# Ad fontes! (To the sources!)  Euclid's Elements, Book VII Definitions for Elementary Number Theory

## 'Greek to English' Translation Master List for Primary Research and Cross Referencing Post-Peyrard 1804 - 2013 i.e. based on the Vatican manuscript (Vat. Gr. 190. P.)

Euclid's 23 Greek* definitions & translations by...
Henry Mendell David Joyce Richard Fitzpatrick Thomas Little Heath
Bold text © Dr Dimitrios E. Mourmouras  Source
Direct translation hyperlinks Tufts University  Source
Dr Fitzpatrick's comments and noting of (parenthetical) text not in the Greek.
Cambridge University Press 1908 & 1926 Reprinted Dover 1956

### 1 Μονάς ἐστιν, καθ' ἣν ἕκαστον τῶν ὄντων ἓν λέγεται. ονάςἐστιν, καθ᾽ἣνἕκαστοντῶνὄντων ἓνλέγεται.

1 Unit is that according to which each of the things which are is one, 1 A unit is that by virtue of which each of the things that exist is called one. 1 A unit is (that) according to which each existing (thing) is said (to be) one. 1 An unit is that by virtue of which each of the things that exist is called one.

### 2 Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.  ἀριθμὸςδὲτὸ ἐκμονάδωνσυγκείμενονπλῆθος.

2 and the multitude composed from units is a number. 2 A number is a multitude composed of units. 2 And a number (is) a multitude composed of units. In other words, a "number' is a positive integer greater than unity. 2 A number is a multitude composed of units.

### 3 Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα. μέροςἐστὶνἀριθμὸςἀριθμοῦὁἐλάσσωντοῦμείζονος,  ὅτανκαταμετρῇτὸνμείζονα.

3 A number is a part of a number, the smaller of the larger, whenever it measures the larger, 3 A number is a part of a number, the less of the greater, when it measures the greater; 3 A number is part of a(nother) number, the greater, when it measures the greater. In other words, a number a is part of another number b if there exists some number n such that n a = b. 3 A number is a part of a number, the less of the greater, when it measures the greater;

### 4 Μέρη δέ, ὅταν μὴ καταμετρῇ.μέρηδέ, ὅτανμὴκαταμετρῇ.

4 and parts whenever it does not measure, 4 But parts when it does not measure it. 4 But (the lesser is) parts (of the greater) when it does not measure it. In other words, a number a is parts of another number b (where a < b) if there exist distinct numbers, m and n, such that n a = mb. 4 but parts when it does not measure it.

### 5 Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.πολλαπλάσιοςδὲὁμείζωντοῦἐλάσσονος,  ὅτανκαταμετρῆταιὑπὸτοῦἐλάσσονος.

5 and the larger is a multiple of the smaller whenever it is measured by the smaller. 5 The greater number is a multiple of the less when it is measured by the less. 5 And the greater (number is) a multiple of the lesser when it is measured by the lesser. 5 The greater number is a multiple of the less when it is measured by the less.

### 6 Ἄρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος. ἄρτιοςἀριθμόςἐστιν ὁδίχαδιαιρούμενος.

6 The number which is divided in two is an even number, 6 An even number is that which is divisible into two equal parts. 6 An even number is one (which can be) divided in half. 6 An even number is that which is divisible into two equal parts.

### 7 Περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ [ὁ] μονάδι διαφέρων ἀρτίου ἀριθμοῦ.  περισσὸςδὲὁμὴδιαιρούμενοςδίχαἢὁμονάδιδιαφέρωνἀρτίουἀριθμοῦ.

7 and the number which is not divided in two is odd, or the number which differs from an even number by a unit. 7 An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number. 7 And an odd number is one (which can)not (be) divided in half, or which differs from an even number by a unit. 7 An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

### 8 Ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν. ἀρτιάκιςἄρτιοςἀριθμόςἐστινὁὑπὸ ἀρτίουἀριθμοῦμετρούμενοςκατὰἄρτιονἀριθμόν.

8 The number measured by an even number taken in groups of an even number is an even-times even number, 8 An even-times-even number is that which is measured by an even number according to an even number. 8 An even-times-even number is one (which is) measured by an even number according to an even number. In other words, an even-times-even number is the product of two even numbers. 8 An even-times even number is that which is measured by an even number according to an even number.

