Sect. III.

The Diameter of any Circle being given, to find the Area, (or any part thereof) in Inches, or in Ale or Wine-Gallons.

1. For the whole Area in Inches.

The Area of a Circle is equal to the Fact, or Rect-Angle or half the Diameter into half the Circumference; that is to say, if half the Diameter be multiplied by half the Circumference, the Product will be the Area.

    Thus, when the Diameter is 1, the Circumference is 3.141592, the half of this is 1.570796, which multiplied by half the Diameter (viz. 5.) the Product will be the Area of that Circle, whose Diameter is 1, (viz.) .785398

    The Area’s of all Circles are in proportion one to another, as the Squares of their Diameters (2.12. of Euclid.)

    Therefore, As the Square of the Diameter of any Circle is to the Area of that Circle;
    So is the Square of the Diameter of any other Circle to the Area thereof.

    In the Circle above-mention’d the Diameter is 1, and the Area .785398: Now the Square of 1 being but 1, it must hold: As 1 is to .785398, So is the Square of the Diameter of any Circle, to the Area thereof.  So .785398 is a fixed Multiplicator; and if an Unit with Cyphers be divided by .785398, the Quotient will be 1.27324, a fix’d Divisor: and by either of these fix’d Numbers the Area of any Circle may be found, either by Multiplication or by Division.  For if the Square of any Diameter be
 

the Area in Square Inches, Feet or Yards, according as the Diameter was measur’d by Inches, &c.

    But with more expedition by the Instrument, Thus:

    Set 1 (a Diameter) upon the Line D, to .785398 (the Area thereof) upon C.

    The Rule being thus set, the Lines are like a Table of Circles Areas to all Diameters; for against any Diameter upon the Line D, you have the Area thereof upon C.

Example: Let the Diameter be 20.

    Set 1 upon D, to .785398 upon C, and then against 20 upon D, is 314.159, the Area required upon C; and as the Rule now stands, I also find, that if
 

    Contrariwise, when the
 

2. For the Area in Gallons.

    The Area in Inches divided by 282 or 231, gives the Area in Ale or Wine-Gallons respectively, and so for any other measure expressed in the former Tables; but without knowing the Area in Inches, the Area in Gallons may be found thus: Divide .785398
 

    The Quotients are the Area of a Circle, whose Diameter is 1, in Ale or Wine-Gallons, and are fixt Multiplicators for finding the Area of all Circles in either of these Measures; for if the Square of the Diameter of any Circle be multiplied by either of these Numbers, the Product is the Area in Ale or Wine-Gallons respectively.

    If you would effect this Division, the several Divisors are thus found: Multiply the Divisor for finding the Area in Inches, viz. 1.27324
 

    The Products are the Divisors sought.  And the Square of the Diameter of any Circle divided by one of these, quotes the respective Area.

     Having thus found the several fixed Multiplicators and Divisors for finding the Area of a Circle in Inches and Gallons, I shall here represent them all together in the following Table.
 

    But the Area of any Circle may be more readily found by the help of certain fixt Numbers, called Gage-Points, and these fixt Numbers are the Diameters of those Circles, whose Content at one Inch deep is equal to the respective Gallon to which they belong; thus the Gage-Point for the Ale-Gallon is 18.95, which is the Diameter of that Circle, whose Contents is 282, the Square Inches in the Ale-Gallons.

    The several Gage-Points, are the Square-Roots of the Divisions last mention’d, and by the Rule are all found at once by the Lines C and D, for (setting these Lines even at the end)
 

    To which are set the Letters a.g. and w.g. upon the Rule.

    Now by these Gage-Points the Area of any Circle may be thus found:

For Ale-Gallons,

   Set 18.95 (the Gage-Point for Ale-Gallons) upon D, to 1 upon C, then against any Diameter upon D, you have the Area upon C.

    So if the Diameter were 40 Inches, the Area will be 4.45 Ale-Gallons: And the Rule being thus set, the Lines are in effect a Table of Circles Areas: For I likewise find, that if the Diameter
 

    The like for Wine-Gallons, by the proper Gage-Point.

   Note, When the Area of any Circle is sought in Ale-Gallons, if the Diameter be more than 18.95, and less than 100, set the Gage-Point upon D, to 1, at the beginning of C; then against any Diameter, from the Gage-Point to 100, upon D, you have the Area upon C.  Thus the Rule being set for Ale-Gallons, you will find that when the
 

    When the diameter is less that the Gage-Point, or more than 100, then set the Gage-Point to 1, in the middle upon C; than against these
 

    And without moving the Rule, if 10 at the beginning of the Line D, be 100, the Area against it will be 27.851: So against this Diameter, viz. 200, the Area is 111.4 Gallons; against 300, it is 250.7, and so on to 600 Inches Diameter, against which you have 1002.6, &c. by this you may further observe, that if the Diameter be increased by Tens, the Area will increase by Hundreds: Thus
 

    The like for Wine-Gallons by the proper Gage-Point.

Lastly, To find any part of the Area of a Circle, in Ale or Wine-Gallons.

    Set the Gage-Point 1/3, 1/2, or any other part of 1; then against the Diameter you have the like part of the Area.

Example: Let it be required to find the third part of the Area of a circle in Ale-Gallons.

    Set the Gage-Point to 1/3 of 1, viz. .333, then against any Diameter you have 1/3 of the Area; thus against 100 is 9.283, which is 1/3 of 27.85, the whole Area.

    Understand the like for Wine-Gallons, Ale of Beer-Barrels.

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This page was last updated on 29 March 2000.