Stupid modern Greek math academics call these by the name "ρητοί αριθμοί" where ρητοί means *explicit* or *mentionable* (as in a name) or *recite* as in poetry. Unfortunately most of them have never understood the Elements of Euclid. I know of only one Greek PhD who agrees with me. It's a shame because modern Greeks are not even a shadow of their ancestors.

In Book 7, Def. 1 we see Euclid attempt to define the abstract unit. That Euclid and the Ancient Greeks knew exactly what they were doing, was not understood by any one who came after Euclid until I revealed these things.

This Definition is vague and confusing:

Μονάς ἐστιν, καθ᾿ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.

Literal translation: Single is for each of the beings called.

More meaningful: Each single object is called a unit.

Exact mathematical definition by me: A unit is a ratio of equal magnitudes chosen as the standard of measure. In geometry there is no ideal unit, but in algebra, we talk about the distances and theorems in geometry by transferring these (to algebra) through the abstract unit.

The idea of unit is derived directly after the ratio of magnitudes is established in Book 5.

Def. 3 Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητά ποια σχέσις.

Literal translation: Ratio is two magnitudes homogeneous or toward quotient what relationship.

More meaningful: A ratio is the comparison of two magnitudes with a particular relationship.

The word πηλικό today means quotient, but any astute scholar will see that this is not meant in the same sense as quotient was interpreted then because the idea of quotient comes after ratio is established. For example. two magnitudes may have no relationship or proportion at all, ie, circle circumference : diameter or square diagonal : square side, etc. So, πηλικό refers to a possible proportion, but not necessarily one that is known.

All the arithmetic operations were established geometrically before Book 7. Now, in chronological order, Euclid would introduce the abstraction of measure from geometry to algebra through the abstract unit in Book 7, Def. 1.

We see fraction being defined in definition 3:

Book 7, Def. 3: A number is part of a(nother) number, the lesser of the greater, when it measures the greater.

Greek: Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.

THE ABOVE IS THE DEFINITION OF A FRACTION.

Example: A number (4) is part of (another) number (12), the lesser (4) of the greater (12), when it measures the greater (12).

Can you guess the fraction in the above example?

If you guessed 1/3, then BRAVO to you!!!

The mistaken Sir Thomas Heath thought in a foot note in his translation:

"In other words, a number p is part of another number q if there exists some number n such that n p = q."

which is obviously nonsense because the general definition of MAGNITUDE is given in book 5 (Defs 1 and 2) in very similar text:

Def. 1 **A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.**

Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.

Heath's definition is also laughable because multiplication is derived AFTER the QUOTIENT (not DIVISION which is thought to be synonymous) is established. Division is established after the QUOTIENT and then comes multiplication. Heath's definition would also be circular because n = p/q implies that the rational numbers have already been established.

Def. 2 **And the greater (magnitude is) a multiple of the lesser when it is measured by the lesser.**

Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάττονος.

The above definition uses the word Πολλαπλάσιον which originally meant more the one of the same kind. The word multiple has its origins in this word and today is only considered in the sense of a special category of rational numbers called natural numbers or integers. However, it can apply to a magnitude equally well as stated in Book 5, Def. 1.

We could write Book 7, Def. 3 also as:

A number (12) is greater than (another) number (4), when equal parts of the greater (12), measures the smaller (4).

Can you guess what fraction is described by the above?

If you guessed 3/1, then you deserve a gold star! Bravo! Chuckle. 3/1 is also a special rational number called a "natural number".

Apparently Euclid didn't bother either in book 5 or book 7 to state the converse because it was to them pretty obvious. Anyone who denies this is true, is a fool.

Yes, I do know better. I have always known better. I am a genius and the lot of you are nothing but fucking jealous morons. It's a shame. Really a shame that scum like you wastes resources on the planet. You would do humanity a favour by immediately resigning your posts as "educators" (LMAO) and picking up litter from the streets. Even better, stop breathing!

On 6/2/2021 7:56 AM, Eram semper recta "I am always a rectum" wrote:
> Stupid modern Greek math academics call these by the name "ρητοί αριθμοί" where

Shut up imbecile.

On Wednesday, 2 June 2021 at 11:03:35 UTC-4, Bandersnatch Cabbagepatch wrote:
> On 6/2/2021 7:56 AM, Eram semper recta "I am always a rectum" wrote:
> > Stupid modern Greek math academics call these by the name "ρητοί αριθμοί" where
> Shut up imbecile.

Go fuck yourself asshole!