### 9 Ἀρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.ἀρτιάκιςδὲπερισσόςἐστινὁὑπὸἀρτίουἀριθμοῦ μετρούμενοςκατὰπερισσὸνἀριθμόν.

9 and the number measured by an even number taken in groups of an odd number is an even-times odd number, 9 An even-times-odd number is that which is measured by an even number according to an odd number. 9 And an even-times-odd number is one (which is) measured by an even number according to an odd number. In other words, an even-times-odd number is the product of an even and an odd number. 9 An even-times odd number is that which is measured by an even number according to an odd number.

### 10 [Περισσάκις ἄρτιός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν.] **Περισσάκιςἀρτιόςἐστινὁὑπὸπερισσοῦ ἀριθμοῦμετρούμενοςκατὰἄρτιονἀριθμόν.

10 [and the number measured by an odd number taken in groups of an even number is an odd-times even,] ** 10 Greek Omitted 10 Greek Omitted 10 Greek Omitted

### 11 Περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν. περισσάκι&iigmaf;δὲπερισσὸςἀριθμόςἐστινὁὑπὸ περισσοῦἀριθμοῦμετρούμενοςκατὰπερισσὸνἀριθμόν.

11 and the number measured by an odd number taken in groups of an odd number is an odd-times odd number. 10 An odd-times-odd number is that which is measured by an odd number according to an odd number. 10 And an odd-times-odd number is one (which is) measured by an odd number according to an odd number. In other words, an odd-times-odd number is the product of two odd numbers. 10 An odd-times odd number is that which is measured by an odd number according to an odd number.

### 12 Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος. πρῶτοςἀριθμόςἐστινὁμονάδι μόνῃμετρούμενος.

12 The number measured only by a unit is a prime number. 11 A prime number is that which is measured by a unit alone. 11 A prime^  number is one (which is) measured by a unit alone. ^ Literally, “first”. 11 A prime number is that which is measured by an unit alone.

### 13 Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ. πρῶτοιπρὸςἀλλήλουςἀριθμοί εἰσινοἱμονάδιμόνῃμετρούμενοικοινῷμέτρῳ.

13 Prime numbers relative to one another are those measured only by a unit as a common measure. 12 Numbers relatively prime are those which are measured by a unit alone as a common measure. 12 Numbers prime to one another are those (which are) measured by a unit alone as a common measure. 12 Numbers prime to one another are those which are measured by an unit alone as a common measure.

### 14 Σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος. σύνθετοςἀριθμόςἐστινὁἀριθμῷ τινιμετρούμενος.

14 Compound number is a number measured by a number, 13 A composite number is that which is measured by some number. 13 A composite number is one (which is) measured by some number. 13 A composite number is that which is measured by some number.

### 15 Σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ τινι μετρούμενοι κοινῷ μέτρῳ. σύνθετοιδὲπρὸςἀλλήλουςἀριθμοίεἰσινοἱ ἀριθμῷτινιμετρούμενοικοινῷμέτρῳ.

15 and compound numbers relative to one another are those measured by a number as a common measure. 14 Numbers relatively composite are those which are measured by some number as a common measure. 14 And numbers composite to one another are those (which are) measured by some number as a common measure. 14 Numbers composite to one another are those which are measured by some number as a common measure.

WARNING! THE ENGLISH LANGUAGE TRANSLATIONS OF EUCLID'S DEFINITION OF MULTIPLICATION BELOW ARE INCORRECT

### 16 Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις.

ἀριθμὸς  ἀριθμὸν  πολλαπλασιάζειν λέγεται,
ὅταν, ὅσαι εἰσὶν ἐν αὐτῷμονάδες, τοσαυτάκις
συντεθῇ  πολλαπλασιαζόμενος, καὶ γένηταί τις.

