beta Euclid's Elements, Book VII  - Elementary Number Theory Definitions.
A journey from Greek, through Arabic, Sanskrit and English.

Dear friend, 

Here is a quick summary...

• In 300 BCE* Euclid of Alexandria wrote 'The Elements' in Greek. It presents an overview of Greek mathematics at the time. While Euclid wrote 'The Elements' it contained the mathematics of both Euclid and others. Book VII on elementary number theory is considered to be Pythagorean and dates back to around 540 BCE.  
  

• Around 1220 CE^, in what is now modern day Iran, 'The Elements' was critically edited and translated into Arabic by Muslim scholar Nasir al-Din al-Tusi.
  

• From this Arabic edition it was translated into Sanskrit by Jagannātha Samrāt in 1719 CE and called The Rekhāgaṇita.
  In 1902 the Sanskrit Rekhāgaṇita version of Euclid's Elements was edited and issued by K.P. Trivedi. I have extracted the number theory definitions from Vol. 2 of The Rekhāgaṇita. 

• From Trivedi's Sanskrit edition, in 2013 Professor Avinash Sathaye provided a translation at my request.
  
  Thank you for visiting!
  
  Jonathan Crabtree
  Melbourne Australia
  P.S. A variety of English translations of Euclid's number theory definitions (post Heiberg) can be read here.
  

तत्रैकोनचत्वारिंशत्क्षेत्राणि सन्ति
There are 39 sections


अत्राङ्कैर्गणितप्रकारा निरूपिताः
Here techniques of calculations with numbers are described

