Dr. Maria D. Chalkou
Ph.D., M.Sc. Dept. of Mathematics, University of Athens.
State School Advisor.
Home address: Vitolion 159- 18546, Piraeus, Greece.
e-mail: firstname.lastname@example.org or email@example.com
Arithmetical operations, fractions, progressions, linear equations and roots of real numbers, according to the Codex Vindοbonensis phil. gr. 65 of the 15th century.
I will present you some few results of my study on the mathematical content of the anonymous Codex Vindobonensis phil. Graecus 65.This 15th century (1436) Byzantine MS which named Tractatus Mathematicus Vindobonensis Graecus and which I propose to be digitizated, includes the solution of problems of practical arithmetic, algebra, and geometry the roots of which can be traced back to antiquity.
The symbols, which are used in the manuscript are the letters of the Greek alphabet but the calculations are carried out with the new decimal Hindu-Arabic system of numeration. Even though the author is not used to the new symbolisation, it should be emphasised that the use of letters and not numbers does not affect the result, since it concerns a system in which the arithmetical value of a letter depends upon its place . Thus, the author insisted on preservation of the old symbols, whilst other earlier scholars, such as Planudes (1255-1305 A.D.) in Byzantium and Fibonacci (born in 1170), who introduced the new arithmetical symbols in Western Europe, were familiar with the new system. However, the use of the new numbers was not generalized during the Byzantine period because their use created various problems in commercial mathematics.
In the codex the term “milliouni” is mentioned which means a million. According to D. E. Smith, this term first appeared in 1478 in the Italian manuscript “Arithmetic of Treviso”. We therefore have an important indication that the term “milliouni” did not first appear in the Italian “Arithmetic of Treviso” but in Codex 65, which appears to date back to 1436 A.D.
In 1494 Luca Pacioli issued the “Summa” which was the first mathematical encyclopaedia of the Renaissance. The first part includes Arithmetic and Algebra and the second part Geometry, exactly as in our MS. Pacioli used the Hindu numerals  in “Suma” and calls the “crosswise method” of multiplication “crocetta” (little cross). For example in multiplication of 12 with 13, initially the 2 was multiplied with 3 to make 6. Further, the “crosswise” digits of 12 and 13 were multiplied as in the codex 65, and the results are added, so we have 5. The 5 represents the decades and the 6 the units. Further multiplication of the first digits of the numbers 12 and 13 arrives at 1. The 1 represents the hundreds and thus the final result is 156. This procedure is found in the codex 65 too.
In the same work, Pacioli who taught arithmetic and commercial algebra mentions to the method of “four-sided” in multiplication of two 3-digit numbers, in which the number which multiplies is made descending downwards from the number which must be multiplied. However this is exactly how multiplication of three digit numbers is done in Codex 65, which is older than the “Suma” . The similarities of this Codex in relation with the “Suma” and with the “Arithmetic of Treviso” do not stop here since in the second one, the division is done in a similar way to that of Codex 65  .
Of course, the interactions between the Byzantines and Western are undoubted since Planudes makes division using the Fibonacci method, which is also identical with the method used in our MS.
To test the multiplication the anonymous author requires the remainder of the division of 15 by 7, which is 1. Because the remainder of the division of 6 by 7 is 6, multiplication of 1 with 6 placing the remainder in a circle. Finally the remainder is found in division of 90 by 7, which is 6, to be compared with the number, which has been placed in a circle. Since the two results are the same, then the multiplication is correct.
The Hindu used that method, by dividing by 9 instead of 7. Al Khwarizmi (c. 825 A. D.) was familiar with this as well as Al Karkhi (c. 1020 A.D.), who are even more ancient than the actual date of Codex 65. We also know that the Arabs had adopted this using of course the number 7, as well as 8, 9 and 11, but the check by 7 according to Fibonacci, Planudes and others ensures a very little possibility of error . The same opinion was expressed by the author of our MS.
Although this procedure is not in use any more, I found it in a 20th century’s book with title ‘‘A detailed description of Theoretical Arithmetic for Practical Schools’’ of Secondary Education written by N. Nikolaou, which taught in the fifties . This does not mean of course, that the aforementioned method was taught up to that time continuously at all schools. Immediately after the fall of Constantinople, the lower schools taught the ‘‘Arithmetic’’ written by Emmanuel Glyzonios for more than two and a half centuries. In this Arithmetic, the check of multiplication was done by the crosswise method .
