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A Tunn in the form of the Frustum of a Pyramid, whose Bases are in the form of a Pentagon,
The three Coefficients found in the last Ex∣ample, viz. 〈 math 〉〈 math 〉
For Ale Gall.
〈 math 〉〈 math 〉 The Answer = 6099.46920 Ale Gall.
Let another Frustum of a Pyramid of the same Bases and Altitude, stand upon its grea∣ter Base, and the Content in Ale Gallons be demanded.
The Coefficients so found and divided as be∣fore directed, are as followeth:
〈 math 〉〈 math 〉The Answer = 6099.45120
If 50 Inches of the Perpendicular in the first Tunn be wet, and 10 Inches in the last, what is the Content in Ale Gallons?
The first.
〈 math 〉〈 math 〉
The last.
〈 math 〉〈 math 〉
Proof.
〈 math 〉〈 math 〉
A Tunn in the form of a Frustum of a Cone, the Bases being alike and alike situate, as in the precedent Examples, the Dimensions being,
p)A−B(=d.
〈 math 〉〈 math 〉
These three divided severally by the Divisor proper to a Cone, as in the Table mentioned, viz. 359.05 they quote: 〈 math 〉〈 math 〉
〈 math 〉〈 math 〉 The Answer = 2671.0440 Ale Gall.
And that the young Gauger may not be ob∣liged to Dray-men to repleat the horizon with liquor, of such Tunns whose Bases are not posi∣ted parallel thereto, (as indeed most are not be∣ing made with a Drip or Fall) let him take this Example, a b c d e f g h m x, a Cone or Pyramid.
Having the length of each Line in this Dia∣gram, and the Content of the whole Cone or Pyramid in Cubick Inches, Gallons, or Bar∣rels, &c. the Quantity of the Hoof h c b d is found by this Analogy:
As the Cube of the Line a c, to the whole Solidity:
So is the Cube of a Geometrical Mean between a c and a h, to the Content of the Cone or Pyramid cut off:
Which subducted from the whole, the re∣mainder is the Content of the Hoof.
Some of the Lines being given, the rest are to be found.
Example.
There is a Tunn taken as the Frustum of a Cone,
Admitting the Base were raised 3 Inches, as the Line c h, it is then necessary to take another Diameter between c h and e f to find c d.
To find a b the Cones Axe.
As 18, the semi. diff. of Diam. co. ar. 8,744728
to 72, the semi-diam. at the Base: 1,857332
So is 60, the Tuns depth, 1,778151
to 240, the Cones Axe.〈 math 〉〈 math 〉
Log. of the Sum of their q. 〈 math 〉〈 math 〉
The Logarithm of ac 250.567 2,398924
The Logarithm of ab 247.567 2,393692 〈 math 〉〈 math 〉
A Geometrical Mean 2,396308
To find the Content of the whole Cone.
The Square of 144 4,316724
⅓ of ab = 80 1,903090
359 &c. complement arithm. 7,444842〈 math 〉〈 math 〉
Content in Ale Gallons 4620 ferè 3,664656
The Line a c, the Line a h, and the Content being known, to find the Content of the Fall c b d h.
As the Cube of ac, co.ar. 2,803228
to the whole Content: 3,664656
So is the Cube of the Mean, 7,188924 〈 math 〉〈 math 〉
to the Content of the Cone cut off. 3,656808
〈 math 〉〈 math 〉 82.6 the Content of the Fall.
Or thus:
As the Cube of ac, co. ar. 2,803228
to the whole Cones Solidity: 3,664656
So is the Cube of ah, 7,181076 〈 math 〉〈 math 〉
to the lesser Cones Solidity. 3,648960 〈 math 〉〈 math 〉
The whole Cone = 4620
The Frustum mxhcbd = 164
The being ½ the Fall or Hoof = 82