Translated and annotated by
Ian Bruce


From the Original Editor of a later German edition.


Herewith we hand over to lovers of the art of higher arithmetic a work, of which a Russian translation already has appeared two year ago.

[In 1768 & '69, translated by Inochodtzoff & Iudin; the original German was began in Berlin in 1765, according to examples used in the text, and for the first part finished at the latest in 1767, and the second part ready by 1768. See Heinrich Weber, in the introduction to the Opera Omnia edition of 1911.]

The intention of the world famous author by the same work was to make a textbook, from which anyone without any assistance could easily grasp and master the fundamentals of Algebra.

The loss of his eyesight awakened these ideas in him, and through the constant activity of his genius, he did not delay long to put his plan into action. To this end, he had himself chosen a young man, whom he had taken into his service in Berlin, and who could do arithmetic reasonably well, but otherwise had not the slightest idea about mathematics: he was a tailor to trade, and concerning his ability, he could not be placed amongst the average minded. On account of which, not only did this young man understand well enough everything his illustrious Master said or dictated to him, but also in a little while through being placed in that position, he was able to pursue and resolve in succession all the forthcoming difficult algebraic questions all by himself with much proficiency.  

This provides so much more praise about the discussions and teaching methods of the present work ; since the apprentice who has written it down, has understood and pursued it, without the slightest help otherwise from anyone other than that enjoyed from his truly famous but sightless teacher.

Besides the great advantage from these already, the expert can read with pleasure and admiration especially the teaching of logarithms and their association with the remaining ways of calculating, as well as the given methods for resolving of cubic and biquadratic equations. But lovers of Diophantine problems can rejoice over the last section of the second part, in which these tasks are treated in a pleasing coherent manner, and requisite stratagems for all their resolutions are to be explained.

Translated by Ian Bruce



PART I : Section I.

Click here  for Chapters 1 to 8 :


Ch. 1 (§1−§7): General math. study; Ch. 2: (§8−§22):  + & − signs;

Ch. 3: (§23−§36): Simple multiplication ; Ch. 4: (§37−§44): whole nos. & factors ;

Ch. 5: (§45−§57): Easy division ; Ch. 6: (§58−§67): Whole nos. & divisors ;

Ch. 7: (§68−§84): Fractions in general ; Ch. 8: (§85−§84): Properties of fractions.  


Click here  for Chapters 9 to 14 :


Ch. 9 (§94−§100): Add. & subtr. fractions. study; Ch. 10: (§101−§114): Mult. & div. fractions ;  Ch. 11: (§115−§122): Square nos. ; Ch. 12: (§123−§138): Sq. roots & irr. nos ;

Ch. 13: (§139−§151): Sq. roots & im. nos; Ch. 14: (§152−§157): Cubic nos. ;


Click here  for Chapters 15 to 23 :


Ch. 15 (§158−§167): Cu. roots & irr. nos ; Ch. 16: (§168−§179): Powers in gen. ; 

Ch. 17: (§180−§188): Calc. with powers ; Ch. 18: (§189−§194): Roots of powers ;

Ch. 19: (§195−§205): Fract. exponents; Ch. 20: (§206−§219): Cal. with powers;

Ch. 21: (§220−§231): Logs in gen.; Ch. 22: (§232−§241): Log. tables ; Ch. 23: (§242−§155): Expressing logs.


PART I : Section II.

Click here  for Chapters 1 to 6 :


Ch. 1 (§256−§262): Addition with composite magnitudes; Ch. 2: (§263−§269):  Subtraction with composite magnitudes;

Ch. 3: (§270−§281): Multiplication with composite magnitudes; Ch. 4: (§282−§288): Division by composite magnitudes ;

Ch. 5: (§289−§305): The resolution of fractions into infinite series ; Ch. 6: (§306−§316): The squares of composite magnitudes ;


 Click here  for Chapters 7 to 13 :


Ch. 7 (§317−§325): The extraction of square roots in composite magnitudes; Ch. 8: (§326−§332):  Calculating with irrational numbers;

Ch. 9: (§333−§339): Cubes and the extraction of cube roots; Ch. 10: (§340−§360): The magnitudes of higher composite powers ;

Ch. 11: (§352−§360): On the permutation of the letters on which the proof of the preceding rule is based ;

Ch. 12: (§361−§369): On the generation of irrational powers in infinite series ;

Ch. 13: (§370−§377): On the expansion of negative powers.


PART I : Section III.