On Wednesday, 2 June 2021 at 10:56:39 UTC-4, Eram semper recta wrote:
> Stupid modern Greek math academics call these by the name "ρητοί αριθμοί" where ρητοί means *explicit* or *mentionable* (as in a name) or *recite* as in poetry. Unfortunately most of them have never understood the Elements of Euclid. I know of only one Greek PhD who agrees with me. It's a shame because modern Greeks are not even a shadow of their ancestors.
>
> In Book 7, Def. 1 we see Euclid attempt to define the abstract unit. That Euclid and the Ancient Greeks knew exactly what they were doing, was not understood by any one who came after Euclid until I revealed these things.
>
> This Definition is vague and confusing:
>
> Μονάς ἐστιν, καθ᾿ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.
>
> Literal translation: Single is for each of the beings called.
>
> More meaningful: Each single object is called a unit.
>
> Exact mathematical definition by me: A unit is a ratio of equal magnitudes chosen as the standard of measure. In geometry there is no ideal unit, but in algebra, we talk about the distances and theorems in geometry by transferring these (to algebra) through the abstract unit.
>
> The idea of unit is derived directly after the ratio of magnitudes is established in Book 5.
>
> Def. 3 Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητά ποια σχέσις.
>
> Literal translation: Ratio is two magnitudes homogeneous or toward quotient what relationship.
>
> More meaningful: A ratio is the comparison of two magnitudes with a particular relationship.
>
> The word πηλικό today means quotient, but any astute scholar will see that this is not meant in the same sense as quotient was interpreted then because the idea of quotient comes after ratio is established. For example. two magnitudes may have no relationship or proportion at all, ie, circle circumference : diameter or square diagonal : square side, etc. So, πηλικό refers to a possible proportion, but not necessarily one that is known.
>
> All the arithmetic operations were established geometrically before Book 7. Now, in chronological order, Euclid would introduce the abstraction of measure from geometry to algebra through the abstract unit in Book 7, Def. 1..
>
> We see fraction being defined in definition 3:
>
> Book 7, Def. 3: A number is part of a(nother) number, the lesser of the greater, when it measures the greater.
>
> Greek: Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.
>
> THE ABOVE IS THE DEFINITION OF A FRACTION.
>
> Example: A number (4) is part of (another) number (12), the lesser (4) of the greater (12), when it measures the greater (12).
>
> Can you guess the fraction in the above example?
>
> If you guessed 1/3, then BRAVO to you!!!
>
> The mistaken Sir Thomas Heath thought in a foot note in his translation:
>
> "In other words, a number p is part of another number q if there exists some number n such that n p = q."
>
> which is obviously nonsense because the general definition of MAGNITUDE is given in book 5 (Defs 1 and 2) in very similar text:
>
> Def. 1 **A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.**
>
> Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.
>
> Heath's definition is also laughable because multiplication is derived AFTER the QUOTIENT (not DIVISION which is thought to be synonymous) is established. Division is established after the QUOTIENT and then comes multiplication. Heath's definition would also be circular because n = p/q implies that the rational numbers have already been established.
>
> Def. 2 **And the greater (magnitude is) a multiple of the lesser when it is measured by the lesser.**
>
> Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάττονος.
>
> The above definition uses the word Πολλαπλάσιον which originally meant more the one of the same kind. The word multiple has its origins in this word and today is only considered in the sense of a special category of rational numbers called natural numbers or integers. However, it can apply to a magnitude equally well as stated in Book 5, Def. 1.
>
> We could write Book 7, Def. 3 also as:
>
> A number (12) is greater than (another) number (4), when equal parts of the greater (12), measures the smaller (4).
>
> Can you guess what fraction is described by the above?
>
> If you guessed 3/1, then you deserve a gold star! Bravo! Chuckle. 3/1 is also a special rational number called a "natural number".
>
> Apparently Euclid didn't bother either in book 5 or book 7 to state the converse because it was to them pretty obvious. Anyone who denies this is true, is a fool.
>
> Yes, I do know better. I have always known better. I am a genius and the lot of you are nothing but fucking jealous morons. It's a shame. Really a shame that scum like you wastes resources on the planet. You would do humanity a favour by immediately resigning your posts as "educators" (LMAO) and picking up litter from the streets. Even better, stop breathing!

Μέρος means "part" and what is part of a number usually called?

[A] {} as in the bullshit of set theory
[B] Some notion you learned from your teacher
[C] It means FRACTION

Your choice morons?

