5.

To find the Area of a Circle-Multiply half the circumference by half the diameter.-Or, multiply the diameter by itself, and the product by 7854. Or, multiply the circumference by itself and the product by '07958.

6. To find the Surface of a Cylinder, Pillar, &c.-Multiply the circumference by the length.

7. To find the Solidity of a Cylinder, Pillar, &c.-Multiply the square of the diameter by 7854, and the product by the length or height.

8.

To find the Surface of a Cylindrical Ring-Add to its thickness, its inner diameter, then multiply the sum by the thickness and that product by 9.8696.

9. To find the Solidity of a Cylindrical Ring-Add to its thickness its inner diameter, then multiply the sum by the square of the thickness and that product by 2.4674.

10. To find the Area of a Flat Ring-Deduct the area of the inner circle from the area of the outer circle.

11. To find the Surface of a Globe or Ball—Multiply the circumference by the diameter.-Or, multiply the square of the diameter by 3·1416.

12. To find the Solidity of a Globe or Ball-Multiply the cube of the diameter by 5236.—Or, multiply the cube of the circumference by '01688. 13. To find the Diameter of a Circle from the circumference-Multiply the circumference by 7, and divide by 22.-Or, multiply the circumference by 113 and divide by 355.-Or, multiply the circumference by 318309.

14. To find the circumference of a Circle from the diameter.Multiply the diameter by 22 and divide by 7.-Or, multiply the diameter by 3.1416.-Or, multiply the diameter by 355 and divide by 113. The result of the first is about 2400th part, and of the last, about 10,000,000th part from the truth.

15. To find the side of a Square of equal Area to a given Circle.Multiply the diameter by 8862269.-Or, multiply the circumference by ⚫2820948.

16. To find the Side of a Square inscribed within a given CircleMultiply the diameter by 7071068.-Or, multiply the circumference by •2250791.

17. To find the Contents of Circular Cisterns, Pipes, &c. in Gallons, -Refer to Table 1. Or, square the diameter and multiply by 00283257, -or divide by 353-03622 for the number of gallons per inch deep, then multiply by the depth of wet inches.-Or, find the area in cubic inches (by Rule 5)-multiply that by the depth of wet inches, and divide by 277.274 for gallons, or convert the square inches of the last product into gallons by Table 16. Refer to Table 18 for contents of pipes.

18. To find the Contents of Square Cisterns or Tanks in Gallons— Multiply length, breadth, and depth for cubic inches, and convert them into gallons by dividing by 277-274, or by multiplying by 00360654, or by referring to Table 16.

19. To find the Contents of a Full Cask-Refer to Table 2. Or, multiply the difference between the head and bung diameters, if less than 6 inches, by 68 (if more, by 7) and add the product to the head diameter as a mean diameter of a cylinder, then proceed as for a circular cistern by Rule 17.

20. To find the Contents of a Cask partly filled-Divide the wet inches by the bung diameter, if lying, (or by the length if standing) to three places of decimals, and if the quotient exceed 500, add to it one-fourth of such excess; but if less than 500, deduct one-fourth of the deficiency; then multiply the number of gallons which the cask contains when full by the increased quotient, if lying, or, by the diminished quotient if standing; the product will be the contents of the partly filled cask in gallons.

21. To find the cubic measure of Grindstones by the customary Rule -Add the diameter to the thickness, and divide by 8, the result is feet and eighths.

22. To find the value of Tin Plates by their weight-Refer to Table 4. Or, divide the weighing by the weight of the full box to three places of decimals, and multiply the price of the whole box by that quotient.

23. To find the value of Tin Plates from their number-Refer to Table 5. Or, divide the given number by the number of the full box, to three places of decimals, and multiply the price of the whole box by that quotient.

24. To Tale Nails-Tale as many as, by the statement below, will give the required number per thousand; then, if the weight be in drams, weigh them so many ounces per thousand, if it be in ounces weigh them so many pounds. Ex.-55-880, 56-896, 57=912, 58—928, 59=944, 60—960, 61 976, 62-992, 63-1008, per thousand.