---

As the order of the definitions vary, it is better to read down the Greek definitions and then read down the Sanskrit definitions. The English translations below are © Dr Henry Mendell. Source.
α (Elements Definition 1) Μονάς ἐστιν, καθ' ἣν ἕκαστον τῶν ὄντων ἓν λέγεται Unit is that according to which each of the things which are is one.	१ (Rekhāgaṇita Definition 1) अङ्को नाम रूपाणां समुदायः। तन्मते रूपेऽङ्कत्वाभावः। अन्ये तु गणनायोग्यमङ्कं वदन्ति तन्मते रूपेऽप्यङ्कत्वमस्ति गणनायोग्यत्वात्         A number is a collection of units. In that sense, the unit (1) does not have the property of being a number. Others say that what can be counted is a number, and in their opinion unit has the property of being a number, for it can be counted.
β (Elements Definition 2) Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος and the multitude composed from units is a number.	२ (Rekhāgaṇita Definition 2) यत्र लघ्वङ्को बृहदङ्कादसकृत् शोधितः सन् बृहदङ्को निःशेषः स्यात् तदा लघ्वङ्को बृहदङ्कस्यांशोऽस्ति।बृहदङ्को गुणगुणितलघ्वङ्कतुल्योऽस्ति When a small(er) number is severally removed from a big(ger) number and the big number becomes without remainder, then the small number is a factor of the big number.  (In this case) the big number is the multiple times the small number.
γ (Elements Definition 3) Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα A number is a part of a number, the smaller of the larger, whenever it measures the larger.	३ (Rekhāgaṇita Definition 3) यस्य भागद्वयं समानं भवति स समाङ्को ज्ञेयः One which has two equal parts should be known as an even number.
δ (Elements Definition 4) Μέρη δέ, ὅταν μὴ καταμετρῇ and parts whenever it does not measure	४ (Rekhāgaṇita Definition 4) यस्य भागद्वयं समानं न भवति स विषमाङ्को ज्ञेयः One which does not have two equal parts is an odd number.
ε (Elements Definition 5) Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος and the larger is a multiple of the smaller whenever it is measured by the smaller.	५ (Rekhāgaṇita Definition 5) समाङ्को यद्येकेन हीनोऽधिको वा भवति सोऽपि विषमाङ्को ज्ञेयः When an even number is reduced or augmented by 1, it is also known as an odd number.
ϛ (Elements Definition 6)  Ἄρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος The number which is divided in two is an even number,	६ (Rekhāgaṇita Definition 6) समाङ्को द्विविधः। एकः समसमः  ८। एकः समविषमः ६   An even number is of two types: one even-even (8), another odd-even (6).
ζ (Elements Definition 7) Περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ [ὁ] μονάδι διαφέρων ἀρτίου ἀριθμοῦ and the number which is not divided in two is odd, or the number which differs from an even number by a unit.	७ (Rekhāgaṇita Definition 7) समसमो यथा। समाङ्कः समेन ह्रियमाणः समा लब्धिः प्राप्यते स समसमः Even-even is thus: an even number when it gives an even result after division by an(y) even number, then it is even-even.
η (Elements Definition 8) Ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν The number measured by an even number taken in groups of an even number is an even-times even number,	८ (Rekhāgaṇita Definition 8) यः समाङ्कः समेन ह्रियमाणः विषमा लब्धिः प्राप्यते स समविषमो ज्ञेयः  An even number when it gives an odd result after division by an even number, it is known as  even-odd.
θ (Elements Definition 9)  Ἀρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν and the number measured by an even number taken in groups of an odd number is an even-times odd number,	९ (Rekhāgaṇita Definition 9) अथ विषमविषमाङ्कलक्षणम्।  विषमाङ्को विषमेण ह्रियमाणः विषमा लब्धिः प्राप्यते स विषमविषमाङ्कः। यथा नवमाङ्कः  ९ त्रिभक्तः त्रयं प्राप्यते  Now the definition of odd-odd. An odd number, when it gives an odd result when divided by an odd number is an odd-odd number. For example 9 when divided by 3, one gets 3.
ι (Elements Definition 10)  [Περισσάκις ἄρτιός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν.] [and the number measured by an odd number taken in groups of an even number is an odd-times even,]	१० (Rekhāgaṇita Definition 10) योऽङ्को रूपातिरिक्ताङ्केन निःशेषो न भवति स प्रथमाङ्को ज्ञेयः। यथैकादशाङ्कः  A number, which when divided by a number bigger than 1 does not leave a zero remainder; it is to be known as a prime number. The number 11 for example.
ια (Elements Definition 11) Περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν and the number measured by an odd number taken in groups of an odd number is an odd-times odd number.	११ (Rekhāgaṇita Definition 11) यो रूपातिरिक्तेन विभागार्हः स योगाङ्को ज्ञेयः  One that can be divided by a number bigger than 1; it is to be known as a composite number.
ιβ (Elements Definition 12) Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος The number measured only by a unit is a prime number.	१२ (Rekhāgaṇita Definition 12) यावङ्कौ रूपातिरिक्तेन भक्तौ निःशेषौ भवतस्तावङ्कौ मिलितसंज्ञौ ज्ञेयौ  Two numbers, which give a zero remainder when divided by a number bigger than 1 should be known as joined. [Professor Sathaye comment - In modern terminology, these are non-coprime.]
ιγ (Elements Definition 13) Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ Prime numbers relative to one another are those measured only by a unit as a common measure.	१३ (Rekhāgaṇita Definition 13) यावङ्कावेकातिरिक्तः कोऽपि हरो निःशेषं न करोति तौ भिन्नाङ्कौ ज्ञेयौ Two numbers which don't give zero remainder by any any number bigger than 1 are to be known as separated numbers. [Professor Sathaye comment - In modern terminology, these are coprime.]