On Wednesday, 2 June 2021 at 11:25:33 UTC-4, Eram semper recta wrote:
> On Wednesday, 2 June 2021 at 10:56:39 UTC-4, Eram semper recta wrote:
> > Stupid modern Greek math academics call these by the name "ρητοί αριθμοί" where ρητοί means *explicit* or *mentionable* (as in a name) or *recite* as in poetry. Unfortunately most of them have never understood the Elements of Euclid. I know of only one Greek PhD who agrees with me. It's a shame because modern Greeks are not even a shadow of their ancestors.
> >
> > In Book 7, Def. 1 we see Euclid attempt to define the abstract unit. That Euclid and the Ancient Greeks knew exactly what they were doing, was not understood by any one who came after Euclid until I revealed these things.
> >
> > This Definition is vague and confusing:
> >
> > Μονάς ἐστιν, καθ᾿ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.
> >
> > Literal translation: Single is for each of the beings called.
> >
> > More meaningful: Each single object is called a unit.
> >
> > Exact mathematical definition by me: A unit is a ratio of equal magnitudes chosen as the standard of measure. In geometry there is no ideal unit, but in algebra, we talk about the distances and theorems in geometry by transferring these (to algebra) through the abstract unit.
> >
> > The idea of unit is derived directly after the ratio of magnitudes is established in Book 5.
> >
> > Def. 3 Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητά ποια σχέσις.
> >
> > Literal translation: Ratio is two magnitudes homogeneous or toward quotient what relationship.
> >
> > More meaningful: A ratio is the comparison of two magnitudes with a particular relationship.
> >
> > The word πηλικό today means quotient, but any astute scholar will see that this is not meant in the same sense as quotient was interpreted then because the idea of quotient comes after ratio is established. For example. two magnitudes may have no relationship or proportion at all, ie, circle circumference : diameter or square diagonal : square side, etc. So, πηλικό refers to a possible proportion, but not necessarily one that is known.
> >
> > All the arithmetic operations were established geometrically before Book 7. Now, in chronological order, Euclid would introduce the abstraction of measure from geometry to algebra through the abstract unit in Book 7, Def. 1.
> >
> > We see fraction being defined in definition 3:
> >
> > Book 7, Def. 3: A number is part of a(nother) number, the lesser of the greater, when it measures the greater.
> >
> > Greek: Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.
> >
> > THE ABOVE IS THE DEFINITION OF A FRACTION.
> >
> > Example: A number (4) is part of (another) number (12), the lesser (4) of the greater (12), when it measures the greater (12).
> >
> > Can you guess the fraction in the above example?
> >
> > If you guessed 1/3, then BRAVO to you!!!
> >
> > The mistaken Sir Thomas Heath thought in a foot note in his translation:
> >
> > "In other words, a number p is part of another number q if there exists some number n such that n p = q."
> >
> > which is obviously nonsense because the general definition of MAGNITUDE is given in book 5 (Defs 1 and 2) in very similar text:
> >
> > Def. 1 **A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.**
> >
> > Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.
> >
> > Heath's definition is also laughable because multiplication is derived AFTER the QUOTIENT (not DIVISION which is thought to be synonymous) is established. Division is established after the QUOTIENT and then comes multiplication. Heath's definition would also be circular because n = p/q implies that the rational numbers have already been established.
> >
> > Def. 2 **And the greater (magnitude is) a multiple of the lesser when it is measured by the lesser.**
> >
> > Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάττονος.
> >
> > The above definition uses the word Πολλαπλάσιον which originally meant more the one of the same kind. The word multiple has its origins in this word and today is only considered in the sense of a special category of rational numbers called natural numbers or integers. However, it can apply to a magnitude equally well as stated in Book 5, Def. 1.
> >
> > We could write Book 7, Def. 3 also as:
> >
> > A number (12) is greater than (another) number (4), when equal parts of the greater (12), measures the smaller (4).
> >
> > Can you guess what fraction is described by the above?
> >
> > If you guessed 3/1, then you deserve a gold star! Bravo! Chuckle. 3/1 is also a special rational number called a "natural number".
> >
> > Apparently Euclid didn't bother either in book 5 or book 7 to state the converse because it was to them pretty obvious. Anyone who denies this is true, is a fool.
> >
> > Yes, I do know better. I have always known better. I am a genius and the lot of you are nothing but fucking jealous morons. It's a shame. Really a shame that scum like you wastes resources on the planet. You would do humanity a favour by immediately resigning your posts as "educators" (LMAO) and picking up litter from the streets. Even better, stop breathing!
> Μέρος means "part" and what is part of a number usually called?
>
> [A] {} as in the bullshit of set theory
> [B] Some notion you learned from your teacher
> [C] It means FRACTION
>
> Your choice morons?

I would like to hear WM once again say what he thinks because he always thought that natural numbers came first. LMAO.

On Wednesday, June 2, 2021 at 10:56:39 AM UTC-4, I am Super Rectum (aka "John Gabriel") wrote:
> Stupid modern Greek math academics...

Unlike you, Troll Boy, proving 2+2=4 is a trivial exercise for them, but a seemingly intractable problem in your goofy little system.

Face it, math never was your thing. Time to cut your losses and move on.

