**Euclid’s Elements from the West to the East**

**Mao Dan***Study on 1533 Greek First Prints of Elements: Terminologies and Compares *

The two original prints from which the earliest Chinese *Elements* were translated, namely, that compiled by Clavius in Latin and that translated by Sir Henry Billingsley into English, are both of some ambiguity concerning their own originations. As for the latter, theories favoring its Greek origin were dominant for most of the time, supported by its Chinese translator Alexander Wylie among other scholars such as Augustus De Morgan in 1837 and G. B. Halsted in 1879.

It’s not until 1950 when R. C. Archibald published his treatise comparing Billingsley’s notes on Simon Grynaeus’ 1533 Greek prints and 1558 Campanus-Zamberti’s, that the theory of Latin origin became persuasive (though Zamberti’s Latin version was strictly translated from Greek). Later it was adopted and further developed by Dr. Cao Jingbo in her doctoral thesis in 2018.

But there’s still logical problems that need to be dealt with. Archibald denied the possibility of 1533 edition being what Billingsley’s translation was based, mainly on the basis that Billingsley took much fewer notes on this copy than on 1558 version of Campanus-Zamberti’s edition, and that Billingsley in his comprehensive citations didn’t include (later as Dr. Cao Jingbo argured, there are actually two notes concernning Zamberti anyway) any of Theon-Zamberti’s. If, however, these arguments could alone be regarded as decisive, why bother to compare the other translations that Billingsley’s *Elements* cited, in their construction and in their way of proof? They didn’t even bear a single note of Billingsley’s, after all, neither did they be excluded from citation.

So, we should add the first Greek prints of *Elements* in 1533 back into the above-mentioned comparisons, for which to be made more forceful.

The further comparison could be carried out in 3 ways. Firstly, terminologies: to what extent the 1533 system of expression coincides with Billingsley’s, comparing with other translations cited, and finally reflected in the Chinese translation? Secondly, the already practiced comparison in construction and way of proof; Finally, characteristics that might be traceable during process of translation such as grammatical inconsistencies or choices between synonyms.

The 16th century witnessed rise of interests to translate directly from Greek rather than through media languages. That’s partly the reason why the theory of Greek origin had been prevailing for long. And caution should be applied in judging whether we’ve swung too far to the other direction.

**Tian Chunzhi***Viewing Pacioli’s Study of Euclid’s Elements from Summa*

Luca Pacioli,as a famous mathematician and “The Father of Accounting and Bookkeeping”in Europe,published his first influential compendium of 15th century mathematics in 1494 in Venice.The book called “Summa de Arithmetica,Geometria,Proportioni et Proportionalita”(hereinafter reffered to as Summa),which covers essentialy all of the Renaissance mathematics.Summa includes practical arithmetic,algebra,commercial mathematics(business and trade)and elementary geometry.In this book,Pacioli quoted so much knowledge from Fibonacci,Euclid,Piero della Francesca etc.Besides,he published a latin translation of Euclid’s Elements.Therefore,he must be much more familiar with Elements and cited geometrical part in Summa.This paper tries to explore Pacioli’s quotation,understanding and application of Elements.

The following two parts will be discussed in the paper:**Pacioli’s quotation of Elements**. The paper aims to investigate Pacioli’s standers of quoting axioms,postulates,definitions and propositions in order to explore the reasons why he cited these.Some think Summa as a textbook,others think it was mainly written for,and sold mostly to merchants.To a large degree,readers’purchasing power depends on the degree of difficulty of the contents.Knowing about which axioms,postulates and definitions is helpful to explore the market for Summa.**Pacioli’s understanding and application of Elements**. Summa marks the beginning of the movement of algebra in 16^{th}century,which tends to use the logical argumentation and theorems in the study of algebra,following the model of classical Greek geometry founded by Euclid.In algebraic part,how he used the geometrical model to solve algebraic problems.

**Kostas Nikolantonakis***The Elements and their transmission *

Euclid’s *Elements* is the most published and translated treatise. This can be explained by the fact that the *Elements* have had two careers: 1. Provide a synthesis of “basic” knowledge in a given field, required for a “first” learning, as a monograph on geometry and arithmetic (theory of numbers) which knowledge is very important for learning mathematics and 2. Simultaneously embody a model of reasoning, in our case, a paradigm of demonstrative and deductive reasoning which should be followed from all scientists (mathematicians, physicians etc.) in their scientific treatises.