25. To convert Shillings, &c. to Decimals of three places-Refer to Table 12. Or, take half the shillings for the first place, and the number of farthings contained in the rest for the second and third places-increased by 1 if 24, by 2 if 48, and by 3 if 72.

26. To convert Decimals of a Pound into Shillings, &c.-Refer to Table 12. Or, double the first place, and add one if the second and third places be 50 or more, for the shillings-then take the second and third places, or their excess over fifty, as so many farthings, first deducting 1 if 25.

27. To Compute Simple Interest-Take a penny a month for every pound, for 5 per cent. Or, multiply the principal by the number of days, and divide by 365, the quotient will be shillings at 5 per cent.; deduct a fifth for 4 per cent. and so on. Or, multiply the principal, time, and rate, decimally.

28. To Compute Compound Interest-decimally-Multiply the amount of one pound for the first payment, by itself, and that product again by the amount of one pound for the first payment, and so on till the number of payments are exhausted-then multiply the last product by the principal for the amount, and deduct the principal from this product for the interest only.

29. To find the Present Value of any sum improved at Compound Interest, due some time hence-Ascertain, by Rule 28, the amount of one pound at compound interest for the term, and divide unit by that amount, the quotient will be the value of one pound, which must be multiplied by the sum to be received at the end of the term for its present value.

30. To Discount Bills or to Discount Debts, due some time hence, but agreed to be paid immediately-Multiply the sum by 100, and divide by 100 increased by the product of the time and rate at which money can be employed at simple interest.

31. To Compute Discount and Profit-Refer to Table 11, then for amounts of 2/7 to 20/11 with odd, pence, add the amount opposite the shillings to that opposite the pence: or, in some instances, double the principal and halve the result; in others, halve the principal and double the result. For amounts of 21/ to £13-reduce the sum to shillings, and consider so many of them as pence as will equal the nearest number in the column "Eq." then reduce the sum opposite that number to pence and let them be so many shillings and each farthing a threepence-obtain the rest of the sum by one of the preceding methods, and add the two results together. The above is readily performed without pen or pencil. Larger amounts may be worked as follows:-multiply the sum by the decimal placed at the head of each rate in the table-that over "Dis." will give the discount only, and that over" Gro." will give the gross amount to sustain the reaction. Or, multiply the sum by the number of fives in the rate, and let the pounds of the product be shillings, each 1/8 a penny, and each 5/ threepence. Or, multiply the amount by the rate, and divide by 100.

32. To find the rate per cent of Profit on the Cost of any Goods or Transactions-Multiply the profit by 100, and divide by the gross amount. 33. To find the amount of a Bankrupt's Composition in the pound -Divide the net amount of assets by the sum of his liabilities. Rates and taxes may also be determined by this rule.

34. To divide a sum of money amongst a number of Partners, or otherwise, in definite but unequal proportion-In such cases the sum is supposed to have been divided into a certain number of shares, say, for example 20-one party is to have 5, another 6, and another 9 of these shares or twentieths. The rule then is, to multiply the sum seperately by each party's number of shares, and divide by the total number of shares for each party's amount.

35. To divide the Profit or Loss of a Partnership, time equal; also to divide the effects of a Bankrupt-Divide the whole profit or loss by the whole capital, and multiply each partner's capital by that quotient for his share of the profit or loss.

36. To divide the Profit or Loss of a Partnership, time unequal— Multiply each partner's capital, or each advance of the same partner, by the time of its continuance-add all these products into one sum, and divide the whole profit or loss by it, then multiply the quotient by the

product of each partner's capital and time for his share. The three foregoing rules may be proved by adding the shares together, their sum will equal the amount to be divided if the work be right.

37. To find at a given sum per day, how much per year-Reduce the sum to pence, add to it the half of itself, and let the pence be pounds, and each farthing 5/, then add five times the daily sum for the year's amount.

