tho' I don't fee the Neceffity of the Querift's mentioning "a Tun a Minute.'

Mr. John Rickerby, of Wooburn, Bucks, fays, he has spent great Part of his Life among the beft Paper-Mills in the Nation; and obferves, that a Swing-Wheel, which receives its Force of Water eight Inches above its Breaft-Center, exceeds an Overr-fhot Wheel; provided the Current of Water and Fall are alike : And fays, tho' Milwrights differ in their Opinions concerning the true Pitch of the Water-Wheel, that this Opinion of his own is true. He speaks of a Pen, to give the Difcharge of Water the greater Force, at that Part of the Fall where the Water-Wheel receives its Impetus, or depreffing Force; and computes the Diameter of fuch a Wheel to be twenty-fix Feet and eight Inches; but on Principles a little doubtful.

XI. QUESTION 357 anfwered by Mr. James Hartley, of Yarum,

ET AB be the Length of the whole Seale

equally ascend from d towards c; so that de 2 Let M be the Place of the Water and Mercury at a middle State of the Atmosphere; then per Queft. MH MK=55. Put d Diam. greater Tube,

=

y= Diam. leffer, and let c i =x; then+xx d2

21 S

4

2
+xxy; and, as Mercury is about

fourteen Times as heavy as Water, 14 x X d2

218

4

xxy, whence x. Now, if in

52

H

K

ftead of x, in the first Equation, we fubftitute its Value, and affume

7 137

1, we get, by Reduction, y 2 = whence y,243 ; if therefore d be taken at Pleasure, it will be as 1,243::d: y. W. W. R.

The Rev, Mr. Baker's Solution is thus: s to 1 the specific Gravity of Mercury to that of any Fluid in a leffer Tube; r to the Ratio of the Tubes; a given Variation in the common Barometer, and x the correspondent Variation in the leffer Tube. Then r2:x:: 1 2 : x Variation at the upper Surface C of the greater Tube (being reciprocally as Squares of the Tubes Diameters); the whole Variation r2 x + x of the leffer Tube= the Variation of the Mercury's Surface

Surface at C, in the greater Tube, the fame with that of the Water in

the fame Place, is = Variation at B, of the fame Diameter.

2 T

The whole Variation of the greater Tube B G =

of the Mercury and Water together upon the Air at K is from the Length of the Tubes, the contained Fluids in GLC and Cm I being fufpended in Equilibrio; the Variation in the Preffure of the different 2xxxx 2 x

Columns depend on their Weights

2

and : Say, : 1 ::

Column of Mercury of the fame Length;

a

Variation in the common Barometer; whence x =

2.

2 237 - I

i. e, x::: r2s: 2s - go He w 2 I; and per Quest. 10:1:: 14 r :28 I, (here s 14 nearly) whencer 33541, and the Diameters of the Tubes are as 3.3541 to 1. W. W. R. COR. I. If 1, 2, 3, 4, 5, 6, &c. and r√

✔✔✔T, &c. correípondent; or if s=

251, the Variations in this Barometer will thofe in the sit I

common Sort..

COR. II. If s=1, 2, 3, 4, &c. and r≤√ I, √ 3, √ 5, √7,

&c. or if s

infinita. Hence,

2

orr = √25 --- 1, the Variation will be

may have any

COR. III. The Scale of Variation in this Barometer affignable Ratio to the Variation in the common Barometer. Mr. J. Williams fays, that Mr. Rovning (in his Compendious Syftem, Page 112) determines the Ratio of the Variation of x, in the leader

XI. QUESTION 358 answered by &IAO-pwn. being the tranfverfe, and c the conjugate Diameters of the inner Ellipfis, of a "folid elliptical Ring,

of p Inches Diameter; then

whofe Circumference is circular, and cp will be the tranfverfe

and conjugate Diameters of the Ellipfis paffing thro' the Middle of

the

the Ring, whofe Circumference is in the Center of Gravity; which Circumference puta; then, fince the Solid or Surface generated is equal to the Product of the generating Plane, or circular Line, refpectively multiplied into the Way made by the Center of Gravity, therefore 7854 ppa is the Solidity, and 3,1416 p a the Surface, of the elliptical Ring, required,

The fame answered by Mr. John Hartley, of Yarum: If d be in the Question, and m 3,1416, also A, B, C,

taken =

c. the next preceding Term, in the following Series, then

m2 p2t

N. B. The above Series are taken from Mr. Emerson's Excellent Doctrine of Fluxions, Page 174.

There is a fmall, tho' ufeful, Introduction to the Doctrine of Fluxions fold by J. Noon, in the Poultry, which might be called The Child's Guide to Fluxions, proper to be read by Beginners before they engage with Mr. Emerfon's Teftament of the Art.

XII. QUESTION 359 anfwered by Mr. John Honey, of Redruth, Cornwall.

P

UT a and b 42 and 26 Inches, the tranfverfe and conjugate Diameters of the Hoop above; and c and d 48 and 29, the Dimenfions of those below; alfo m 12, the Fruftum of the Ellipfoid's Altitude; and n = 18 Inches, the elliptical Cylinder's Altitude: ab + c d + √ a b c d x m 882,36

Then, by a known Theorem,

= 50,54

Wine Gallons, the Content of the Hoop's Concavity; and

1

cdn

294,12

= 85,18 Wine Gallons, the Content of the cylindrical Concavity; whence the Concavity of both = 135,72 Wine Gallons.