In this paper we are going to describe the transmission of the text to Greek, Arabic and Latin mathematical traditions but also the editions in different languages (among these the Chinese) in different historical periods.

**Cao Jingbo***Studies on the commentaries in Billingsley’s English Element 1570*

The work on Euclid’s Elements of Renaissance European Scholars is investigated by reading their comments and annotations collected in Billingsley’s Elements. The age of the commentators who is quoted by Billingsley’s Elements spans from ancient Greece to Renaissance, especially in the 100 years before the birth of Billingsley’s Elements. In the first six books of Billingsley’s Elements quotes large numbers of commentators, for the Euclid’s Elements had been studied by successive scholars in one and a half millennium after its birth. These comments and annotations on the section of number theory, namely the Book VII, VIII and IX of Euclid’s Elements, shows how these ancient scholars studied the concept of “number” and “quantity”, which is originated from ancient Greece. They tried to extend the propositions of quantity in the first six books to “number”. Some significant thought rises from the study, such as the thought of infinite descent from Campanus. The annotations on the tenth books of Billingsley’s Elements explain the nomenclature of the 13 irrational lines, so they become valuable guides of people to understand the tenth book of Elements. The section of solid geometry of Billingsley’s Elements contains most of the comments and annotations. The long annotation of John Dee states the significance of solid geometry in mechanics and engineering. The supplement of Flussates reflects an atmosphere of occultism in European Mathematics in the sixteenth century.

**Ji Zhigang***How Did the Knowledge of Geometry Affect China: A Cultural Interpretation of Ricci’ Preface of Jihe yuanben*

The year of 1607 saw the Chinese translation of Euclid’s *Elements*(*Jihe yuanben*). This event is considered as a new epoch of the communication between East and West. Chinese Scholar Liang Qichao(1873-1929) has even praised the* Jiheyuanben* “each word is like a pure gold or a beauty jade, and the book would kept for more than thousand years. ” But the real significant of *Jihe yuanben* should not be coved by those over-praised words.

For the reason that Euclid’s* Elements* is some of alien culture to Chinese Scholars, so Mateo Ricci wrote a long preface, which formed an important medium to express his opinions about Western-Learning and to promote the transmission of the Geometry in China. This paper tries to explore the significances of Ricci’s prefaces, and also to show the Chinese scholars’ enthusiasm echoes to *Jihe yuanben* in the late Ming and early Qing.

The following three aspects will be discussed in this paper.

(1)** Knowing the Nature Principle from the Geometry** At the beginning his preface, Ricci pointed that if a Confucian scholar wants to extend his knowledge, he should to investigate the principle of nature, but the nature principle implies in the mathematics. “*So in order to making the extending knowledge to be depth and solidity, nothing surpasses the knowledge of mathematics*.”**(2) Doing the Practical Matters by the Geometry** For emphasizing the practice of geometry, Ricci listed more than ten aspects, such as astronomy, measurement, creating instruments, architecture, agriculture, geography and so on. So Ricci gave his opinion that *“All those skills fall directly under the realm of mathematics. [Moreover], as far as all the various professions are concerned, the important principles and the subtle touches depend to a considerable extent on mathematical theory.”***(3) Emphasizing the Multiple Power of the Geometry** Except for the civil, the military affairs, which is the basis of national security and of the major affairs of state, are some utmost need of mathematics. Ricci even told the story that Archimedes, with a small and weakened group of soldiers, resisted against the Rome army. This story proves that only the power mathematic can gain the victory with a small and feeble army over a strong and large army. So Ricci declared “ *a wise and courageous general will be deeply convinced of the importance of the study of mathematics.*”

As a co-translator, Xu Guangqi is a first Chinese scholar who knew the utmost value of *Jihe yuanben*. Xu said “ *Jihe yuanben is the ancestor of measures and numbers, ……In truth it can be called the pleasure-garden of the myriad forms, the erudite ocean of the hundred schools*” Xu even expressed his hope that “*after hundred years everybody will study it and that they will them realize that they have started learning too late*.”

All those word tell us the knowledge of Geometry did affect China.