Πολλαπλάσιον δε τό μείζον του ελάττονος, όταν καταμετρήται υπό του ελάττονος.
And the greater (magnitude is) a multiple of the lesser (magnitude) when it is measured by the lesser (magnitude).
If p and q are magnitudes with q less than p, then we say that q measures p if and only if p divided by q leaves no remainder. In other words p is a multiple of q.
Book 5 is the most important book of all the 13 books of Euclid's Elements. It is also the hardest because it appears before Book 6, which in my opinion should have come before Book 5 since much of Book 6 explains the important concepts of Book 5 using the geometry established thus far. ... https://www.youtube.com/watch?v=juU2ne0KwPg
In this video I show you some useful ways to study the propositions in Euclid's Elements, especially with a CAS such as Geogebra, MATLAB, Maple, Mathematica, etc.
Also included are some tips for learning and understanding the original Greek text.
Link to applet:
https://drive.google.com/file/d/1Nb4tEhXkHO6iL0GfSGCKD4p88mU3xOYc
...
https://www.youtube.com/watch?v=IRrs-iQCbMs
The mean value theorem is about a very special arithmetic mean that is calculated using ALL the y ordinate lengths in a given interval for a specified SMOOTH function. Note that the methods of calculus are both NULL and VOID unless one is using a SMOOTH function.
The mean value theorem says that the arithmetic mean f'(c) of all the y - ordinates of a function f' can be found from its antecedent function f by the quotient: [f(b)-f(a)]/(b-a).
The lengths are considered as magnitudes (not nonsense such as "real number") because many have no number that describe the length of their measure.
None of the mainstream theorems (extreme value or Rolle's theorem) are required to prove the mean value theorem. In fact, even the bogus concept of "real number" has no influence on the working of the mean value theorem.
The mean value theorem explained using positional derivatives:
https://drive.google.com/file/d/0B-mOEooW03iLZG1pNlVIX2RTR0E
The positional derivative is a concept I invented to formulate a constructive of the mean value theorem using the flawed mainstream definitions and concepts:
https://drive.google.com/file/d/0B-mOEooW03iLSFVKRmx6c0tPWms
Constructive proof of the mean value theorem using the rigorous New Calculus:
https://drive.google.com/file/d/0B-mOEooW03iLblJNLWJUeGxqV0E
Britannica entry:
https://www.britannica.com/science/mean-value-theorem
It is FALSE that one has to know anything about "real number" or real analysis.
You can learn the New Calculus here:
https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO
My historic geometric theorem of January 2020 (discovered as a result of my New Calculus, but not a part of my New Calculus which is far simpler and rigorous):
https://drive.google.com/file/d/1RDulODvgncItTe7qNI1d8KTN5bl0aTXj
Fixing the broken mainstream formulation using my geometric identity:
https://drive.google.com/file/d/1uIBgJ1ObroIbkt0V2YFQEpPdd8l-xK6y
Exactly how stupid are mainstream math professors?
https://drive.google.com/file/d/1520NjhgiakcrssQxtbxRCDXus_aHXpI9
The BIG LIE: "Calculus was made rigorous...":
https://drive.google.com/file/d/1peL7lzXsmZ4AVsgAwDLvn8QIZbWy8Ebm
There are no postulates or axioms in sound mathematics:
https://drive.google.com/file/d/1vlU-PJeIk672bFwZyULD1ASTRFF3jXg8
Pi is not a number:
https://drive.google.com/file/d/1FFg_9XCkIwTZ9N1jbU4oMYfHHHuFHYf3
Is the "real number line" real?
https://drive.google.com/file/d/0B-mOEooW03iLMHVYcE8xcmRZRnc
How a genius discovers the concept of number:
https://drive.google.com/file/d/1hasWyQCZyRN3RkdvIB6bnGIVV2Rabz8w
...
https://www.youtube.com/watch?v=FIx0KrDi2so
π is a symbol given as a name to a constant (rational number!) with the adjective 'rational' being redundant because to be a number implies rationality and vice-versa. The expression "irrational number" is an oxymoron because it is like saying "an irrational rational number". No, I am sorry but mainstream math academics are unbelievably stupid and do not know that you cannot define a concept in terms of attributes it lacks.