Click here  for Chapters 1 to 7 :


Ch. 1 (§378−§389): Arithmetical proportions, or the differences between two numbers;

Ch. 2: (§390−§401): Arithmetical proportions;

Ch. 3: (§402−§411): Arithmetical progressions; Ch. 4: (§412−§424): Summation of A.P's ;

Ch. 5: (§425−§439): Polygonal figures or numbers ; Ch. 6: (§440−§450): Geometrical proportions ;

Ch. 7: (§451−§460): Greatest common divisor between two given numbers ;



Click here  for Chapters 8 to 13 :


Ch. 8 (§461−§476): Geometrical proportions;

Ch. 9: (§477−§487): Notes on proportions and their uses; mainly dealing with monetary exchanges;

Ch. 10: (§488−§504): Compound ratios; Ch. 11: (§505−§524): Geometrical progressions ;

Ch. 12: (§525−§539): Infinite decimal fractions ; Ch. 13: (§540−§5620): Calculating interest ;


PART II : Section I.


Click here  for Chapters 1 to 4 :


Ch. 1 (§1−§10): On the general solution of equations;

Ch. 2 (§11−§22): Equations of the first order and their solutions;

Ch. 3 (§23−§42): The solution of some relevant questions;

Ch. 4 (§43−§60): The solution of two or more equations of the first order.



Click here  for Chapters 5 to 9 :


Ch. 5 (§61−§75): On the solution of pure quadratic equations ;

Ch. 6 (§76−§93): On the solution of mixed quadratic equations;

Ch. 7 (§94−§106): On the extraction of the square roots of polygonal numbers;

Ch. 8 (§107−§126): On the extraction of the square roots of binomials;

Ch. 9 (§127−§143): On the nature of quadratic equations.



Click here  for Chapters 10 to 12 :


Ch. 10 (§114−§156): On the solution of pure cubic equations ;

Ch. 11 (§157−§171): On the solution of complete equations;

Ch. 11 (§172−§188): On Cardan’s Rule.



Click here  for Chapters 13 to 16 :


Ch. 13 (§189−§203): On the solution of equations of the fourth degree, or biquadratic equations ;

Ch. 14 (§204−§211): On Bombelli's Rule…..

Ch. 15 (§212−§222): On a new method of solving fourth degree equations.

Ch. 16 (§223−§240): On the solution of equations by approximations.


PART II : Section II.

Click here  for  :

Ch. 1 (§1−§23): On the solution of simple equations in which more than one unknown number is present;


Click here  for  :

Ch. 2 (§24−§30): On the solution of more general equations in which more than one unknown number is present. The basis if modular arithmetic, essentially.


Click here  for  :

Ch. 3 (§31−§37): Concerning a number of indeterminate equations taken together, in which one of the unknown numbers  only is of the first degree. Here whole positive number solutions are found for some equations of increasing complexity.


Click here  for  :  Ch. 4  (§38−§62) & Ch. 5  (§63−§78) : These chapters are devoted to finding the circumstances whereby a quadratic form is a perfect square, and conversely the circumstances in which such a form cannot be a perfect square.


Click here  for  :  Ch. 6  (§79−§95) ;  Ch. 7  (§96−§111) ;  Ch. 8  (§112−§127);  Ch. 9  (§113−§146): These chapters are devoted to finding the circumstances whereby higher order formulae can be perfect squares, and conversely the circumstances in which such a form cannot be a perfect square.


Click here  for  :  Ch. 10  (§147−§161) ;  Ch. 11  (§162−§180) ;  Ch. 12  (§181−§201);  Ch. 13  (§202−§211): These chapters are devoted to finding the circumstances whereby higher order formulae can be perfect cubes and biquadratics, and conversely for the roots.


Click here  for  :   Ch. 14  (§212−§240): This chapter is concerned with numerous examples in which the methods devised in the previous chapters are used, mainly for finding squares and fourth powers or biquadratic numbers.



Click here  for  :   Ch. 15  (§240−§250): This chapter concludes the work and deals with cubic equations. I should perhaps point out that this translation if from the original German of Euler, rather than via the French translation of Hewitt, which is often a paraphrase of a paraphrase, and so is occasionally far from the original wording, which I have tried to maintain, apart from the obvious inversions required. This work probably concludes my translations of Euler's text books, although I intend from time to time to add individual papers.


Ian Bruce. Aug. 2nd , 2016 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish by clicking on my name on the index page, especially if you have any relevant comments or concerns.