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MR. Gunter in his first Book, Section 4, hath shewed the Making of the Line of Solids on his Sector: but this Rule shews the proportion of the Diameters in weight: having a Shot of one pound 2 pounds or 3 pounds weight of the Metal or Stone assigned; if it be of a pound, divide the Diameter into 4 equal parts, and 5 such parts will make a Diam. for a Shot of the said Metal or Stone that shall weigh just two l.
And divide the Diameter of a Shot that weighs just 2 l. into 7 equal parts, and 8 such parts will make a Diam. for a Shot of 3 l. weight; And divide the Liamet. of a Shot of 3 l. weight into 10 equal parts, and 11 such parts will make a Shot for 4 l. weight.
And divide the diam. for a Shot of 4 l. weight into 13 parts, 14 such parts will make a diam. of a Shot for 5 l. weight.
And divide the diam. of a Shot of 5 l. weight into 16 equal parts, 17 such parts will make a diam. for a shot that will weigh 6 l. and so dividing each next diam. into 3 equal parts more then the next lesser was Divided into, and it will with one part added from a diamet. of a shot that will weigh just 1 l. more; and so you may proceed infinitely, in∣creasing or decreasing, by taking one part less, as is appointed to be Divided, for one l. less, and the next into 8 l. less, to abate 1 for the Remainder, infinitely decreasing it.
First you must have exactly the diamet. of a shot that weigheth one pound, and then describe a Circle, whose diamet. shall be just equal thereunto; and Divide it into 4 Qua∣drants,
with 2 diamet. cutting each other in the Centor orthogonally. Then take the Chord of the whole Quadrant 90 degr. BC in your Compasses, and lay it from the Centor of the first Shot one pound D to 2,* 1.1 and so A 2 will be the Diameter of a Shot of 2 pound; Then extend the Compasses from 2 to the Chord C, and lay that distance from D to 3, so will A 3 be the Diameter of a Shot of 3 l. And so likewise extend the Compasses from 3 to C, it will reach from D to 4, and from 4 to C, and it reaches from D to 5, and from 5 to C, lay it still always from D to 6; and so continuing till you have proceeded as far as you will: You shall find that if AB were the Diameter of one pound, A 2 is the Diameter of 2 pound, and A 3 is the Diam. of 3 l. and A 4 the Diam. of 4 l. and A 5 the Diam. of 5 l. A 6 the Diam. of 6 l. and lastly, A 8 is the Diam. of 8 l. and so you may proceed in like manner infinitely.
Likewise having the Diameter of a Shot of any weight, the double of the Diam. is the Diam. of a Shot which weighs 8 times as much. Thus the double of A 1, which is A 8, makes the Diameter of a Shot of 8 pound; and so the double of A 2, which is the Diameter of a Shot of 2 l. makes A 16, the Diameter of a Shot of 16 pounds, that is 8 times 2 pounds; and so the double of A 3 makes the Diameter of a Shot of 24 pounds, and the double of A 4 makes the Diameter of a Shot of 32 pounds, four times 8 being 32; and so you may proceed as you please, and find the bigness of any Shot.
This you may do also, having the Diameter of a Shot of one pound, double that diam. it will make a diam. of 8 pound; and treble the diameter of one pound, will make a diameter of a Shot of 27 pound, and quadruple or 4 times the same, will make a diam. of a Shot of 64 pounds, and 5 diameters will make a Ball of 125 l. and 6 diameters of a Shot of one l. will make a diameter of a Shot that will weigh 216 l.
Now it is convenient to shew how to find the Mean-divisions between these extremes; as for the diameter of a Shot of 2 l. 3 l. 4 l. 5 l. 6 l. 7 l. or what more you will; so as by such progression you may proceed from pound to pound, until you come to the last term of 216 pound; nevertheless the same manner of working will proceed infinitely. Lay the forementioned 6 diameters upon one and the same right Line; you must at the end of them draw another Right-line orthogonally, and set therein the diamet. of 2 such Shot given as at C, and from thence draw another Right-line parallel to the first, as GH, and then draw a Quadrant as A B, and from the Centre G draw right Lines through all the divisions of the diam. marked upon the right Line AF which are all equal, so shall you have 6 divisions to be divided; the first being divided already, and is the diam. of a Shot of 1 l. but the second division is to be in the Circumference or Quadrant divided into 7 parts equally, because it containeth the second diameter unto 8, for adding 1 to 7 it makes 8; the third division is into 19 equal parts, which being added to 8, makes 27; the fourth shall be divided into 37 equal parts; which together with 27, makes 64; the fifth shall be divided into 61 equal parts, which added to 64, makes 125; and lastly, the sixth place must be divided into 91 equal parts, unto which adding 125, you shall make a diameter of a Shot of 216 pound justly.
Now forasmuch as these divisions are difficult to make well, within so small a Qua∣drant: you may therefore describe a greater, as the Quadrant LM, and there the di∣visions are more distinct, and larger than in the lesser they can be; Further, you may note, that Fire-balls, Granadoes, and other Globous Artifices, must have the same pro∣portion to their Grandures from their Ball of one pound, which may be exactly consi∣dered; and so by this Method you may make Balls of Lead, Brass, Stone, and Granadoes, Fire-balls, and all other Spherical Fire-works, of what weight you will, having one of one pound first, to lead you accordingly.
The Diameter of a Shot of 1 pound is 1 Inch 94/100 parts.