There is no actual length or mass or volume π because we realise π through an ABSTRACT UNIT, not a physical unit.
Learn this and much more by downloading your free copy of the best book ever written on the number concept:
https://www.academia.edu/105399167/The_Ultimate_Book_of_Numbers
Become a follower here:
https://independent.academia.edu/JohnGabriel30
Feeling generous? Buy me a cup of coffee or a nice meal here:
https://gofund.me/af8a5312
Video I referred to in my video:
https://www.youtube.com/shorts/WpJlJ8DROWY
Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR
Merchandise Store:
https://new-calculus.printify.me/products
...
https://www.youtube.com/watch?v=V_aC1YkbCcU
In part 4 of the 5-part series called 'There are no axioms or postulates in mathematics', I define parallel lines and prove the sum of angles in a triangle is equal to two right angles (180 degrees).
https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
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https://www.youtube.com/watch?v=UBF8AsMHup8
In this video I show you how to find the derivatives of e^x and ln(x) without any use of mainstream calculus such as the garbage of limit theory.
Binomial theorem for beginners:
https://www.academia.edu/50921215/The_Binomial_Theorem_Explained_for_Dummies
Link to the article:
https://www.academia.edu/111250851/Derivative_of_e_to_the_x_and_ln_x_without_any_use_of_limit_theory_or_the_garbage_of_mainstream_calculus
Link to applet used:
https://drive.google.com/file/d/1l5zAIXLVSLtrFcb7Ec6eF2rThU0Vvq4T
Become a follower on Academia.edu:
https://independent.academia.edu/JohnGabriel30
Donate here:
https://gofund.me/af8a5312
Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR
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...
https://www.youtube.com/watch?v=FF3Tw9sjMuc
Definition 4.
Λόγον έχειν πρός 'αλληλα μεγέθη λέγεται, α δύναται ολλαπλασιαζόμενα αλλήλων υπερέχειν.
(Those) magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.
Of all the definitions in Euclid’s Elements, I have always found this one to be the most problematic in terms of understanding. I do not agree with Thomas Heath’s interpretation or anyone else’s that I have come to know.
Perhaps the most laughable interpretation is the one by cranky Prof. David Joyce of Clark University:
"This definition limits the existence of ratios to comparable magnitudes of the same kind where comparable means each, when multiplied, can exceed the other. The ratio doesn’t exist when one magnitude is so small or the other so large that no multiple of the one can exceed the other. This definition excludes the ratio of a finite straight line to an infinite straight line and the ratio of an infinitesimal straight line, should any exist, to a finite straight line." - David Joyce
The Ancient Greeks rejected any ill-formed concept such as infinity and infinitesimal. Moreover, the only objects that were recognised to be numbers by the Ancient Greeks were the rational numbers. Irrational magnitudes were never called (or understood to be) numbers of any kind.
Here’s my interpretation:
Magnitudes in such a ratio, will when multiplied with each other, exceed the original magnitudes. Such magnitudes are greater than the chosen or standard magnitude of measure.
Simple version: Euclid is talking about magnitudes greater than the lowest common divisor or the unit magnitude (whatever it might be) in the process of measure. For example, the magnitudes 2/10 and 8/10 do not qualify since their product 4/25 is less than both 2/10 and 8/10. Also, 0.8 and 3 have a product of 2.4 meaning that neither 0.8 nor 3 are in such a ratio.
“Whoa!”, you may say, “Mr. Gabriel, what you used are numbers but magnitudes are not numbers.”
You would be right, except that I am speaking in the context of a chosen or standard magnitude of measure. I only used numbers to more easily illustrate the fact. Multiplication in geometry only uses magnitudes.
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https://www.youtube.com/watch?v=ofpp3X6hDXs
The link to the video removed for imagined violations of code:
https://www.bitchute.com/video/IAFx7jmQx1Pk
Note that I am a US citizen living in the US and I am not subject to EU rules.
My video clearly proves I am not bullying or harassing anyone else.
The video contains damning proof that the mainstream is trying to suppress but which I have proclaimed over the last two decades.
