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Note: I do have an analogous post up on the mathematics stack exchange here

Copy-Pasted from there: "I've been going through and writing up solutions for Euler's Elements of Algebra Practice Questions in the 1822 English translation by John Hewlett in my free time, and I can't seem to find the path to Euler's Provided Solution. Here's Euler's question:

One Sessa, an Indian, having first invented the game of chess, shewed it to his prince, who was so delighted with it, that he promised him any reward he should ask; upon which Sessa requested that he might be allowed one grain of wheat for the first square on the chess board, two for the second, and so on, doubling continually, to 64, the whole number of squares. Now, supposing a pint to contain 7680 of those grains, and one quarter to be worth 1 l. 7s. 6d., it is required to compute the value of the whole sum of grains. Ans. 64481488296 l.

I know from problem 2, the conversion for money is 20 s. to 1 l., and 12 d. per 1 s. ( pence/pennies := d. , shillings := s., and pounds := l.) I have been utilizing the assumption that there are 2 pints per quarter, though I'm wondering if due to the age of the text that I am under a wrong volume assumption. This has been frustrating me the past couple of days, and I don't want to proceed to the next section without solving this mystery first.

Any insight into where I could be going wrong here would be very helpful! Certainly, Euler has the correct solution here, and it would be greatly historically interesting if this was a case of myself succumbing to modern volume assumptions.

Thanks!"

So far, I believe the mathematics are 'correct' in that I have the right summation value. The issue is a matter of conversion. I am unaware of what a quarter is. Is a quart equivalent to a quarter in Europe in 1765? Does this differ depending on the region? Euler utilizes pounds (l.), shillings (s.), pence (d.) and farthings (f.) where the conversions are upheld as 1 l. -> 20 s. 1 s. -> 12 d. 1 d. -> 4 f. These are the correct conversions, as they provide the solution for the previous problem in this chapter.

In sum: What is a "quarter" in 1765 Europe as a conversion for pints. How many pints of the period make up a quarter?

Any insight is greatly appreciated, this problem has been driving me nuts!!

Further Context Below: (Thanks so much everyone for the wonderful insight!! Happy to know that this post has inspired people to perform their own calculations :] )

As the question appears in English translations, with the practice questions of the text themselves being an addition by the English translations, Euler's Elements of Algebra maintains the solution as

64481488296 l.

Which notably mathematically does not utilize a 1 quarter to 512 pints conversion.

However, in an 1831 Mathematics text A Course of Mathematics: For the Use of Academies as Well as Private Tuition: in Two Volumes, Volume 1 by Charles Hutton, added onto by Olinthus Gregory and Robert Adrian, on page 144 question 27 we see the solution that matches utilizing the conversion of 1 quarter to 512 pints, given as:

Ans. 6450468216285 l. 17 s. 3 d. 3* (32757)/(32768) q.

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    @MCW Thank you! Adjusted the title for clarity, I appreciate it!
    – Fuzzy Trex
    Commented Jun 25 at 23:27
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    Leonhard Euler published most of his works in Latin, and the Vollständige Anleitung zur Algebra in German, so your quote is obviously a translation. Also, as a Swiss living in St Petersburg, there is absolutely no reason why he would use English currency in a problem. That leads me to the conclusion that the numbers and units are not named by Euler, but by the translator. To understand what that translator was referring to, we need to know who he was, when the translation was made, and what audience he was writing for. Please cite the precise source of that question.
    – ccprog
    Commented Jun 26 at 1:28
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    @ccprog: Bottom of page 225 (numbered page 170) here
    – Pieter Geerkens
    Commented Jun 26 at 1:48
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    @ccprog Thank you! In discussion on the math stack exchange post, I found that these are not in fact Euler's problems. The original publication in Russian did not have problem sets for the reader. The English translations from the previous French translation follow from one by John Hewlett, and another by Charles Tayler. This problem specifically comes from Hewlett's translation. Hence, why the English currency is used! Thank you so much for the help! :]
    – Fuzzy Trex
    Commented Jun 26 at 1:57
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    @PieterGeerkens So the reference is London 1822 for the first edition of Hewlett's translation.
    – ccprog
    Commented Jun 26 at 2:01

3 Answers 3

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This is solved by reading through the Wikipedia definition of a quarter:

These copies describe the "London quarter" as notionally derived from eight "London bushels" of eight wine gallons of eight pounds of 15 ounces of 20 pennyweights of 32 grains of wheat, taken whole from the middle of an ear;

So a quarter equals eight bushels of (x) eight gallons of (x) eight pounds...