In the MS the way of defining a fraction is based on the condition that the numerator must be smaller than the denominator. The same notion is extended, within the same Codex to all type of fractions. The most unusual thing is that in the Arithmetic of Pagani written in 1591 A. D. the numerator is less than the denominator, whilst all the other type of fractions is considered according to some researchers to be a subsequent discovery .
In Codex 65 the operations between fractions are carried out using methods similar to those of today. This is another indication of the unbroken tradition of mathematical methods until today .
In another chapter the author deals with problems, which are easily solved today by using linear equations, despite the fact that he himself however solves them with practical arithmetic. As is well known, the problems of equations of first order have there roots in antiquity . It is worthwhile noting, that these problems were found in Arithmetic books which were considered more advanced than the usual ones . This indicates that Codex 65 was probably a worthy Arithmetic of its time.
A customary method used at that time was the one of “false assumption” which leads the author, as is to be expected, to a false conclusion result, so he reaches the correct answer by applying the qualities of proportions .
The method of “false assumption” was particularly beloved by Diophantus, and was taught at schools in Europe and America up to the 19th century. It seems that it was very well known in Medeaval times since Fibonacci related to it in his works  and used it often in problems .
Another type of problem relates to movements for meeting or removal of ships or persons.
Metrodoros is considered as the main creator of these problems, which belong to recreational mathematics, and, as Smith asserts , they first appeared in the West in 1483 and were found in the manuscript ‘‘Suma’’ of Luca Pacioli, written in 1494. If Smith’s assertions are correct, it is very likely that Codex 65 is the source from which Pacioli drew subjects, when he wrote his Suma.. The question therefore arises, concerning the relationship of Codex 65 with the other two manuscripts, namely the Suma and the Arithmetic of Treviso.
Of course, the Suma was not known for new discoveries in mathematics. However it gives us information about the mathematical knowledge up to its time and is considered that it laid the foundations for the further development of algebra in the 16th century. The Arithmetic of Treviso like the codex 65 contained problems of the four operations, problems on coins’ conversion, progressions, interests, undetermined analysis, equalization as well as assignation of the perfect number. It also contained geometry problems.
On the other hand it is certain that many Latin scholars who knew ancient Greek read Greek manuscripts and were influenced by them.
Thus in this case in order to reach certain conclusions, a comparison between the contents of those Italian works and that of Codex 65 is required.
In our codex the material of algebra includes the roots of real numbers, equations up to fourth level, and the system of equation up to second level.
In accordance with the methods of calculation of the square root it appears that the root of 30 is equal to 5 5/11 (chapter 123, f. 64v). The preferred method is the same as that of Omar Khayyam. If the calculation of the root of 30 is done with the method used by Planudes, which is based on the formula of the Hero of Alexandria , we will have as result 5+5/10 and not 5+5/11.
From a comparison between the method of the author of Codex 65 and that of Rabdas, at first glance it appears that the latter used Hero’s formula, and that also he further considered that if A had been the higher approximation of the root, then the A1=30/A was the less approximation, and the rate (1/2) (A +A 1) was considered from Rabdas as the better of these .
According to this formula the better approximation would be the number 5 21/ 44.
We observe that, when in the codex 65 is given approximately the root of 30, then the number 5 21/44 is found as the second approximation of this root (chapter 123, f. 64v, 65r), which agrees with the second approximation which is found by Rabdas, although their values for the first approximation do not agree; in the codex 65 is found the number 5 5/11 while Rabdas gives 5 5/10.
The methods of calculating a square root, which I referred to above, seem to have been abandoned within the years, and finally in the year 1494 Luca Pacioli gives a method, similar to the this one which was taught at schools of secondary education some years ago in Greece. Later, in 1546, Cataneo reaches more this method , which reminds the art of division and raises particular difficulties, for the students, in memorizing.
I have presented to you some few results of my study on the mathematical content of the published part (f. 11r-126r) of the Codex Vind. Phil. gr. 65 (Tractatus Mathematicus Vindobonensis Graecus or TractMathVindGr). This 15th century (1436) Byzantine MS includes as I have said the solution of problems of practical arithmetic, algebra and geometry, the roots of which can be traced back to antiquity and their comparison with modern mathematical solutions reveals –apart from some differences- many identities and similarities showing the unbroken continuity of mathematical tradition through the centuries. Moreover, my research has revealed so far some important results according to which we are probably in the position to give to the TractMathVindGr the title of the Byzantine encyclopaedia of Mathematics.