The ultimate truth that dishonest mainstream academics continue to reject is simply this:
[f(x+h)-f(x)]/h = f'(x) + Q(x,h) where Q(x,h)=0 implies and is implied by h=0.
is EXACTLY EQUIVALENT to:
f'(x) = Lim_{h \to 0} [f(x+h)-f(x)]/h
So it is obviously flawed in many respects, not just division by 0, but also circularity wrt limit theory verifinition.
...
https://www.youtube.com/watch?v=kI8k77O6vsk
Discovering the concept of number:
https://www.academia.edu/44820487/Discovering_the_concept_of_number_a_personal_journey
Proposition 1.
᾿Εάν ή οποσαούν μεγέθη οποσωνούν μεγεθών ίσων το πλήθος έκαστον εκάστου ισάκις πολλαπλάσιον, οσαπλάσιον εστίν εν των μεγεθών ενός, τοσαυταπλάσια έσται καί τα πάντα των πάντων.
Heath: If there are any number of magnitudes whatsoever (which are) equal multiples, respectively, of some (other) magnitudes, of equal number (to them), then as many times as one of the (first) magnitudes is (divisible) by one (of the second), so many times will all (of the first magnitudes) also (be divisible) by all (of the second).
Heath claims in a remark:
“In modern notation, mα + mβ + · · · = m(α + β + · · ·)”.
Not only does Heath misunderstand the proposition which does NOT say what he remarks, but he also forgets that in this particular book, the notion of number has not yet been established. So there is no addition or multiplication defined as yet for magnitudes. Moreover, Heath incorrectly states that the sums can be “infinite” through implication of the ellipsis - a misleading and also FALSE interpretation of the proposition. One cannot talk about “many times” in the context of infinity (an utterly junk concept that was rejected by the Ancient Greeks). If for arguments sake, the interpretation was that pertaining to an infinite series, then it could not be proved except as a partial sum and then transferring the property of the part over to the whole.
The only measure available at this stage is that of division, that is, the measure of each magnitude using one of its equal parts. The proposition says that given any two line segments divided into the same number of equal parts, then both the line segments are measured by their equal parts exactly the same number of times. The identity mα + mβ = m(α + β) arises in algebra from this proposition, but it is stated using addition and multiplication which have not been formally defined as yet. The difference and sum of line segments is at this stage only understood in terms of line segment lengths (magnitudes) which have no specified measure.
Given any two line segments as shown below with the vertical bars indicating partitions:
___|___|___
_____|_____|_____
Proposition 1 tells us that ___ measures ___|___|___ the same number of times that _____ measures _____|_____|_____. The actual number is irrelevant suffice to say it is the same in either case. This and NOTHING else.
Professor David Joyce (Clark University) ignorantly states:
Book V is on the foundations of ratios and proportions and in no
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https://www.youtube.com/watch?v=8W6MQ44mMvE
The true story of arithmetic is more fascinating than you ever imagined.
In this video I reveal everything you ever wanted to know about fractions and arithmetic.
Far too long, highly ignorant mainstream mathematics academics have peddled lies (sincere or otherwise), confusion and nonsense about mathematics. You cannot understand mathematics without acquiring a very deep understanding of geometry. After all, everything you learn in algebra, calculus, mathematical statistics, engineering and any other knowledge that matters, comes from geometry (without ANY exception!).
Were it not for my brilliant ancestors and their genius descendant (that would be me!), you would know NOTHING. You would still be living in caves, fighting and killing each other. I'd say you owe us royalties and every Greek should be allowed to do what a Greek does best and that is to think. All other nations should provide food and services to the Greeks for at least the next 2500 years. After that, we'll forget about the royalties. Chuckle.
You can do your part if you are well off and offer me what has been denied to me and my ancestors. I am currently homeless with shelter but if you expect me to live much longer, you will have to help me. Please, no small change. Contact me only if you can help change my lot in a very big way. Chuckle.
Link to presentation used in the video:
https://drive.google.com/file/d/1pt3KvwhfirrapJZLvTfSxWoLfxhIQNtL
Link to applet used in the video:
https://drive.google.com/file/d/1vVhVTK3oJOc57asItXP07PRiHbgNg0qA
Download my free eBook - the most important mathematics book ever written:
https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO
...
https://www.youtube.com/watch?v=m_hAhJ4p4os