7.155, and AC= 8.944. W. W. R. With which Mr. T. Allen's, of Gofberton School in Lincolnshire, and Mr. Widd's Solutions agree.

Mr. Thomas Cowper, Teacher of the Mathematics, at Wellingborough, computes thus: Put x= Sine, and y Cof DA B DOH; then 1:4::: 4x0I=0H. And 1:2::y:2y

x 2

=OH.. ===,5, the Tang. of OA I = 26° . 33′ · 54′′,2; whence A B = 5.3665626; B C = 7.1554168; and A C 8.944271. W. W. R.

Mr. Cottam, at his Grace the Duke of Norfolk's, answers it elegantly in the fame Numbers; as likewife does Mr. John Williams, Mr. James Hartley of Yarum, Mr. William Bevil, Mr. Robert Butler, Mr. Holling fworth, Mr. F. Holden, Mr. Richard Gibbons, Mr. Charles Tate of Hull, Mr. John Adams, (who conftructs the Solution) Mr. William Honnor, our old Friend Mr. John Ramsey, (Mathematician and Enigmatist) Mr. John Himpfon, Mr. Alexander Roe, Mr. Jofeph Hilditch, Mr. Brownbridge, M. J. Hirft, Mr. Stephen Hartley, Mr. Honey; and, in a beautiful and incomparable Hand-Writing, Mr. Thomas Huntley folved the fame in Latin Diction; as he has done many other Problems in our Diary, fit for Pofterity to look at! putting at the Bottom of his elegant Latin Letter, Dabam Burfordiæ, in Comitatu Oxonienfi, pridie Kalendas 4ti Menfis, Anno 1752.

Mr. Jofeph Orchard, Mr. Thomas Allen, Mr. Charles Smith, Mr. Thomas Cowper, Mr. James Terey, Mr. James Hartley, Mr. William Bevil, Mr. Holden, and fome others, add to the Value and Correctness of their Performances by Propriety of Diction, and Hand-Writing, being clear, full, and concife; from whofe Compofitions we find Pleasure to collect; as we do to encourage all useful and correct Correfpondents in general.

VI. QUESTION 352 answered by Mr. Henry Watfon, of Golberton School in Lincolnshire.

x the Number of Fruit-Trees unknown; then

PUT =

x

x x 20

12

++50=, and, by Transposition, - ++ 2

Number of Fruit-Trees required.

The fame concifely and elegantly anfwered by Mr. John Fife,

44 11

I

Therefore T 50 Trees; confe

C 4

quently

quently the Whole 600 Trees. Now

300 Apple-Trees, = 150 Pear-Trees, 7 = 100 Plumb-Trees, whofe Sum = 550, to which adding 50 Trees, the Sum-Total = 600 Trees, the Proof.

Mr. John Nicholfen, of Rochefter, answers it in the fame concife and eafy Manner: As did Mr. James Hartley of Tarum, (who folved all the Problems) Mr. F. Holden, Mr. Hollingworth, Mr. Peer Brooke, Mr. Thomas Trimingham of Hull, Mr. Richard Gibbons, Mr. John Adams, Mr. John Williams, Mr. John Ramsay, Mr. Jofeph Hilditch, Mr. Robert Butler, Mr. William Cottam, Mr. John Hampson, (who alfo fent the Times of Eclipfes for Leigh) Mr. A. Brook, Mr. T. Cowper, Mr. Brownbridge, Mr. John Potter of Duke's-freet in Southwark, Mr. John Peachy, Mr. Honnor, Mr. Henry Watson, and Mr. Thomas Allen of Gofberton School in Lincolnshire, and others.

But Mr. Thomas Huntley, of Burford, putting 12 x = Trees; then 6x+3x+2x+50 — 12 x; whence x 50, and 12 x = 600, required.

VII. QUESTION 353 anfwered by Mr. Henry Watson, of Gofberton School in Lincolnshire.

cc, then
d

4

dd

x++ dx x = aa, or

aa; whence xx= +√ -aa81, and

dd

4

*=13. 2. E. F.

= 9, whence y = 5, ∞ = 15, and

Moft of the Gentlemen before-mentioned folved this Queftion, and particularly Mr. Thomas Allen and Mr. John Williams; all agreeing with Taptinos, the Propofer,'s Solution: And Mr. Thomas Huntley folved it in Latin.

VIII. QUESTION

VIII. QUESTION 354 anfwered by Mr. Terey, of Portsmouth.

Trapezium D F GE, to be a Maximum. In Fluxions and reduced 8 a (let b be what it will). Now, fubftituting this Value of x in F B above, F B= b, and alfoa, for b (per Quest.) FB ➡ 4: But DC+FB x B C =

2

12

8

8

a + 3 a × — a = — a × 3 α a = 16 And a = 18

A C,

:4; the Area of the Parabola

bxa = 27 a b.

Parabola is to the greateft infcribed Trapezium, as 9 to 8. W. W. R.

