Recap: The most interesting and important characteristic of ratios that we realise, is about their quotientness (πηλικότητα), which tells us much more about their relationship than just the fact that they are somehow related. For example, the ratio c (circumference) : 2r (diameter) is a valid ratio with a relationship (*), however, there is no quotientnes, that is, no common measure or divisor exists (as in well formed magnitude) that can relate the aliquot parts in a meaningful way, after all, a ratio is just a comparison of two magnitudes in its most primitive form. (*) While c and 2r have no common divisor, if any attempt is made to measure c using 2r as the unit, we realise a CONSTANT (σταθερό) because every approximate measure yields the same constant, that is, 3; 3.1; 3.14; 3.141; 3.1415; 3.14159, etc. There is no precise measure meaning that there is no number to describe the measure of c (which is the symptom of π) using any circle whatsoever. A poll was conducted on LinkedIn to see how many people actually understood what is pi. Over 80% of them chose the wrong option and closer to 95% did not understand: https://www.academia.edu/66662818/Answers_to_LinkedIn_Pi_Poll_What_exactly_is_Pi_in_mathematics Definition 14. Σύνθεσις λόγου εστί λήψις του ηγουμένου μετά του επομένου ως ενός πρός αυτό το επόμενον. Heath: A composition of a ratio is a taking of the (ratio of the) leading plus the following (magnitudes), as one, to the following (magnitude) by itself. In other words, if p:q then the composed ratios are: p+q:q or p+q:p. By itself, such a definition is worthless, but in a ratio of two ratios such as 2:3 and 4:6, then: 2+3:2 is in proportion with 4+6:4 and 2+3:3 is in proportion with 4+6:6 Definition 15. Διαίρεσις λόγου εστί λήψις της υπεροχής, ή υπερέχει το ηγούμενον του επομένου, πρός αυτό το επόμενον. Heath: A separation of a ratio is a taking of the (ratio of the) excess by which the leading (magnitude) exceeds the following to the following (magnitude) by itself. By itself, this definition is worthless, but in a ratio of two ratios such as 2:3 and 4:6, then: 2-3:2 is in proportion with 4-6:4 and 2-3:3 is in proportion with 4-6:6 Euclid only discussed the positive case (“exceeds”) in Definition15, that is: 3-2:2 is in proportion with 6-4:4 and 3-2:3 is in proportion with 6-4:6 Definitions 14 and 15 are used in propositions 17 and 18. Definition 16. Αναστροφή λόγου εστί λήψις του ηγουμένου πρός τήν υπεροχήν, 'η υπερέχει το ηγούμενον του επομένου. ... https://www.youtube.com/watch?v=vxndEKtepDs

Arithmetic with ratios of magnitudes came long before algebraic arithmetic and algebra itself. In fact, the most important book of Euclid's Elements(Book V), lays the foundation of everything we know in arithmetic and algebra. Euclid however, did not explain this arithmetic as I have done for you! Using the article and applet that follows, you can for the first time understand number, arithmetic and how we finally got algebra. https://www.academia.edu/44820487/Discovering_the_concept_of_number_a_personal_journey Link to free applet (you can run in browser or in Geogebra): https://drive.google.com/file/d/1TvexyEe2bs_YhdOjELY69WEfmkVmOBj_ My older video: https://www.youtube.com/watch?v=e86aVojMWqA Nikos Papoulas' video in Greek: https://www.youtube.com/watch?v=mjpvJWRvUf8 Help me live my last few years in a humble place I can call my own: https://gofund.me/af8a5312 If you can't contribute, please spread the link on every social channel that you can. I gave the New Calculus (https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus) to humanity free of charge and have explained many mathematical concepts improperly understood by any in the last 2300 years. So far, my reward has been nothing but persecution from vile and jealous mainstream mathematics academics. ... https://www.youtube.com/watch?v=NkLkuCq1fsI

In this video I discuss the sheer intransigence of the mainstream academics which transcends ethnicity. I recently asked if any Greek were willing to translate my free eBook into Greek and the responses are shown at this link: https://mathematica.gr/forum/viewtopic.php?f=40&t=65174 https://www.mathematica.gr/forum/viewtopic.php?f=9&t=65422 The hostility, especially from the chief primate Mihalis Lambrou is quite noticeable. He was traveling on the last leg of his journey at Athens airport and instructed his sub-primate "Dimitris" to instantly close the threads. Given there was only incorrect opinions and assertions with respect to my request for a party interested in translating my work, I decided to post a second comment explaining why f'(x)=f'(x)+Q(x,h) is exactly the same as [f(x+h)-f(x)]/h, which is one of the assertions by the administrators. This is proved in my video: https://youtu.be/O35x6HS1bps Download my free eBook here: https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view Remember to click LIKE, become a subscriber and share the NEWS! ... https://www.youtube.com/watch?v=iCdQQ2Dfo8M

I want to destroy the mythical foundations of mathematics created by moron mathematicians of the last 2300 years. I want even the janitors to understand the meaning of number and how it is perfectly derived. I want every child to love mathematics and it's not possible unless they understand basic philosophy and the concept of number which is derived perfectly using philosophy. I am the great John Gabriel, discoverer of the first and only rigorous formulation of calculus in human history. More advanced alien civilisations probably know of it already. ... https://www.youtube.com/watch?v=7iby3JDxxx0