Eight pounds=1 gallon. This is the same formulation we use today, 8 pints = 1 gallon. This is confirmed in another wiki entry here, where the pound is expressed in grains of wheat:

...At that time, the pound unit in use in England was the Tower pound, equal to 7,680 Tower grains (also known as wheat grains).

There is a familiar number which ties back to the ops' question. So a pint is the equivalent of a pound = 1/8 of a gallon. Therefore the Quarter = 8x8x8=512 pints/pounds of grain.

Concerning comments wondering if the measures were still in place at the time of the writing of the OPs' source, the article on the Assize of Bread and Ale mentions that there were amendments to this law in 1822 and 1836, and it was not repealed until 1863:

The law was amended by the Bread Acts of 1822 and 1836, which stipulated that loaves should be sold by the pound, or multiple thereof, and finally repealed by the Statute Law Revision Act 1863 (26 & 27 Vict. c. 125).

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  • "at that time" in your quote refers to "the Middle Ages". Hewlett's edition was published 1822. Do these definitions still apply in mid-19th century?
    – ccprog
    Commented Jun 26 at 2:03
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    @JustCal YES. Thank you!! The mathematics works out with 512 pints. I was crunching powers of 2 blindly, made it up to assuming it was 256 before seeing your answer!
    – Fuzzy Trex
    Commented Jun 26 at 2:04
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Now, supposing a pint to contain 7680 of those grains, and one quarter to be worth 1 l. 7s. 6d., it is required to compute the value of the whole sum of grains. Ans. 64481488296 l.

According to Wikipedia, a "quarter" (the unit of volume, not the unit of mass) was 64 gallons, and a gallon was/is 8 pints. Thus, a quarter is 64×8 = 512 pints. (While the definition of a gallon changed between the Magna Carta and the Weights and Measures Act 1824, the ratio between quarter, gallon, and pint never changed.) So if a pint contains 7680 grains of wheat, then a quarter contains 512×7680 = 3932160 grains.

The total number of grains on the chessboard is 1 + 2 + 4 + 8 + ... + 2^63 = 2^64 - 1 = 18446744073709551615. Divided by 3932160 grains in a quarter, we get 4691249611844.267 quarters of grain.

The monetary value of "1 l. 7s. 6d." is equivalent to £1.375. Multiplying this price per quarter by the number of quarters, we get a total value of £6450468216285.87. But this is over 100 times higher than the value given in the book, so either I or the author made an arithmetic error.

As for which pint would have been meant in 1824, I would assume Winchester measure, which was used for grains and dry goods, in contrast to Queen Anne's wine gallon (the basis of the modern US gallon) or the ale gallon, which were used for measuring liquids. A Winchester gallon was 268.8025 in3, making a pint (1/8) 33.6003125 in3 ≈ 550.61 mL. This works out to a bulk volume of 71.7 µL per grain of wheat.

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    That (namely £6 450 468 216 285.87) is the figure I got last night also - which is 100.03 times bigger than that (namely £64 481 488 296) in the text. Check your digit counts again.
    – Pieter Geerkens
    Commented Jun 26 at 23:46
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    @PieterGeerkens: You're correct. I'll have to see if a different "quarter" was in use that explains the discrepancy.
    – dan04
    Commented Jun 27 at 0:48
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    copied from my post on the math stack exchange (depending on the source you are using, the answer could be wrong): "So, taking our sum of 264−1 and dividing by 7680∗512=3932160 to get a unit amount and convert via 330d. per unit, we arrive at your suggested correct solution of 6450468216285 pounds 17 shillings 3 pence and 332757/32768 farthings! "
    – Fuzzy Trex
    Commented Jun 27 at 3:38
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A "quarter" as a unit of measurement comes to around 8 UK gallons, which is 64 pints or 30 litres. The British imperial measurement system was first standardized by Queen Elizabeth the First sometime in the 1550s to 1580s when the monarch made statutes to make the units the same size all over Britain, which is also why the standard land mile is sometimes called the "statute mile". By 1827 however, the British imperial system was updated, possibly to calibrate with its rival measurement system, the French metric system. The isolated United States was left out of this recalibration of the imperial system, which is why the US imperial system is slightly different from the version of the imperial system used in the United Kingdom, Canada, and other former British territories.

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