Sources in the history of algebra: arithmetical and recreational problems

Conspectus of problems from: Liber Abbaci.
Transcription by: Sigler, Laurence (2000) Fibonacci’s Liber Abaci. A Translation into Modern English of Leonardo Pisano’s Book of Calculation, Springer, Heidelberg.

FLA001 If you wish to separate 10 into two parts so that their product makes one fourth of the greater part multiplied by itself, then you put for the greater part the root that you will call the thing. (p. 558)
FLA002 Also I separated 10 into two parts, and I multiplied one of them by itself, and that which resulted I multiplied by 7/9 2, and that which resulted was 100, namely the square of 10. (p. 559)
FLA003 Also I separated 10 into two parts, and I divided the greater of them by the lesser, and that which resulted was s 1/3 2. (p. 559)
FLA004 I separated 12 into two parts, and I multiplied one of them by 27, and that which resulted was equal to the square of the other part. (p. 559)
FLA005 I multiplied 2/3 of one number plus one by 3/4 of the same number plus one, and 73 results. (p. 559)
FLA006 I separated 10 into two parts, and I added together the squares of them, and ½ 62 resulted. (p. 560)
FLA007 And if the square of the lesser part is subtracted from the greater part, then 50 remains. (p. 560)
FLA008 I separated 10 into two parts, and I divided the first part by the other, and I divided the second part by the first, and the sum of the quotients was 1/3 3. (p. 561)
FLA009 Again I separated 10 into parts, and I multiplied one of them by 6, and that which resulted I divided by the other part, and a third of this that resulted I added to the product of the first part by 6, and the total that was constructed was 39. (p. 561)
FLA010 I divided 60 by a number of men, and each had an amount, and I added two more men, and I divided the 60 by all of them, and there resulted for each ½ 2 denari less than that which resulted first. (p. 562)
FLA011 Also I divided 20 by a number of men, and there resulted an amount. and I added three men, and I divided 30 by all of them, and for each there was 4 less than before. (p. 563)
FLA012 Also I divided 20 by a number of men, and an amount was obtained for each man, and I added two men, and I divided the number of all of them into 60, and there resulted 5 denari more for each one than occurred before. (p. 563)
FLA013 Also I divided 60 by a number of men, and each had some amount, and I add three men, and I divided 20 by the number of all of them, and there resulted for each of them 26 less than occurred before. (p. 564)
FLA014 Also I divided 10 by a number of men, and there resulted an amount for each, and I added 6 men, and I divided the number of all of them into 40, and there resulted for each the same that resulted before. (p. 565)
FLA015 I separated ten into two parts, and I multiplied one of them by itself, and there resulted thirty-two times the other part. therefore the square of the first e part is equal to the multiplication of the second part by the 32. (p. 565)
FLA016 I bought, I know not how many, cheap things for 36 denari, and I bought some expensive things for the same amount, namely 36 denari. And the price of each of the expensive things was 3 denari more than the price of the others, and there were 10 things in all. (p. 565)
FLA017 I separated 12 into two parts, and I multiplied one by the other, and that which resulted I divided by the difference between the parts, and ½ 4 resulted. (p. 566)
FLA018 Again I separated 10 into two parts, and I divided the greater part by the lesser, and that which resulted I added to the 10. and I multiplied this total by 10 yielding 115. (p. 566)
FLA019 Also I separated 10 into two parts, and I divided the greater by the lesser, and that which resulted I added to the 10, and afterwards I divided the lesser by the greater, and that which resulted I added to the 10, and I multiplied the first sum by the second, and 2/3 122 resulted. (p. 566)
FLA020 I separated 10 into two parts, and I divided the first by the second, and the second by the first, and I added the two quotients, and that which resulted was 1/6 2. (p. 566)
FLA021 Also the division of the greater part by the lesser is added to the 10, and the division of the lesser part by the greater is subtracted from the 10, and that which results is multiplied, and from this multiplication results 1/3 107. (p. 567)
FLA022 Again I separated 10 into two parts, and I divided the first part by the second, and the second by the first, and those that resulted from the divisions I added to the 10, and I multiplied this sum by the first part, and 114 resulted. (p. 567)
