To measure girt timber, globes, and weight of shot by arithmetic

To measure round timber, called girt measure, as suppose a piece of round timber whose diameter is 15 inches from A to B, and 16 feet long from C to D. I desire to know the square feet contained in such a piece. Work thus. Multiply your diameter 15 by 15, and it produceth 225. Having so done, multiply 225 by 11. It yields 24 75, and then divide this 2475 by 14 and the quotient is 176 11/14 and now have you the square inches of the diameter 15. Having so

done, turn your length which is 16 feet into inches by multiplying it by 12 inches, and it produceth 192.

Having so done, multiply your breadth of inches which was 176 11/14, by your length 192 inches, and the product will be 33792, which if you divide by 1728 the quotient will prove 19 960/1728 feet, which is something better than 19 feet and a half, and thus is your piece measured, and all other round timber by this method.

The reason why you multiplied your diameter, after it was squared, by 11, and divide it by 14, is that every circle is eleven fourteenths of a square 1 , and so is in proportion to it as 11 is to 14, and why your divisor is 1728, by reason that number of inches is in a foot cubic, so that the second multiplication brought your circle into a square equal to that circle, and your third multiplication brought both length and breadth into square inches, and lastly your division brought out the square feet, 19 960/1728, which was required to be done.

To measure any globe. I suppose a globe whose diameter from A to B is 45 inches. I would know how many square inches is contained in such a figure. Work thus. Multiply your diameter 45 by 45, and it produceth 2025, then multiply this 2025 by 45, and it produceth 91125. Now multiply this by 11. It yields 1002375, which divide by 21. The quotient will be 47732 3/21 square inches, in this globe whose diameter was 45 inches. And if you would know the square feet, divide . this 47732 by 1728, and the quotient will be 27 1366/1728 feet. And thus are all globical bodies measured.

The reason of your multiplying 45 the diameter by 45 twice is to cube it, and why you multiply that product by 11, and divide that by 21, is because every globe bears proportion to a cube as 11 is to 21 2 , for if your globe figure were square and of the same diameter it would waste from 21 to 11 by cutting off the corners to bring it directly round every way. There is another way to work, by multiplying the diameter of the circumference cubically, and divide the product by 6, and the quotient will give the same number directly as the former. And thus I shall end this

work, hoping that by these two ways you do well understand the measuring of any globical figure, always remembering that they do bear in proportion unto their squares as 11 is to 21, which is the reason of your work.

To know the weight of any shot by proportion. Suppose a shot of 4 inches diameter from A to B. I would know how many pounds such a shot weigheth. You must work thus. Multiply your diameter 4 by 4, which is 16, and that product 16 by 4 and it will be 64. Having cubed your diameter by twice multiplying it, you must add the same number of your cube, and one quarter more. As 64 was your cubed product, add 64 to 64 and it makes 128, and then and 1/4 of 64 which is 16 unto 128 and it makes 144.

This 144 is the true content of ounces in such a shot, which divide by 16 and it will produce in the quotient 9 pounds, and thus have you your content, in pounds, of a shot whose diameter is 4 inches. Again, suppose another shot of 7 inches diameter from C to D. I work as above, saying 7 times 7 is 49, and then multiply 49 by 7 again and it produceth 343, which is the cube of 7.

Having so done, I double the number 343, which is 686. Unto this 686 I add 1/4 part of the cubed number of the diameter, which is 85, under 686, and it produceth 771, the content of ounces of a shot of 7 inches diameter, which, divided by 16, gives the quotient 48 pounds and 3/16 of a pound, which is three ounces.

And now having showed you two demonstrations, I shall not say any more as to this work, it being all performed by this method, in shot of any sizes whatsoever. The reason of the work is plain and needs no more interpretation and so pass unto my other work, hinting only upon every [blank] we may not lose too much time.

To know the content of shot, one from another. In the last you were instructed how to find the weight of any shot by its diameter. In this you shall find them from one another, as suppose a shot of 4 inches diameter weighs 9 pounds. I demand how much one of 7 inches diameter shall weigh. You must work thus. Cube your 4 inches diameter, by saying 4 times 4 is 16, and 4 times 16 is 64. Having done with that, cube your diameter of the shot required, which is 7 inches, by saying 7 times 7 is 49, and 7 times 49 is 343.

Having cubed both the shot's diameters, work thus by the rule of three, saying if 64 gives 9 pounds, what shall 343 give? So that if you multiply 343 by 9, and its product is 3087, which divided by 64. The quotient is 48 15/64 pounds, the weight of the shot whose diameter is 7 inches. In observing this work, you may resolve all questions both of shot and powder it being all wrought by this proportion. And the allowance - powder, is common to allow 3 1/2 ounces to every hundredweight the gun weighs which shoots such shot, which, if you well observe, all art of gunnery is brought to pass by this method of proportions. And therefore I leave you to your own practice in whatsoever you please to make trial of.