Even at his advanced age (60+?), John Gabriel is STILL struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:

"There are no points on a line."
--April 12, 2021

"Pi is NOT a number of ANY kind!"
--July 10, 2020

"1/2 not equal to 2/4"
--October 22, 2017

“1/3 does NOT mean 1 divided by 3 and never has meant that”
-- February 8, 2015

"3 =< 4 is nonsense.”
--October 28, 2017

"Zero is not a number."
-- Dec. 2, 2019

"0 is not required at all in mathematics, just like negative numbers."
-- Jan. 4, 2017

“There is no such thing as an empty set.”
--Oct. 4, 2019

“3 <=> 2 + 1 or 3 <=> 8 - 5, etc, are all propositions” (actually all are meaningless gibberish)
--Oct. 22, 2019

No math genius our JG, though he actually lists his job title as “mathematician” at Linkedin.com. Apparently, they do not verify your credentials.

Though really quite disturbing, interested readers should see: “About the spamming troll John Gabriel in his own words...” (lasted updated March 10, 2020) at https://groups.google.com/forum/#!msg/sci.math/PcpAzX5pDeY/1PDiSlK_BwAJ

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog a http://www.dcproof.wordpress.com

On Wednesday, June 2, 2021 at 6:23:32 PM UTC+1, Dan Christensen wrote:
> On Wednesday, June 2, 2021 at 10:56:39 AM UTC-4, I am Super Rectum (aka "John Gabriel") wrote:
> > Stupid modern Greek math academics...
>
> Unlike you, Troll Boy, proving 2+2=4 is a trivial exercise for them, but a seemingly intractable problem in your goofy little system.
>
> Face it, math never was your thing. Time to cut your losses and move on.
>
> Even at his advanced age (60+?), John Gabriel is STILL struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:
>
> "There are no points on a line."
> --April 12, 2021
>
> "Pi is NOT a number of ANY kind!"
> --July 10, 2020
>
> "1/2 not equal to 2/4"
> --October 22, 2017
>
> “1/3 does NOT mean 1 divided by 3 and never has meant that”
> -- February 8, 2015
>
> "3 =< 4 is nonsense.”
> --October 28, 2017
>
> "Zero is not a number."
> -- Dec. 2, 2019
>
> "0 is not required at all in mathematics, just like negative numbers."
> -- Jan. 4, 2017
>
> “There is no such thing as an empty set.”
> --Oct. 4, 2019
>
> “3 <=> 2 + 1 or 3 <=> 8 - 5, etc, are all propositions” (actually all are meaningless gibberish)
> --Oct. 22, 2019
>
> No math genius our JG, though he actually lists his job title as “mathematician” at Linkedin.com. Apparently, they do not verify your credentials.
>
>
> Though really quite disturbing, interested readers should see: “About the spamming troll John Gabriel in his own words...” (lasted updated March 10, 2020) at https://groups.google.com/forum/#!msg/sci.math/PcpAzX5pDeY/1PDiSlK_BwAJ
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog a http://www.dcproof.wordpress.com

Now, now Dan C 140, you should show more respect to JG 108.

As the world's greatest mathematician, Mr Gabriel will undoubtedly have profound and original advice on how to find and prove the general natural number solution (n , m) of Brocard's Equation:

n! + 1 = m^2

The solutions (4 , 5) , (5 , 11) , (7 , 71) are known, but the completeness of this set of solutions is not. This problem baffled a simpleton like Paul Erdos, but he didn't have the essence of Archimedes flowing through his veins.

After that I am sure JG 108 will offer us a rigorous derivation of the basic Euler-Lagrange equation of the calculus of variations by using his New Calculus methods. Easy for the world's greatest mathematician surely?

Kind Regards

Dan Christensen wrote:

> On Wednesday, June 2, 2021 at 10:56:39 AM UTC-4, I am Super Rectum (aka
> "John Gabriel") wrote:
>> Stupid modern Greek math academics...
>
> Unlike you, Troll Boy, proving 2+2=4 is a trivial exercise for them, but
> a seemingly intractable problem in your goofy little system.

harasho. Not if your 4 lies outside the domain of applicability and your
interest. That's NOT an answer nor a solution to anything.

On Wednesday, 2 June 2021 at 14:24:02 UTC-4, Albert Fullard wrote:
> Dan Christensen wrote:
>
> > On Wednesday, June 2, 2021 at 10:56:39 AM UTC-4, I am Super Rectum (aka
> > "John Gabriel") wrote:
> >> Stupid modern Greek math academics...
> >
> > Unlike you, Troll Boy, proving 2+2=4 is a trivial exercise for them, but
> > a seemingly intractable problem in your goofy little system.
> harasho. Not if your 4 lies outside the domain of applicability and your
> interest. That's NOT an answer nor a solution to anything.

Do you realise that you are responding to a dumb oaf? Christensen is the joke of sci.math and its chief troll. He has even replaced Archimedes Plutonium as chief crank.

Do not feed the troll please.