16 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. 15 A number is said to multiply a number when the latter is added* as many times as there are units in the former. (* See comment below.) 15 A number is said to multiply a(nother) number when the (number being) multiplied is added (to itself) as many times as there are units in the former (number), and (thereby) some (other number) is produced. 15 A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
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CONTINUED FURTHER BELOW

AN ANALYSIS OF EUCLID'S DEFINITION OF MULTIPLICATION

* Between 1997 and 2012 Dr David Joyce's excellent Euclid website gave the following standard and incorrect translation of Euclid's definition of multiplication with the incorrect 'added to itself' phrase: "A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other." 1997 Source: http://bit.ly/1etvbYY  2012 Source: http://bit.ly/14lonea During this 15 year period, Dr Joyce's Book VII Definition page became the standard English language reference on the Internet for Euclid's Elements. I am  thankful to Dr Joyce for sharing his considerable knowledge with me via email. I am also grateful for his editing the English language version of Euclid's definition of multiplication on his website at http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html and improving the definition to become: "A number is said to multiply a number when that which is multiplied is added as many times as there are units in the other."

Dr Joyce also provided further clarification about the unary nature of Euclid's definition of multiplication, writing: "Definition 15 (multiplication) defines multiplication in terms of addition as a kind of composition. For instance, to multiply 4 and 5, add the number 5 4 times, 5 +5 +5 +5, since 4 consists of 4 units. Note that Euclid did not think of addition as a binary operation, but as an operation with any number of arguments." Source: http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII15.html

The correct expression for the correct definition of 'two multiplied by three' on a number line starting at zero is: 0 + 2 + 2 + 2 = 6.  Yet Euclid lived about 750 years BEFORE zero was given the status of number in India.  Euclid could not have added anything to zero as zero did not exist as a number in 300 BCE.  So what did Euclid do when he explained multiplication?  The answer is Euclid was putting, setting, taking and placing line segments as UNARY operations.  What Henry (the haberdasher) Billingsley did in 1570 was to switch Euclid's unary definition into an incorrect binary definition. Euclid's Greek, as best we know from the Peyrard discovery of a non-Theonian version, (MS Vat. Gr 190 P.) is

Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις.

EUCLID'S GREEK DEFINITION OF MULTIPLICATION WITH LIVE CLICKABLE TRANSLATION LINKS

## ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐναὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις

The verb συντεθῇ in the above Greek means PLACED or PUT TOGETHER. Euclid's definition was CHANGED by Billingsley from "ab = a placed (or put together) b times" to ab = a added to itself b times!

Therefore for almost four and a half centuries, children have been taught multiplication via repeated addition, very often with the 'added to itself' error include with the pedagogy. So did everyone make the same howler as Henry? No! Let's explore how Euclid's Elements was translated from the 'classic' languages of Greek and Latin into four modern languages in the 16th century.

EUCLID'S DEFINITION OF MULTIPLICATION CORRECTLY TRANSLATED
In 1543 Niccolo Tartaglia (Mathematics teacher in Verona, and Venice. Professor of Euclid in Brescia.)
TRANSLATION
Quel numero se dice esser multiplicato per un'altro, il quale si e assunato tante volte, quante unita e in lo multiplicante.
SOURCE
Euclide Megarense ... solo introduttore delle scientie mathematice
MEANING
Euclid of Megara (wrong should be 'of Alexandria' - a common error) ... only introducer of scientific mathematics.
1) www.jonathancrabtree.com/1500s/1543-Niccolo_Tartaglia_Euclide_Megarense_Title_Page.jpg Title page.
2) www.jonathancrabtree.com/1500s/1543-Tartaglia_multiplication_definition_from_Euclids_Elements.jpg Definition V
3) www.jonathancrabtree.com/1500s/1543-Tartaglia_multiplication_definition_V_from_Euclids_Elements.jpg Definition V from 1569 edition. (Same as 1543.)

