Notes on the Payment of the Public Debt, [ca. 28 March 1793]
Notes on the Payment of the Public Debt
[ca. 28 Mch. 1793]
In how many years will a Debt bearing int. @ 6 pr. Ct. be extinguished by equal annual payments of 7. 8. 9. or 10. pr. ct. on acct of principal & interest?
Answer.
pr ct y m D By annual payments of 7. in 33- 4-22 4 36/693 8. 23- 9-15 57/693 9. 18-10- 7 420/693 10. 15- 8-21 259/693 Example of the calculn, easily performed in 5 minutes. Annual payment 8 Interest pr. Ct. 6 Annual excess 2. principal for do. 33 1/3 Log. = 1.5228787 debt to be redeemed Add 100. makes 133 1/3 Log = 2.1249387 divide by Log. of ratio y m d of int. 1.06 = .0253059) diff. = .6020600 (23-9-15 5/6) 506118 959420 759177 .21088 1/4 ) 200243 note the first divisor divided by 189794 12 gives 210881/4 the division 692.8 ) 10449 for months, which divided by 307/16 6928 gives 692.8 divisor for days 3521 3464 57 remainder
If the question be reversed the solution is equally easy. viz.
What must the equal annual payments be to extinguish a debt in 20 years at 6. per cent? Answer 8.71845 pr. cent on the debt.
See Bache’s paper Mar. 2. 93. The rule thus expressed.
from the log. of 8. (the annual payment)
subtract the log. of 2. (the part of the principal paid the first year)
divide ye remr by the log. of 1.06 (the amount of 1. dollar in 1. year)
the quotient 23.79 or 24. years will be the time required.
MS (DLC: TJ Papers, 96: 16484); undated; entirely in TJ’s hand; endorsed by TJ: “Finance. public debt. 1793. an idea for paying off public debt”; printed literally.
Benjamin Franklin bache’s paper, The General Advertiser, began printing mathematical questions posed by correspondents on 19 Feb. 1793. Readers who answered correctly were invited to pose new questions in their turn. This initially innocent exercise assumed political overtones on 22 Feb. 1793 when, in the midst of the Republican attack in the House of Representatives on Alexander Hamilton’s management of public finances, “E” answered the third question and asked: “In how many years would Congress pay off the whole of their 6 per cent national debt, supposing them to pay every year (as the funding act allows) 8 per cent on the original principal, both on account of principal and interest?”
On Mar. 2. 93. “Q” responded that “If the six per cent national debt be supposed to amount to 100 dollars (and whether more or less will not affect the present question) then the following rule is easily demonstrated, viz.
“From the logarithm of 8 (the annual payment) subtract the logarithm of 2 (the part of the principal paid the first year) and dividing the remainder by the logarithm 1.06 (the amount of 1 dollar in 1 year) the quotient, 23.79, or 24 years, will be the time required.”
The potential of this method of calculation for providing a date certain on which the nation could aspire to be free of public debt—a Republican political imperative—must have appealed to TJ. With characteristic rigor he expanded on the response in the newspaper by posing the same question based on payments of varying size, extending the calculation to give the answer in months, days, and fractions of days, and successfully inverting the method so as to obtain the size of payment which would pay off a debt in a given number of years. For TJ’s continuing interest in rapidly extinguishing the national debt, see Editorial Note on proposals for funding the foreign debt, in Vol. 14: 190–7.
The article in The General Advertiser of Mar. 28. 93. which evidently caught TJ’s eye was a response to another mathematical question with political overtones: “To find the value of a six per cent annuity, with the interest paid quarterly, calculating the market rate at 5, 6, and 7 per cent per annum.” This question had been posed by “A.B.” in the 21 Mch. 1793 issue of the paper.