The classic Body Mass Index, (BMI), developed in the 19^th century by the Belgian mathematician Adolphe Quetelet [1] is an important indicator of the risk of death, of obesity, of negative health consequences, body fat percentage and of the shape of the body. While he BMI is assumed to indicate obesity in sedentary people and in people who do not practice sports, it is undisputed and a consensus among researchers [2][3][4][5][9][25] that Body Mass Index (BMI) is not a good indicator for obesity in people who developed their body through heavy physical work or sport but also in other segments of population such as those who appear to have a normal weight but in fact have a high body fat percentage and obese methabolism. The BMI also does not include all the variables essential for a health predictor. The BMI is not always a good predictor of metabolic disease, people who appear of healthy weight according to BMI have in some cases an obese metabolic syndrome. The BMI was developed as a law of natural sciences and “social physics” [1], as it was called then, before the middle of the 19^th century, and it had been used from the 70s for medical purposes, to detect obesity and the risk of mortality [6][7]. The BMI has a huge importance for modern society, affected by an obesity epidemic [8]. BMI has applications in medicine, sport medicine, sport, fitness, bodybuilding, insurance, nutrition, pharmacology. The main limitation of the BMI is that it does not account for body composition including non fat body mass such as muscles, joints, body frame and makes no difference between fat and non fat components of the body weight. The body composition and the proportion of fat and muscles make a difference in health outcomes [12][13][14][25][26][27][35][36][37] [38][39][40][41][42][43][44]…[100]. Body composition makes a difference also in the level of sport performance for athletes of every level. In nearly two centuries since the Body Mass Index was developed, no formula had been successfully developed to account for body composition and make the difference between muscle and fat in a consistent way. This can be considered a longstanding open problem of major importance for society. The objective of this analysis is to develop new formulae taking into account the health implication of body composition measured through indirect, simple indicators and making the difference between muscles and fat, healthy and non healthy metabolism. The formulae developed in this article are the only formula to successfully generalize BMI and make this difference. I develop a direct generalization of BMI, in the mathematical and physiological sense to account for fat and fat free mass and muscles, small and large body frames. It is the first such generalization because the classic BMI can be determined as a particular case of my formulae in the strict mathematical and practical physiologic sense. No other formula generalized the BMI to make the difference between fat and a large frame and muscles has ever been published in nearly two centuries since the BMI formula had been developed. The formulae I developed explain and generalize the conclusions of a large number of highly cited empirical experiments cited in the reference section. [35][36][37][38][38][39] [40][42][43][44]..[100] Most of the experimental proof I bring in support of my formulae and bodyweight quantification theory comes from many highly cited experimental research publications in medicine, sports medicine, sport science and physiology. My formulae explain also performance in decades of competitive sports and athletics