In 1555 Johann Scheubel (Professor of Mathematics at the University of Tübingen)
TRANSLATION
Ain zal multiplicirt oder meret ain andere / wann die ander / als offt die erst zal ains in jr beschleüßt / genommen vnd zuesamen bracht wirdt. Als 4. multiplicirt oder meret die zal 7. wann die zal 7. vier mal / in ansehen das ains in 4. viermal begriffen ist / genommen vnd zuesamen bracht wirdt.
SOURCE
Das sibend, acht und neünt Büch, des hochberühmbten Mathematici Euclidis
MEANING
The seventh, eighth and ninth book by the renowned Mathematician Euclid.
4) www.jonathancrabtree.com/1500s/1555-JSCHEUBEL_The_Seventh_Eighth_and_Ninth_Books_of_Eulcids_Elements_Title_Page.jpg Title page.
5) www.jonathancrabtree.com/1500s/1555-JSCHEUBEL_Euclids_Elements_Multiplication_Definition.jpg Definition 16

In 1565 Pierre Forcadel (Professor of mathematics at the University of Paris)
TRANSLATION
Un nombre, se dict multiplier un autre nombre, quand autant d'unitez, qu'il y a en luy, autant de fois se compose le multiplie, & en naist un autre.
SOURCE
Les septieme huictieme et neufieme livres des Elemens d'Euclide
MEANING
The seventh, eighth and ninth book of the Elements of Euclid.

EUCLID DEFINITION OF MULTIPLICATION WRONGLY TRANSLATED IN 1570
In 1570 Henry Billingsley (Haberdasher and Citizen of London)
TRANSLATION
A number is sayd to multiply a number, when the number multiplyed, is so oftentimes ADDED TO ITSELFE, as there are in the number multiplying unities : and an other number is produced.  (Bold caps used to highlight the error.)
SOURCE
The Elements of Geometrie of the most auncient philosopher Euclide of Megara (wrong should be 'of Alexandria')
MEANING
The Elements of Geometry of the most ancient philosopher Euclid of Megara.
8) www.jonathancrabtree.com/1500s/1570-HBILLINGSLEY_Euclids_Elements_Title_Page.jpg Title page.
9) www.jonathancrabtree.com/1500s/1570_HBILLINGSLEY_First_English_Translation_Of_Euclids_Definition_Of_Multiplication.jpg Definition 16

THE ANCESTRY OF OUR MODERN (BUGGY) MULTIPLICATION DEFINITION
1570 Henry Billingsley
A number is sayd to multiply a number, when the number multiplyed, is so oftentimes added to itselfe, as there are in the number multiplying unities : and an other number is produced.
1908 Thomas Heath
A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
2013 Collins Dictionary
A binary arithmetical operation defined initially for positive integers in terms of repeated addition, by which the product of two quantities is calculated, usually written a × b, a.b, or ab. To multiply a by integral b is to add a to itself b times; …

The Crabtree dictionary is quite possibly the same edition used by Thomas Little Heath. The dictionary I'm talking about is the Greek - English Lexicon by Henry George Liddell and Robert Scott. Heath wrote his original three volume translation of Euclid's Elements in 1908. The Crabtree family edition is the 1890 edition. It is impossible for the Perseus Project team to list all the meanings of every Greek and Latin word. So in addition to relying on the Perseus Project, I draw my interpretation of Euclid from the lexicon reference now known as Liddell and Scott (L&S).

Noting that Euclid was writing about geometry, even in the (arithmetical/arithmos) number theory book VII, we must blind ourselves to the modern day notion of pure arithmetic (arithmoi) and especially that aspect of arithmetic involving binary operations. Euclid was looking at geometric objects and the only numbers in Euclid's Elements, as we know number today, are the: book numbers, page numbers, definition numbers, proposition numbers and so on..

So let's look at the entry for the problematic Greek word συντεθῇ in L&S pronounced 'sin tuh thay'. Henry Billingsley translated συντεθῇ from the Greek as 'added to itselfe' and Thomas Little Heath repeated this translation, as nearly all English translators had from the 16th century. Yet the first English meaning given to συντεθῇ is "to place or put together."  The word place os provided by the Perseus link for συντεθῇ.

Now further on L&S provide another option for συντεθῇ interpreted through the lens of arithmetic 2200 years later as added together. So what are we to do? We can look to other non-English translations, both 16th and 20th century) yet why not look at how συντεθῇ was translated by Heath BEFORE it appears in Book VII? (Watch out Thomas - it's a trap!!!)

Too late... Gotcha! The word συντεθῇ was translated as PLACED by Heath in VI Prop. 32.

Book VI Proposition 32 text and Heath's translation.

