Definition 17. Δι΄ ίσου λόγος εστί πλειόνων όντων μεγεθών άλλων αυτοίς ίσων το πλήθος σύνδυο λαμβανομένων καί εν τω αυτώ λόγω, όταν ή ως εν τοίς πρώτοις μεγέθεσι το πρώτον πρός το έσχατον, ουτως εν τοίς δευτέροις μεγέθεσι το πρώτον πρός το έσχατον ή άλλως λήψις των άκρων καθ᾿ υπεξαίρεσιν των μέσων. Heath: There being several magnitudes, and other (magnitudes) of equal number to them (which are) also in the same ratio taken two by two, a ratio via equality (or ex aequali) occurs when as the first is to the last in the first (set of) magnitudes, so the first (is) to the last in the second (set of) magnitudes. Or alternately, (it is) a taking of the (ratio of the) outer (magnitudes) by the removal of the inner (magnitudes). In other words, if p, q, r are the first set of magnitudes, and s, t, u the second set, and p : q : r :: s : t : u, then the ratio via equality (or ex aequali) corresponds to p : r :: s : u. Correct definition: A ratio ex aequali arises when, there being two ratios with an equal number of magnitudes (equal or greater than two magnitudes per ratio), then any two magnitudes from either ratio but in the same positions, will be proportionate to each other. Example: 2 : 7 : 3 :: 4 : 14 : 6 2:3 :: 4:6; 3:7 :: 6:14; (7+3):12 :: (14+6):24; etc. In fact, the definition also suggests that more than two magnitudes can be taken from each ratio as long as relative positions are preserved where certain arithmetic operations are concerned. For example: (2+7+3) :12 :: (4+14+6):24, etc. Definition 18. Τεταραγμένη δε αναλογία εστίν, όταν τριών όντων μεγεθών καί άλλων αυτοίς ίσων το πλήθος γίνηται ως μεν εν τοίς πρώτοις μεγέθεσιν ηγούμενον πρός επόμενον, ουτως εν τοίς δευτέροις μεγέθεσιν ηγούμενον πρός επόμενον, ως δε εν τοίς πρώτοις μεγέθεσιν επόμενον πρός άλλο τι, ουτως εν τοίς δευτέροις άλλο τι πρός ηγούμενον. Heath: There being three magnitudes, and other (magnitudes) of equal number to them, a perturbed proportion occurs when as the leading is to the following in the first (set of) magnitudes, so the leading (is) to the following in the second (set of) magnitudes, and as the following (is) to some other (i.e., the remaining magnitude) in the first (set of) magnitudes, so some other (is) to the leading in the second (set of) magnitudes. In ... https://www.youtube.com/watch?v=wX0s3UvOPRU

I was the first in human history to realise these facts. I was also the first to prove then mean value theorem constructively using the flawed mainstream calculus and my New Calculus. Constructive Proof of the Mean Value Theorem: https://drive.google.com/open?id=0B-mOEooW03iLZG1pNlVIX2RTR0E Explaining the concept of positional derivative: https://drive.google.com/open?id=0B-mOEooW03iLVVg3QWtOdkxUbVk Using the rigorous New Calculus: https://drive.google.com/open?id=0B-mOEooW03iLblJNLWJUeGxqV0E ... https://www.youtube.com/watch?v=8OUULDvdJnY