FLA023 I separated 10 into two parts, and I divided the greater by the lesser, and that which resulted I multiplied by the difference between the parts, and 24 resulted. (p. 568)
FLA024 I separated 10 into two parts, and I divided the first part by the second, and the second by the first, and I multiplied the sum of the quotients by the first part, and 34 resulted. (p. 569)
FLA025 I separated 10 into two parts, and I divided the first part by the second, and the second part by the first, and I multiplied the difference between the quotients by the first part, and it was 5. (p. 569)
FLA026 I separated 10 into two parts, and I divided one by the other, and that which resulted I added to the part by which I divided, and there was ½ 5. (p. 569)
FLA027 I separated 10 into two parts, and I divided the first part by the second, and to that which resulted I added the first part, and the total I multiplied by the second part, and it made 30. (p. 570)
FLA028 I separated 10 into two parts, and I divided the first part by the second, and that which resulted, I multiplied by the first part, and it made 9. (p. 570)
FLA029 There is a census from which if 72 is subtracted, then there will remain the root of it. (p. 570)
FLA030 There were two numbers, the greater of which exceeded the lesser by 6, and I divided the lesser by the greater, and 1/3 resulted. (p. 570)
FLA031 There was a number. I cast out one third of it and 4 denari, and of that which remained I cast out one fourth, and that which remained was the root of the first number. you put the census for the number. (p. 571)
FLA032 There was a census, of which I cast out one third, and that which remained I multiplied by three roots of the census, and there resulted the same census. (p. 571)
FLA033 Also there is a census from which I subtracted 3 roots of it, and I added 4 roots of the difference to the root of the census, and there resulted 20. (p. 571)
FLA034 And it is said, there was a census from which I subtracted 8 roots of it, and I added the 8 roots to 10 roots of the difference, and the census plus 21 denari resulted. (p. 572)
FLA035 Also let it be said, there was a census, 4 roots of which I multiplied by 5 roots of it, and that which resulted was four census plus 48 denari. (p. 572)
FLA036 Also there was a census, 1/13 of which was equal to 1/7 of the root of it. (p. 572)
FLA037 Also there is a census which, if you multiply it by quadruple it, yields 20. by this rule, when you multiply it by itself, 5 results. (p. 573)
FLA038 Also there was a census that I multiplied by a third of itself, and 10 resulted. considering this, when you multiply the census by itself, then 30 results. (p. 573)
FLA039 Also there is a census which multiplied by the quadruple of itself yields a third of a denaro. (p. 573)
FLA040 Also there is a census which multiplied by the root of it yields triple the census. (p. 573)
FLA041 Also I multiplied a third of a census plus 1 denaro by a fourth of it plus two denari, and the census augmented by 13 denari resulted. (p. 573)
FLA042 There is a number from which if is subtracted ¼ 1/3 of it and 4 denari, then the root of it will indeed remain. (p. 573)
FLA043 I multiplied a number by 4 roots of it, and there resulted septuple it. when­ever a number will be multiplied by another number, and from the multiplication results septuple the multiplied number, then the number is only multiplied by 7. (p. 573)
FLA044 Also there was a number, and I cast out a fourth of it, and the remaining I multiplied by 4 roots of it, and sextuple it resulted, (p. 574)
FLA045 Also there was a number, and I cast out 4 roots of it, and I took 1/4 of the residue, and it was equal to 4 roots. (p. 574)
FLA046 Also there was a number, and I cast out 3 roots of it, and that which remained was the root of quadruple the number. (p. 574)
FLA047 Again there is a number which when multiplied by 2/3 of it yields 5. (p. 574)
FLA048 Also there is a number from which a third of it and 6 denari are subtracted. if the difference will be multiplied by itself, then double the number will result. (p. 574)
FLA049 And if it is proposed that the multiplication of ad. by itself yields the number .ab. when augmented by 18 denari, then you find a number which multiplies by the number ag. yields the number .ab., and it will be ½ 1 which is the line segment .ae.. (p. 575)
FLA050 Also there is a number, and I cast out a third of it and 6 denari, and that which remained I multiplied by 5, and the same number resulted. (p. 575)