38. To find at a given sum per year, how much per day-State the amount in pounds and decimals. Deduct one third and consider the result as pence, and deduct a farthing for every eighteen-pence, or tenpence for every £3.

39. To find the value of Freehold Premises, &c.-Deduct from the rent all incumbrances, if any, then divide 100 by the rate of interest, at which it is desired to improve the purchase-money, (for the number of years' purchase,) and multiply that quotient by the net rent, for the value of the property.

MISCELLANEOUS FORMS, MEMORANDA, &c.

Acknowledgment of the Balance of Account.

I have examined the above account between myself and C. D., and admit that the balance against me of £ -is a just balance.

Dated this day of

Witness, C. D.

185-.

A. B.

Acknowledgment of Debt to take the Case out of the Statute of Limitations.

I hereby acknowledge that the sum of £, being part of the sum of £, lent to me by C. D. of &c. is still owing and unpaid. Dated, &c. A. B.

Guarantee for Certain Amount.

--- but

I hereby guarantee the payment to Messrs. &c., for such goods as they may supply to J. W. of &c., not exceeding the amount of £this is not intended as a continuing guarantee, but only for the once supplying goods to the above amount. Dated, &c.

Continuing Guarantee for Goods to be delivered.

I hereby guarantee the payment to Messrs. -&c. for all goods which they may from time to time supply to J. W. &c.-this is to be a continuing guarantee. Dated, &c.

Guarantee for Debts Overdue to prevent proceedings. Messrs.of &c. having, at my request, agreed to forego proceedings which they were about to take against Mr. J. W. of &c. to enforce the payment of £ due from him to them, I hereby guarantee the payment to them of that sum. Dated, &c.

£100

Bill or Acceptance.

London, 1st Jan. 185.-.

(Two months) after date pay to (my order) the sum of (one hundred) pounds value received.

To Mr. C. D., Merchant, Liverpool.

A. B.

[Special acceptance across the bill, “Accepted payable at Messrs. B. & L. London-C. D."]

£100

Promissory Note, and "Note of Hand."

Place and date.

I promise to pay to Mr. C. D., or order (two months) after date (or "on demand") the sum of (one hundred) pounds, with lawful interest for the same until paid, for value received.

A. B.

£100

Promissory Note with Surety.

Place and date.

(Six months) after date, we jointly and severally promise to pay A. B. or order, the sum of (one hundred) pounds with lawful interest, value received.

Receipt pro tem. when Stamps are not at hand.

Memorandum,

Place and date.

Received of Mr. A. B., the sum of (Twenty) pounds in discharge of account delivered, (or, as the case may be) for which I undertake to give him a receipt upon a proper stamp.

C. D.

£20

Agreement for putting off a Business.

Memorandum, that it is agreed this- -day of, 18A. B. (vendor) of- on the one part, and C. D. (purchaser) of· the other part, as follows: ;

between

on

and

The said A. B. agrees, that in consideration of the sum of £ of the valuation money hereinafter mentioned, the said A. B shall and will, on or before the day of, relinquish to and in favor of the said C. D. the business of now carried on by the said A. B., at aforesaid, and shall and will transfer to the said C. D. the stock in trade in and belonging to the said business, and the shop fixtures, and other fixtures, implements, and utensils in trade, in and upon the premises of the said A. B. at- -aforesaid; and also shall and will execute to the said C. D. an assignment of the lease which the said A. B. now holds, of his dwellinghouse, shop, and premises at - aforesaid, subject to the rents and covenants therein reserved and contained. And it is agreed, that the [goodwill of the said business and the] said stock, fixtures, implements, and utensils shall, on or before the -day of be valued and appraised by L. M. [or "by two referees or their umpire,"][or "by two appraisers, one to be chosen by each of the said parties, and, in case of disagreement between them, then by a third appraiser, to be by such two appraisers named and appointed]." And the said C. D. agrees, that he will accept the sai