Ἐὰν δύο τρίγωνα συντεθῇ κατὰ μίαν γωνίαν τὰς δύο πλευρὰς ταῖς δυσὶ πλευραῖς ἀνάλογον ἔχοντα ὥστε τὰς ὁμολόγους αὐτῶν πλευρὰς καὶ παραλλήλους εἶναι, αἱ λοιπαὶ τῶν τριγώνων πλευραὶ ἐπ' εὐθείας ἔσονται.

If two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line.

[ © 2013 All Rights Reserved, Jonathan Crabtree - www.jonathancrabtree.com - May be Reproduced in Part or Full With This Line Intact ]

Last Update 27 November 2013

Continued from definitions 1 - 16.
Euclid's 23 Greek* definitions & translations by...
Henry Mendell David Joyce Richard Fitzpatrick Thomas Little Heath

### 17 Ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.ὅτανδὲδύοἀριθμοὶπολλαπλασιάσαντεςἀλλήλους ποιῶσίτινα, ὁγενόμενοςἐπίπεδοςκαλεῖται, πλευραὶ δὲαὐτοῦοἱπολλαπλασιάσαντεςἀλλήλουςἀριθμοί.

17 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, 16 And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another. 16 And when two numbers multiplying one another make some (other number) then the (number so) created is called plane, and its sides (are) the numbers which multiply one another. 16 And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

### 18 Ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.ὅτανδὲτρεῖςἀριθμοὶπολλαπλασιάσαντες ἀλλήλουςποιῶσίτινα, ὁγενόμενοςστερεόςἐστιν,  πλευραὶδὲαὐτοῦ οἱπολλαπλασιάσαντες  ἀλλήλουςἀριθμοί.

18 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. 17 And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another. 17 And when three numbers multiplying one another make some (other number) then the (number so) created is (called) solid, and its sides (are) the numbers which multiply one another. 17 And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.

### 19 Τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ [ὁ] ὑπὸ δύο ἴσων ἀριθμῶν περιεχόμενος.τετράγωνοςἀριθμόςἐστινὁἰσάκιςἴσοςἢὁ ὑπὸδύοἴσωνἀριθμῶνπεριεχόμενος.

19 A square number is the equal-times equal number or the number enclosed by two equal numbers, 18 A square number is equal multiplied by equal, or a number which is contained by two equal numbers. 18 A square number is an equal times an equal, or (a plane number) contained by two equal numbers. 18 A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

### 20 Κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ [ὁ] ὑπὸ τριῶν ἴσων ἀριθμῶν περιεχόμενος.κύβοςδὲὁἰσάκιςἴσοςἰσάκιςἢὁὑπὸτριῶν ἴσωνἀριθμῶνπεριεχόμενος.

20 and a cube is the equal-times equal equal-times or enclosed by three equal numbers. 19 And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers. 19 And a cube (number) is an equal times an equal times an equal, or (a solid number) contained by three equal numbers. 19 And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

### 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.  ἀριθμοὶἀνάλογόνεἰσιν, ὅτανὁπρῶτοςτοῦδευτέρου καὶὁτρίτοςτοῦτετάρτουἰσάκιςᾖπολλαπλάσιοςἢ τὸαὐτὸμέροςἢτὰαὐτὰμέρηὦσιν.

21 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. 20 Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth. 20 Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third (is) of the fourth. 20 Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

### 22 Ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς.ὅμοιοιἐπίπεδοικαὶστερεοὶἀριθμοίεἰσινοἱἀνάλογονἔχοντεςτὰςπλευράς.

22 Similar plane and solid numbers are those having proportional sides. 21 Similar plane and solid numbers are those which have their sides proportional. 21 Similar plane and solid numbers are those having proportional sides. 21 Similar plane and solid numbers are those which have their sides proportional.

### 23 Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.τέλειοςἀριθμόςἐστινὁτοῖςἑαυτοῦμέρεσινἴσοςὤν.

23 A perfect number is one which is equal to all its parts. 22 A perfect number is that which is equal to the sum its own parts. 22 A perfect number is that which is equal to its own parts. In other words, a perfect number is equal to the sum of its own factors. 22 A perfect number is that which is equal to its own parts.
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