FLA051 In a certain negotiation a certain man had 12 pounds of capital, and he made a certain profit in three months; to this total, namely to the capital and the profit, another man added 11 pounds, and on all of this the profit was proportional to the first profit, and at the end of twelve months the profit was some amount, and the total profit of twelve months and three months was 9 pounds; it is sought how much of the profit belongs to each of them, and how much was the profit in each month per pounds. (p. 576)
FLA052 Let there be found a number which multiplied by itself plus the root of 10 makes nonuple the number. (p. 577)
FLA053 And let it be said that the multiplication of .ab., namely the number mul­tiplied by itself, plus the root of 10, yields nonuple the square of the number .ba.. (p. 577)
FLA054 Also there is a number, which multiplied by itself plus the root of 10, yields 20. (p. 578)
FLA055 I multiplied octuple the root of a certain number by triple the root of it, and to the product that resulted I added 20 denari, and the total was equal to the square of the number. (p. 578)
FLA056 Let it be said, I multiplied the root of octuple a certain number by the root of triple it, and to the product I added 20, and from all this resulted the square of the number. (p. 578)
FLA057 Again I multiplied the root of sextuple any number you wish by the root of quintuple it, and I added decuple it plus 20 denari and there were in all as many as the number multiplied by itself. (p. 579)
FLA058 I separated 10 into two parts, and I multiplied one by the other, and that which resulted, I divided by the difference between the parts, and the root of 6 resulted. (p. 580)
FLA059 Also I separated 10 into two parts, and I multiplied one of them by the root of 8, and the other by itself, and I subtracted that which resulted from the multiplication of one part by the root of 8 from that which resulted from the multiplication of the other part by itself, and there remained 40 denari. (p. 581)
FLA060 Also I separated 10 into two parts, and I multiplied one of them by the root of 10, and the other by itself, and those that resulted were equal. (p. 582)
FLA061 To a certain amount I added 10 denari, and I multiplied the sum by the root of 5. of this the root was taken and it was the aforesaid amount. (p. 582)
FLA062 There were two quantities, and one was 5 greater than the other, and I multiplied the greater quantity by decuple it, and I took the root of that which resulted, and it was the same as the lesser quantity multiplied by itself. (p. 582)
FLA063 Also there are two numbers, one of which exceeds the other by 5, and I multiplied the greater of them by the root of 8, and the lesser by the root of 10, and those that resulted were equal. you put the thing for the lesser number, and the greater will be the thing plus 5 denari. (p. 583)
FLA064 When there are three quantities in continued proportion that have some other given quantity in the same proportion to another quantity, the multiplication of the least quantity of the two quantities by the sum of the middle and greatest of the three quantities will be the same as the multiplication of the greatest of the two quantities by the sum of the middle and least of the three quantities. (p. 584)
FLA065 And because we wish to find two numbers, one of which exceeds the other by 5, let the multiplication of the greater of them by the root of 2 be equal to the multiplication of the smaller by the root of 8. (p. 585)
FLA066 I multiplied a certain amount by double it, and I added 2 to the root of the product, and I multiplied the sum by the aforesaid sought number, and 30 denari resulted. (p. 586)
FLA067 I separated 10 into two parts, and I divided the greater by the lesser, and the lesser by the greater, and I added the quotients, and the sum was the root of 5 denari. (p. 587)
FLA068 And if you wish to find the root of the difference of the 225 and the root of 50000, (p. 588)
FLA069 I separated 10 into two parts, and I divided the first by the second, and the second by the first, and I added the quotients together, and the total was the root of 5. (p. 589)
FLA070 I separated 12 into two parts, and I divided each one of the parts by the other, and I multiplied each quotient by itself, and the sum was 4 denari. (p. 591)
FLA071 I separated 10 into two parts, and I divided each part by the other part, and I multiplied each quotient by itself, and I subtracted the lesser from the greater, and 2 denari remained. (p. 592)
FLA072 I separated 10 into two parts, and I divided 10 by each of them, and the sum of the quotients was 50 denari. (p. 594)
FLA073 I separated 10 into two parts, and I divided the first by the second, and the second by the first, and the sum of the quotients was 3 denari. (p. 595)
