppofe AD (in Fig. 16.) to be perpendicular to the Plain of the Horizon, and DE parallel thereto : AD= 43 Yards, DE 47 Yards. Then upon the Point E with the Radius EC 28 Yards let a Circle be defcribed It is required to find the Inclination of that Plain as A C, on which a heavy Body will defcend in the fhorteft Time poffible from A to (fome Point in) the Circle Ccs Q

:

YE

(14.) QUESTION the 70th, by R. D.

E learn'd Adeptifts (Euclid's Sons) by whom
True Alchymistry is de facto us'd;

Not Metals, but Equations, ye tranfmute
By Grand Elixir Analytic Art.)

Vouchfafe th' fubjoin'd* Equations to evolve.

5000. 2

y= b = 3000. S

ΤΗ

"HE Editorof this Mifcellany having propofed a Queftion in the Lady's Diary, 1726, Page the 16th, about finding the Area of an Ellipfis, &c. from three Lines given in Magnitude and Pofition, interfecting each other in one of the Foci: And it being faid, in Page the 7th of the Ladies Diary, 1727, that the faid Question is unlimited; he thinks himself obliged to clear it from that Imputation, and hopes it will not be unprofitable to the Reader, as it is not foreign from his Subject. My Method of Solution is as under. In Figure the 17th, Let AEFPD be the Ellipfis, S the given Focus, H the other Focus fought; SE200. Feet, SF=313.62, SD-872.62; the Angle ESF 25 Degrees, the Angle FSD=65 Degrees. Join the given Points by drawing the Lines DE, DF, EF, which will also be given (because in each of the Triangles 2 Sides and a contained Angle are known) from H draw the Lines HD, HE, HF, and put z tranfverfe Diameter of the Ellipfis AP, SE⇒b; SF=c; SD = d; Then is HD 2-d; HE2-b; HF-2-c; as is plain from the Nature of an Ellipfis: That is, the Problem amounts to this, to find a Point H, from whence the Lines HE, HF, HD being drawn, they fhall have given Differences: Which Problem you may find propofed by me, Queftion the 167th in the Ladies Diary 1732, and analytically folved by me, &c. at Pages the 9th and 10th Lad. Diary 1733; by a Quadratic Equation. According to which Method, the Lines HE, HF, HD will come out 8on.; 686.38 ; and 127.38 and the Length of the Tranfverfe Axis = 1000. The Method of Operation here will be the fame as there; only in the prefent Cafe the Point H falls without the given Triangle EFD, and in the Question propofed in the Diary it falls within. And now HF, HD, being found, and DF before known =793.0 d 1

we fhall find the Angle HFD to be equal 5: 23 and ad

d

ding it to the Angle SFD=93:59 gives the Angle

d SFH 99: 22; again SF, FH and the contained Angle being known, we may find SH the Distance of the Foci to be 800.Feet, and confequently the Semi-Conjugate Axis of the Ellipfis BK is 300. Feet, and the Conjugate 600.

Laftly,

A Question in the Ladies Diary 1726, Answered. 47

Laftly, for the Put SAK=12=500. Km=p,=200. 2BK =m=300. Kp=x,

Length of the
Ditch qt,

Ap=n+x; and Ppn-x; and mpx- p.

Then is

By the Property of the Ellip fis As AK2 (n2): BK2 (m2)

m2 n2-m2x2

(m2

722

But

√mp2 + pt2 =mt. which by the Question must be a √n2x2 - 2n2px+n2p2+m2n2—m2x2

Minimum. Ergo

222

in Fluxions to nothing. n2xx—n2px—m2xx = 0;

n2 p

and by Tranfpofition and Divifion x=

2312.5

therefore mp 112.5 Feet, and mt 259.8; and the Angle of Inclination mt 64d: 21' its Nat. Sine = .90145b; its Cofine.43287g; and finally putting Kr; mr will be = p-x; and the other

Symbols remaining as before gr2 =

But As gp-x: h

2

m2 n2-m2x2

222

bp-bx

g

~=qr. & b2p2 — 2b2 px + b2x2

g2

h2p2—2b2px+b2x2

g2

369.4% 17769. And x 184.716345.09= +127.8; here x=56.9 and mr = 143.1 And qm=330.66; Wherefore the whole Length of the Ditch is 590.46 Feet.

N. B. I alfo had defign'd to have given in this Place a Solution to the Exponential Queftion propofed by Mr. R. Fearnfide, in Page the ift, Ladies Diary, 1732, among the Latin Enigmas, and which has not yet been answered in the fucceeding Diaries, viz. xxxy; and g+y=x; to find the Values of x and y. But for want of Room am obliged to omit it. Only I fhall obferve that x 1.7481 and y. 90 58. 9058.