Mr. John Wigglesworth anfwers it thus: Let à 42 Inches, b=26 tranfverfe and conjugate Diameters above; t=48, c=29 tranfverfe and conjugate Diameters below; b = 12; p = 18, and m,2618, then the Content of the whole Concavity= mbxce+be+ab+ac+3micp

Gallons.

t

231

= 135,7416 Wine

The Solutions by Mr. T. Cowper, Mr. J. Hartley, Mr. J. Adams, and Mr. Cottam at his Grace the Duke of Norfolk's, agree the fame; but they miffed the Secret of Half Mifs Pelly's Solidity, which being

taken

taken from the laft-found Concavity leaves the Concavity fought. Mr. Adams obferves, the Fruftum of the Ellipfoid may be reduced, with Advantage, to the Fruftum of a Cone, and the elliptic Cylinder to à round Cylinder.

The PRIZE QUESTION answered by Mr. F. Holden, at Wefthouse, near Settle, Yorkshire.

AKE a Piece of Wood or a Stone, of a known Superficies,

dipping it into a Veel full melted Tallon,

Pullies

trying the Weight of the Tallow and Dipping-Vessel in a scale before and after dipping, know the Quantity of Tallow, in Weight, taken up by the Piece of Wood or Stone. Then take about 18 or 20 modern Mathematicians, (the more the better) ftrip them stark naked, and fufpend them (like Abfaloms) by the Hair of their Heads, as Chandlers bang their Candles, or elfe by foft Bandages under the Chin and behind the Nape of the Neck, fo that they may be raised or let down without hurting; dip them alfo, one by one, in the fame Veffel where the the Wood or Stone was lately dipped, and mark the Tallow they all take up, by weighing the Vesel and Tallow, before and after they are all dipped, (keeping the Tallow juft melted and of an equal Warmth): Then fay, as the Quantity of Tallow, in Weight, taken up by the Wood or Stone, is to the known Superficies of either, fo is the Weight of Tallow taken up by all the Mathematicians to the Superficies of all the Mathematicians. But, by all Means, take Care that they are kept naked, 'till they are shivering, and almost as cold as the Wood or Stone itself, before they are dipped, elfe this Proportion will not bold good.

When they are all dipped, well scoured with Soap, and cleanfed from the Tallow, let them be weighed, (or they may be all weighed before dipping) and fay, as the Weight of them all in Pounds is to the latefound Superficies of them all in fquare Inches, or fquare Feet, fo is 160 Pounds Weight to the Superficies of the modern Mathematician required to be known, ́( 14 fquare Feet, nearly, as we find by another Method). W W. R.

I

N. B. I don't doubt but there is a fufficient Number of Mathematicians in the Society of which Mr. Pedant is Prefident very proper to make the Experiment upon; and if, by their Means, this important problematic Problem can be affected, it feems to be the only Way of rendering our modern mathematical Conjurers of Service to Pofterity; as it will redound an Honour and Satisfaction to the Discoverer of an Invention fo important and useful to the Public. F. HOLDEN.

REMARK. Immortal. Honour! our Contributor has gained and claims the Prize, without a Competitor! Archimedes facrificed an Hecatomb of Oxen to Jupiter ('tis faid) for infpiring him with the Invention, when he first found out a Method to measure the Solidity of all irregular Bodies by Immerfion: And is not this Discovery, that reduces the Irregularity of all Superficies to one uniform Rule of Menfuration, a vaft and noble Improvement in the Juperficial Science, by Immersion in a clammy Fluid ? Of which, Mud, Paint, Oil, and Lamb-black,

Lamb-black, may be a farther Improvement, as Gum-Water may ferve for nicer Experiments.

But (we baving only put Weight for the Menfuration of the Tallow taken up by the Mathematicians, and set down the Superficies of the known one, 141⁄2 Feet) we fhall not prefume to claim any Share in the Invention, as all Property thereof belongs to the original Author and Proprietor.

We fhall only take Liberty to add, that a Paper Cafe being spread open, in which the human Solid, or any Part of it, has been inclofed, (like a Clock papered from the Duft) will nearly determine fuch Superficies: Which, perhaps, may be alfo determined by affuming a regular Form of Solidity and proportionable Superficies, of the fame Specific Gravity of Substance with the irregular buman Substance and Form.

The last Year's PARADOXES answered.

1. PARADOX anfwered by Mr. Richard Gibbons, of Plymouth.

T

HE Fork was as the Steelyard, Roger's Shoulder as the Fulcrum, fuftaining the Burthen, between the two Powers, acting at both Ends of the Fork.

II. PARADOx anfwered by Mr. Edward Griffiths, of Ellefmere, Shropshire, and others.

Cafe 1.

•A

Piece of pliable Metal being doubled, by applying a round
File to the doubled Edge, and filing a balf-fquare Gap,

on opening the Metal, a fquare Hole will appear.

!

Or, the ingenious Mr. Cato informs us, that, if two Carners and an Edge, at the End of a Mifer's Iron Cheft, be filed away, with a round or any other File, there may be an exact square Hole left.

2

Cafe 2. A cylindrical Body being cut obliquely, the Plane of the Section will be an Oval; and, confequently, a round Body, fituated obliquely in an oval Hole, will completely fill it. Archimedes.

A

RTIFICIAL TEETH fet in fo firm, as to eat with them, and fo exact, as not to be diftinguished from Natural: They are not to be taken out by Night, as is by fome falfely fuggeftéd, but may be worn Years together; yet they are fo fitted, that they may be taken out and put in by the Perfon that wears them at Pleafure, and are an Ornament to the mouth, and greatly help the Speech. Alfo Teeth are cleaned and drawn by Samuel Rutter and William Green, Operators, who apply themselves wholly to the faid Bufinefs, and live in Racquet Court, Fleet-Street, London.