Problem 23.

To find the Difference of the Areas of Ifoperimetrical Figures; that is, of Figures whose Number of Sides and Measure about them is equal, the one to the other.

It is observable that two Figures of the same Number of Sides, and the fame Perimeters, may have their Areas very different from each other. For, suppose, in the two following Parallelograms, the Length of the former, marked A, be 6 Feet, the Breadth 4 Feet, and the Perimeter 20

Feet: And suppose the Length of the latter, marked B, be 8 Feet, the Breadth z Feet, and Perimeter 20 Feet, the fame with the other; then the Area of the one will be 24 Feet, and the Area of the other but 16 Feet, which is equal only to of the former. *

Hence we discover that of all Isoperimetrical Figures having the fame Number of Sides, the nearer their Sides and Angles approach to an Equality, the greater is their Arca or Surface, and that the Perimeter only being given is.. not fufficient to determine the Content.

Stadia, and Lacedæmon no more than 48, and yet that this last City is twice as large as the former, they know not how to believe it; and if any one, designing to increase their Surprize, should affirm, that it is poffible that a Ground, Camp, or City, which contains only 40 Stadia in Perimeter, may be twice as large as another that contains 100 Stadia, they are ftruck with the greatest Amazement. The Caufe of this Surprize arifes from their Neglect of Geometrical Learning in their Youth; and I was rather inclined to take some Notice of these Matters, because not the Vulgar alone, but some even of those who are employed in the Administration of States, or placed at the Head of Armies, are fometimes aftonished, and not able to conceive, that Lacedæmon is a much greater City than Megalopolis, though it be confiderably, less in Perimeter.

Problem 24.

To find the Side of a Cubic Block of Gold, which being coined into Guineas, would pay off the National Debt.

Operation. At the Tower of London 44 Guineas and are coined out of ilb. Troy, or 5760 Grains of Gold; the Standard Weight therefore of I Guinea is 129 Grains.

4382

The Weight of a Cubic Foot of such Gold is 752492116 Grains, or 13061b. 4oz. rodwt. 91 grs. Troy Weight. Out of this Quantity 58135 Guineas may be coined, which is equal in Value to 610421. os. 3d. Sterling.

The National Debt at this Time is about 250,000,000៛. Sterling, which turned into Guineas make 238095238.09, whence we have this Proportion.

Guineas.

If 58135.25 require 1, what will 238095238.09 require?

Answer 4095.539 Cubic Feet.

Consequently, a Lump of Gold equal in Bulk to 4095.539 Cubic Feet coined into Guineas, will pay the National Debt; and the Side of such a Cube will be found to be 16 Feet nearly.

Problem 25.

What Annuity will be sufficient to pay off the National Debt of 250 Millions in 30 Years, at 4 per Cent. Compound Interest?

Operation.

The Annuity of 17. for 30 Years, at 4 per Ct. is 0.0578301 Which multiplied by the Debt

250000000

14-457525

10.000000

4.457525

Consequently, the National Debt, allowing it, this prefent Year 1783, to be 250 Millions, and the Interest yearly paid at 4 per Cent. to be 10 Millions; then will an Additional Sinking Fund of 4-4575251. per Ann. clear off the whole Debt in 30 Years.

If former Ministers had been prudent enough to have discharged this Debt as oft as it had amounted to 40 or 50 Millions, the Annuity to have been raised would have been but small; i. e. between 2 and 3 Millions yearly at moft. Had this Measure been adopted, we should have been at this Time, not only free from the heavy and ruinous Load of Taxes we groan under, but, with our present Resources, should have been a mighty People;-eafy at Home, and

Problem 26.

Of Magic Squares.

Magic Squares are Numbers in progressive Order in a Natural Square so disposed in the Cells of a Geometrical Square, that the Sums, (if the Numbers given are in Arithmetical Progression, -but their Products, if the Numbers are in Geometrical Progreffion) of each Row taken either Perpendicularly, Horizontally, or Diagonally, are equal.

Thus the nine Digits placed in their Order in the Natural Square may be disposed in the following Manner in the two Magic Squares to make the Sum of 15 each Way.

In like Manner the Natural Square of 16 Cells may be disposed 4 different Ways in Magic Squares, making 34 each Way.

Natural Square.

1234

5678

9 10 11 12

13 14 15 16

Magic Squares.

1332 16 12679 8 10 11 5

115 14 4 16 32 134 15 14 1 1267996712 967 12 8 10 11 5 5 10 11 8 5 10 11 8 13 3 2 16 415 141 16 3213 1 15 14 4