A Tale of a TUB, or The Greenwich PROBLEM.  diagram  
GREENWICH, the Twenty sixth of July, 1673.  
THe Dimensions of Mr. Peter Cooper his third Round Tun, taken in Inches by N. G. and M. D.  
Data The Diameters at the top 118 and 120.5; the Geometrical Mean 119.24 = A A.  
The Diameters at the bottom 121.5 and 127; the Geometrical Mean 124.21 = B B.  
The Perpendicular from the top to the bottom 46.5 = P P.  
After these Dimensions were taken in the dry Tun, there was (upon the Fifth of December, 1674.) water 2: 1: 5 two Barrels, one Firkin, five Gallons, put in to replete the Horizon; and then at the Extremities of the Diameter C C, on the Surface of the water, there was taken wet the Depths C D = 2 and C D = 2.  
Ques. The Frustum of Air A A C C A?  
Solu. The Frustum of Air A A C C A = 50: 3: 7  
But the Frustum A A C C A found by Art = 50: 3: 7 and the Frustum C C B B C measured by water = 2: 1: 5 is the whole Content of this Tun in Beer Measure = 53: 1: 3 not 54: 0: 3.  
The Dimensions of the said Tun as they were taken in Inches the Fifth of December, 1674.  
Data The Diameters at the top 117.3 and 120.5; the Geometrical Mean 118.88 = A A.  
The Diameters at the bottom 125.5 and 126.75; the Geometrical Mean 126.12 = B B.  
The Perpendicular from the top to the bottom 46.4 = P P.  
After these Dimensions were taken in the dry Tun, there was water 2: 1: 5 put in to replete the Horizon; and then at the Extremities of the Diameter C C, on the Surface of the water, there was taken wet the Depths C D = 2 and C D = 2.  
Ques. The Frustum of Air A A C C A?  
Solu. The Frustum of Air A A C C A = 51: 1: 6.  
But the Frustum A A C C A found by Art = 51: 1: 6 and the Frustum C C B B C measured by water = 2: 1: 5 is the whole Content of this Tun in Beer Measure = 53: 3: 2 not 54: 1: 3.  
Object. There were several other Depths as well as C D, taken in the Perimeter of the bottom of this Tun, and therefore this way of Casting up this Tun cannot be true.  
Answ. Although there were several Depths, as well as C D, taken in the Perimeter of the bottom of this Tun, yet there was no other pair of Depths diametrically opposite and equal, but these C D and C D; and if the Perimeter of the bottom (viz. the Perimeter at the Crows, I do not mean the whole Surface of the bottom) be granted to be in one and the same Plane, I do aver, that this way of Casting up of this Tun is true to any way or every way of Casting of it up.  
Object. But the Crows of this Tun are of a Snaky Form or Figure, and consequently not in one and the same Plane.  
Answ. Then there is no way afforded for any Analogy between the Diameters taken on the bottom (viz. the Perimeter at the Crows, I do not mean the whole Surface of the bottom) and those on the Surface of the water: So then, if the Crows be not granted to be in one and the same Plane, there is no way afforded to get the Diameters on the Surface of the water, but by taking of them actually; which all Practical Gaugers judge very uncertain and impracticable.  
The reason that the Casting of December 5. 1674. exceeds that of July 26. 1673. is, for that the Diameters of the bottom are not Rectangular Conjugates, and I judge may be said to be an insufficient Data, because July 26. shows a longer and a shorter upon the same bottom.  
By MICHAEL DARY.   

Printed for William Shrowsbury at the Sign of the Bible in Ducklane. 1674.