Ad fontes! (To the sources!)

# THE CENTURIES OLD DEFINITION OF MULTIPLICATION...

# "to multiply a by integral b is to add a to itself b times"

# IS WRONG!

As this page demonstrates, the faulty phrase, 'added to itself' was never in Euclid’s original Greek definition of multiplication. Therefore, after centuries of confusion, more than fifty years of peer-reviewed papers, dozens of book chapters and even a conference on the topic of defining multiplication via repeated addition, it all comes to this... Multiplication was never defined as repeated addition, until 1570, when a haberdasher changed Euclid's correct unary definition of multiplication into a buggy binary version. Addition has never been mentioned in any English translation of Euclid’s Book VII propositions and proofs reliant on a definition of multiplication. The list below presents the translators of every printed English edition of Elements Book VII, none of whom use ‘add’, ‘added’’ adding’ or ‘addition’, within Euclid’s multiplicative propositions and proofs.

1570 Henry Billingsley

1660 Isaac Barrow

1661 John Leeke and George Serle

1731 Edmund Stone

1788 James Williamson

1908 Thomas Little Heath

2008 Richard Fitzpatrick

#17 If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

#18 If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

#19 If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.

#22 The least numbers of those which have the same ratio with them are prime to one another.

#24 If two numbers be prime to any number, their product also will be prime to the same.

#25 If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

#26 If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

#27 If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another.

#30 If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

#33 Given as many numbers as we please, to find the least of those which have the same ratio with them, and

#34 Given two numbers, to find the least number which they measure.

* Text as per The thirteen books of Euclid's elements: translated from the text of Heiberg. Vols. 1-3, Thomas Little Heath Cambridge: At the University Press, 1908, Vol. 2, pp. 296-344

“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; …”

Source: www.collinsdictionary.com/dictionary/english/multiplication

WRONG! ab does not equal (the sum of) ‘a added to itself b times’.

This buggy dictionary definition of multiplication on the positive integers, quoted by mathematics professors for centuries, (as they thought Euclid said it) neither computes nor commutes.

FAILURE TO COMPUTE ALGEBRAICALLY

Any number a, multiplied by zero (a × 0) is not ‘a added to itself zero times’ or a, and (a × 1) is not ‘a added to itself one time’ or 2a. The bug introduced by the London haberdasher, Henry Billingsley, emits an excess a. The sum of, a taken or placed b times, will always have one less a than, a added to itself b times, as one ‘unary take’ partners with one number at a time while one ‘binary addition’ partners with two numbers at a time.

AN ARITHMETICAL REALITY CHECK (Why 'two multiplied by three' does not equal 'two added to itself three times'.)

Two added to one twice (in succession) is the expression 1 + 2 + 2, which equals 5.

Two added to one three times is the expression 1 + 2 + 2 + 2, which equals 7.

So given two added to one three times is seven, how can two added to itself (two) three times be six? It isn't!

Just as two added to one three times is seven, two added to itself three times is eight. 2 (itself) + 2 + 2 + 2 = 8.

The sum of the unary statement ‘one taken two times’ or 1, 1 is one less than the successive binary accumulation of ‘one added to itself two times’ or 1 + 1 + 1, yet it is the latter which is our current definition of one (a) multiplied by two (b).

FAILURE TO COMMUTE

Our modern definition of multiplication also disobeys the law of commutativity, as 'one added to itself two times', does not commute as 'two added to itself one time'.

One added to itself two times, ie one (itself) + 1 + 1 = THREE and two added to itself one time, ie two (itself) + 2 = FOUR.

Post-Peyrard 1804 - 2016 i.e. based on the Vatican manuscript (Vat. Gr. 190. P.)

- Euclid defines points, lines, units and numbers, yet did not define addition in the Elements. Therefore, as per the doctrine of Aristotle, (you can't define something with an undefined concept), he could not have defined multiplication with the undefined concept of 'addition'.
- Euclid's Book VII is about 'arithmos' and line segments. It is NOT about 'arithmoi' or pure arithmetic. Euclid wrote a powerful, proof based, 'picture-story book' on geometry without numbers. Nicomachus wrote 'Arithmetike eisagoge' (Introduction to Arithmetic) which approached arithmetic as a different subject to geometry. Nowhere in Euclid's Elements will you find out how to add, subtract, multiply or divide numbers. Only in Nicomachus' book will you find a Greek 'times table' and yet it is Euclid's geometric, line-based definition of multiplication that we use for our definition of multiplication on positive integers today!
- Neither the word 'added' nor the phrase 'added to itself' are in the multiplication definition in the oldest extant manuscript Greek (MS D'Orville 301 888 CE) or the Vatican MS detailed below circa 950 CE.
- From the 888 MS and the 950 MS Euclid gave a unary definition of multiplication, with the one-to-many proportional meaning.

a) CLICK HERE to see the definition of multiplication in the oldest extant manuscript attributed to Euclid.

b) CLICK HERE to see how Euclid's definition of multiplication was INCORRECTLY translated into English in 1570, by London Haberdasher, Henry Billingsley.

c) CLICK HERE to see a transmission of Euclid's Elementary number theory from Greek to Arabic to Sanskrit to English.

d) CLICK HERE to see correct 16th-century translations of Euclid's definition of multiplication into Italian, German and French.