ιδ (Elements Definition 14) Σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος Compound number is a number measured by a number,	१४ (Rekhāgaṇita Definition 14) योऽङ्कः स्वेनैव गुणितः फलं तस्यैव वर्गो भवति   A number multiplied by itself; its result is a square (number).
ιε (Elements Definition 15) Σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ τινι μετρούμενοι κοινῷ μέτρῳ and compound numbers relative to one another are those measured by a number as a common measure.	१५ (Rekhāgaṇita Definition 15) योऽङ्कः स्ववर्गेण गुणितः घनसंज्ञो भवति   A number multiplied by its square is known as a cube.
ζ (Elements Definition 16) Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις A number is said to multiply a number whenever as many units as there are in it, so many times the multiplied number is added and becomes some number. RESEARCH NOTE: Refer to this page to discover why this translation is incorrect.	१६ (Rekhāgaṇita Definition 16) गुण्याङ्कगुण्काङ्कयोर्घातो गुणनफलं क्षेत्रफलं भवति  The result of multiplication of the number to be multiplied and the multiplier is called an area (measure of a field).
ζ (Elements Definition 17) Ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί Whenever two numbers multiply one another and make some number, the number which results is called plane, and it sides are the numbers multiplying one another,	१७ (Rekhāgaṇita Definition 17) गुण्यगुणकौ भुजसंज्ञौ भवतः   The one to be multiplied and the multiplier are called parts (arms).
ζ (Elements Definition 18) Ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί and whenever three numbers multiply one another and make some number, the number which results is called solid, and its sides are the numbers multiplying one another.	१८ (Rekhāgaṇita Definition 18) क्षेत्रफलं केनचिदङ्केन् गुणितं क्षेत्रफलं भवति   An area multiplied by a number becomes volume.
ζ (Elements Definition 19) Τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ [ὁ] ὑπὸ δύο ἴσων ἀριθμῶν περιεχόμενος A square number is the equal-times equal number or the number enclosed by two equal numbers,	१९ (Rekhāgaṇita Definition 19) यत्र प्रथमाङ्को यद्गुणितो द्वितीयाङ्कतुल्यो भवति तद्गुणगुणितस्तृतीयाङ्कश्चतुर्थाङ्कतुल्यो भवति तदा तेऽङ्काः सजातीया भवन्ति  When a first number multiplied (by some multiplier) becomes the second number; and a third number multiplied by the same multiplier becomes the fourth number; then they become similar (proportional ) numbers.
ζ (Elements Definition 20) Κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ [ὁ] ὑπὸ τριῶν ἴσων ἀριθμῶν περιεχόμενος and a cube is the equal-times equal equal-times or enclosed by three equal numbers.	२० (Rekhāgaṇita Definition 20) क्षेत्रफलघनफले ते सजातीये भवतो ययोर्भुजावेकरूपौ सजातीयौ भवतः   Area and volumes become similar (proportional) when their parts (arms) are proportional in the same manner. [Professor Sathaye comment - The same manner is not explained, but probably means using the same multiplier.]
ζ (Elements Definition 21) Ἀριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ δευτέρου καὶ ὁ τρίτος τοῦ τετάρτου ἰσάκις ᾖ πολλαπλάσιος ἢ τὸ αὐτὸ μέρος ἢ τὰ αὐτὰ μέρη ὦσιν Numbers are proportional whenever the first is an equal multiple or the same part or the same parts of the second as the third of the fourth.	२१ (Rekhāgaṇita Definition 21) योऽङ्कः स्वलब्धियोगतुल्यो भवति स पूर्णसंज्ञो ज्ञेयः। यथा षट्  A number which is equal to the sum of its parts (factors) is known as complete (perfect).
ζ (Elements Definition 22) Ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς Similar plane and solid numbers are those having proportional sides.
ζ (Elements Definition 23) Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν A perfect number is one which is equal to all its parts.

---

You will see that Jagannātha Samrāt does not provide a definition of multiplication to match Euclid's. This may be because the Arabic definition of multiplication in al-Tusi's text may have differed significantly from Euclid's.

The Arabic definition of multiplication** contained in a 1594 edition of the Elements reads as follows: "Multiplication is when one makes one of two numbers a unit of another number.  So the unit of the units of the multiplicand is assigned [the value of] the multiplier, and what is gathered is the result of the multiplication." This was not intended to be a literal translation of Euclid's Greek, but rather a modification of Euclid's book for a medieval 16th century Arabic reading audience.

The 1594 Arabic text attributed to al-Tusi (yet written several decades fter his death) containing Euclid's elementary number theory definitions is provided below.

1594 Arabic edition of Euclid's Elements, mostly likely by a
student or associate of Nasir al-Din al-Tusi, a.k.a. Psuedo -Tusi )

The author of this page, Jonathan Crabtree, is most grateful to Professor Avinash Sathaye (Sanskrit scholar and professor of mathematics) for providing the above translations from Sanskrit into English of Euclid's definitions of number theory in Book VII of the Elements.

Thank you for your interest!