Mr. James Hartley folves it thus: The Ratio of the Abfciffa and Semiordinate being as 3 to 2, the shortest Side of the greatest inscribed Trapezium will be the Parameter, and its Area will be to the Square, whofe Side HG will be double the Parameter. Put 2 x

DC, then 3 x AC, and from the Nature of the Curve

DC 2

AC

4x FG= 3 DCBG DV=

8 x

x

And AB= but AC AB=BC=-·

And

3

3

8x 3

256; whence x 6: Confequently

DE 24, F G = 8, and FD-GE=√ 256 +64 = 17,8885, and lastly, A C=18. W. W. R.

The above Solutions are fhort and elegant, as the Propofer's Solution; as is likewife the Solution by Mr. Cottam, at his Grace the Duke of Norfolk's, and those by Mr. Charles Smith, Mr. Jofeph Orchard, Mr. F. Holden, and feveral others, which 'tis needless to publish, being of a Species with the two Solutions above, fufficient to fatisfy the Curious.

IX. QUESTION

IX. QUESTION 355 anfwered by Mr. T. Cowper, Teacher of the Mathematics, at Wellingborough.

FIRST,

I

6797.4

69,5

970.48', the Distances of the two Places,

39′.

in the fame Parallel, Half of which 48°. 54. By Spherics, As Sine Diff. Long. 72°. 7'. Sine the Dist. = 48° : 54′ :: Rad. Sine 52°. 24′ comp. Lat. Hence the Lat. = : 37° Again, as Radius: Cotang. Lat. ( 37°. 39′) :: Cof. — Diff. Logg. (720.72 Sun's femidiurnal Arch: Tang. Sun's Declination 21°. 42′ almoft; anfwering to the 29th of November and the 12th of January, alfo the 29th of May and the 14th of July, N. S. W. W. R. Mr. Peter Brooke found the fame, as did Mr. Jobn Williams.

Mr. F. Holden folves it thus:

6797,4 Miles
69,5

= 97°. 48' . 15′′, the Dift. of the Cities in Degrees. If a Perpendicular be let fall from the Pole, it will bifect that Dist. and also the Diff. of Long. Therefore, by oppofite Sides and Angles, as Sine Diff. Long. 72° . 7′ 11⁄2 : Sine Dift. 48°. 54: Rad. Cof Lat. 37°. 38'. 38". Now, the Sun's femidiurnal Arch Diff. Long.=72°. 7′ 2• Therefore by right-angled Spheric Triangles, as Tang. Lat. Rad.:: Cof. 72°. 7: Tang. Sun's Declin. = 21° . 42'.

I

2

I

I

But if the apparent Time of Rifing to one City and Setting to the other be required, Say, as Sine Dift. Cities 48°. 54: Sine Sun's Refraction 33' : Rad. 43′ . 48", which being added to the abovefound Declin. gives 22°. 25′. 48" Declin. at the Time of Rifing and Setting required.

Mr. James Hartley, of Yarum, folves it in the very fame Magner, and determines the Situation of the Cities upon the Globe thus,

29th of May and 14th of July, the Sun rifes at Japan when he fets in Spain.

1753 12th of January and 30th of November, the Sun fets at Japan when he rifes in Spain.

And vice verfâ. With which Mr. Richard Gibbons agrees, and Mr. Thomas Allen nearly.

Mr. Cottam, at his Grace the Duke of Norfolk's, determines the fame Latitude, and Days of the Year, very near; and properly obferves, that, in the Question, there fhould have been. expreffed, on what Days, instead of on what Day, in the Year, &c. But, we obferve, as the Declination can feldom, if ever, correfpond with the true Time of Sun-rifing or fetting in the required Latitude, there is no Propriety in this Sort of Queftions, which only admit of Answer near the

Truth.

Truth. And, for the fame Reafon, the 349th Queftion (where the Days of Sun-rifing at Petersburgb and Jerufalem, at the fame Inftant, are required) contains the fame geometrical Abfurdities; fince the required Declination is hardly ever poffible to bit the true Time of Sun-rifing in both thofe different Latitudes; the Declination being fill variable, every Moment of Time. And, if the Sun's Declination be fuppofed the fame for the Space of twenty-four Hours, in this Sort of Queftions, ftill the Declination and Time of un-rifing, on a particular Day of the Month, cannot exactly correfpond, according to Computation of the Sun's Place for that Day, at Noon, or Time of Rifing, by Aftronomical Tables.

2

X. QUESTION 356 answered by Mr. William Cottam, at his Grace the Duke of Norfolk's.

If the Water be conveyed by the Trough B B, then 'tis called an Under-fbot Mill; and the greater the Diameter of the Wheel, the greater will be its Force, and confequently will have more Force than the Over-fhot Wheel.- And, on fecond Confideration, I make the Force of the Water drawn into the Radius of the Over-shot Mill a Maximum, i. e. 1 x √2- x, a Maximum; whofe Fluxion 2 -3x x

2axx

2

2 =0, being reduced x=3a=103, and therefore

21,25 Feet Wheel's Diameter.

But Mr. James Hartley, of Yarum, folves it thus: Let DP= 16 Feet a; 16 Feet C; x" Time of Defcent of the Water from D to r; then cx2= Dr, and 2 cx = Velocity at r.

But

a-cx2

2

Semidiam. of the Wheel, which multiplied into the Velocity is—a cx — c2 x 3, and its Fluxion made o, and