Printable Links:

a) www.jonathancrabtree.com/800s/888_Euclid_Multiplication_Definition_Marked_Up.jpg

b) www.jonathancrabtree.com/1500s/1570_HBILLINGSLEY_First_English_Translation_Of_Euclids_Definition_Of_Multiplication.jpg

c) www.jonathancrabtree.com/euclid/elements_book_VII_definitions_via_Jagannatha_Samrat_The_Rekhaganita.html

d) www.jonathancrabtree.com/euclid/elements_book_VII_definitions.html#correct_definitions_of_multiplication

Henry Mendell | David Joyce | Richard Fitzpatrick | Thomas Little Heath | |

Bold text © Dr Dimitrios E. Mourmouras Source Direct translation hyperlinks Tufts University Source | © Dr Henry Mendell Source | © Dr David Joyce Source | © Dr Richard Fitzpatrick 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 | 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 | 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 | 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. | 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 | 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 | 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 | 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 | 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 | 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.] ** | 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; δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ | 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. πρῶτος ἀριθμός ἐστιν ὁ μονάδι | 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 | 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. σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ | 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.σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ | 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, ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, | 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. |

All copyrights in this section remain the property of their respective owners. |

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, in modern terms, "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.

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.

"That number is said to be multiplied by another, which is (combined / summed / taken) as many times as there are units in the multiplier."

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.

Title page www.jonathancrabtree.com/1500s/1543-Niccolo_Tartaglia_Euclide_Megarense_Title_Page.jpg

Definition V www.jonathancrabtree.com/1500s/1543-Tartaglia_multiplication_definition_from_Euclids_Elements.jpg

Definition V www.jonathancrabtree.com/1500s/1543-Tartaglia_multiplication_definition_V_from_Euclids_Elements.jpg (From 1569 edition. Same as 1543.)

GERMAN

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.

"A number multiplies or makes more (times) another number, if the other - as often as the first number is included in it (i.e. the other number), is taken and brought together."

SOURCE

Das sibend, acht und neünt Büch, des hochberühmbten Mathematici Euclidis

MEANING

The seventh, eighth and ninth book by the renowned Mathematician Euclid.

Title page www.jonathancrabtree.com/1500s/1555-JSCHEUBEL_The_Seventh_Eighth_and_Ninth_Books_of_Euclids_Elements_Title_Page.jpg

Definition 16 www.jonathancrabtree.com/1500s/1555-JSCHEUBEL_Euclids_Elements_Multiplication_Definition.jpg

FRENCH

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.

"A number is said to multiply an other number, when as many units, there are in it, as many times the multiple is composed, and another number is born."

SOURCE

Les septieme huictieme et neufieme livres des Elemens d'Euclide

MEANING

The seventh, eighth and ninth book of the Elements of Euclid.

Title page www.jonathancrabtree.com/1500s/1565_French_Pierre_Forcadel_Euclids_Elements_Book_VII_VIII_IX_Front_Page.jpg

Definition 16 www.jonathancrabtree.com/1500s/1565_French_Pierre_Forcadel_Euclids_Elements_Book_VII_Definitions_15_16_17.jpg

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) Title page www.jonathancrabtree.com/1500s/1570-HBILLINGSLEY_Euclids_Elements_Title_Page.jpg

9) Definition 16 www.jonathancrabtree.com/1500s/1570_HBILLINGSLEY_First_English_Translation_Of_Euclids_Definition_Of_Multiplication.jpg

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 is 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.

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. | 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 | 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 | 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 | 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 | 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 | 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 | 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. |

All copyrights in this section remain the property of their respective owners. |

ENDNOTES AND ACKNOWLEDGEMENTS

* (From the Vatican manuscript (Vat. Gr 190. P.) discovered by Peyrard, then used by Heiberg to recreate the edition of the Elements closest to Euclid, subsequently used by: Heath 1908, Stamatis (Greek), Mendell, Joyce, Fitzpatrick and others. The actual Greek text above is via Dimitrios E. Mourmouras who used Evangelos Stamatis' edition of Euclid's Elements (1953) with the same Greek as the Heiberg Book VII definitions of the 1880s that was created from the Vatican manuscript (Vat. Gr. 190 P.) discovered by Peyrard.

** Some manuscripts and printed editions of Euclid's Elements delete this definition. Therefore the number associated with Euclid's definition of multiplication can vary. Euclid originally wrote lengthy paragraphs, These paragraphs were subsequently split into what are now called 'definitions' which is why they often seem to spill over from one to the next.

My work would not be possible without the knowledge and generous assistance of many people. For this page, I would like to acknowledge: Nick Woodhouse, David Joyce. Henry Mendell, Richard Fitzpatrick, Henry Mason, Dimitrios Mourmouras and Ulrich Reich. I also want to acknowledge the Clay Mathematics Institute for hosting the digital MS D'Orville 301 888 CE and the Tufts University Perseus Project team for the hyperlinked translations from the Greek. I also acknowledge my father Peter Crabtree for his encouragement and patience reviewing this and other material I have assembled over the years.

Thank you for your interest!

Jonathan Crabtree

Mathematics Historian/Researcher

Melbourne Australia

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Last Update 5 August 2019

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www.bit.ly/euclids-elements-book-VII-definitions

Last Update 5 August 2019

www.JonathanCrabtree.com/mathematics