FLA074 Or in another way, you separate the aforesaid 3 into two parts, one of which you multiply by the other making 1. (p. 595)
FLA075 From a certain amount I subtracted two roots of it and 4 denari, and I multiplied the difference by itself, and there resulted octuple the desired number. (p. 596)
FLA076 There is a certain amount for which two roots of it plus the root of one half of it plus the root of one third of it is equal to the amount. (p. 597)
FLA077 There is a certain amount for which 2 roots plus the root of one half of it plus the root of one third of it is 20 denari. (p. 597)
FLA078 And if you will say : from a certain amount I subtracted two roots of it, and the root of one half of it, and the root of one third of it, and there remained 20 denari, (p. 598)
FLA079 And let us say to you : add to a certain amount 4 roots of it, and the root of one half of it, and the root of 2/3 of it, and there will be 10 denari. How much is the amount? (p. 598)
FLA080 Also to a certain amount I added the root of it and the root of one half of it, and that which resulted I multiplied by itself, and 20 denari resulted. (p. 599)
FLA081 Also to a certain amount I added the root of one half of it, and I multiplied the sum by itself, and quadruple the amount resulted. (p. 599)
FLA082 I multiplied a certain amount plus the root of 3 by the same number plus the root of 2 denari, and 20 denari resulted. (p. 599)
FLA083 To a certain amount I added 7 denari, and I multiplied the sum by the root of triple the amount yielding decuple the amount. (p. 600)
FLA084 To each of two unequal quantities, one of which is triple the other, I added the root of it, and I multiplied one of the sums by the other, and there resulted decuple the greater quantity. (p. 600)
FLA085 I took the root of a certain amount, and the root of the root of it, and the root of 2 times it, and the root of quintuple it, and the sum of all these makes 10 denari. (p. 601)
FLA086 There are three unequal quantities. if the least is multiplied by the greatest then the product will be the same as the middle multiplied by itself. and if the greatest is multiplied by itself it will be as much as the sum of the smallest multiplied by itself and the middle multiplied by itself. and the smallest multiplied by itself yields 10. (p. 601)
FLA087 And let it be said, I separated 10 into 3 parts, and the least multiplied by the greatest was the same as the middle multiplied by itself. and the multiplication of the least by itself, plus the middle part multiplied by itself, was as much as the multiplication of the greatest part by itself. (p. 603)
FLA088 And if we wish to find the greatest part, then you put one denaro for it, and the root of the thing for the middle part, and the thing for the least part, so that the thing multiplied by the denaro is the same as the root of the thing multiplied by itself. (p. 604)
FLA089 And let us say, I separated 10 into two parts, and I subtracted from the greater part two roots of it, and to the lesser part I added two roots of it, and those that resulted were equal. (p. 607)
FLA090 I shall separate 10 into two parts (p. 610)
FLA091 Also I separated 10 into two parts, and I divided the 10 by each of the parts, and I multiplied one of the quotients by the other, and ¼ 6 resulted (p. 611)
FLA092 And let us say, I made two parts of 10, and I divided 20 by each of them, and the sum of the quotients was ½ 12. (p. 612)
FLA093 And let us say, I made two parts of 10, and I divided 20 by each part, and I multiplied one quotient by the other, and 25 resulted. (p. 613)
FLA094 Again I separated 10 into two parts, and I divided 40 by one of them, and 50 by the other, and I multiplied one of the quotients by the other, and 125 resulted. (p. 614)
FLA095 And let us say to you, I made two parts of 10, and I divided 10 by each of them, and the sum of the quotients I multiply by itself yielding ¼ 20. (p. 615)
FLA096 And suppose it was said, I separated 10 into two parts, and I divided 10 by each of them, and that which resulted I multiplied by itself, and 30 denari resulted. (p. 615)
FLA097 Also I separated 10 into two parts, and I divided 40 by each of them, and the sum of the quotients I multiplied by itself, and 625 resulted. (p. 615)
FLA098 I separated 10 into two parts, and I divided 10 by one of them, and that which resulted I multiplied by the other part, and ¼ 20 resulted. (p. 615)
FLA099 And let us say to you, I multiplied 30-fold of a census by 30, and that which resulted was equal to the sum of 30 denari and 30-fold the same census. (p. 615)