Snellius versneld

Practically all computations of the value of ? before 1600 were done using Archimedes' method. As is well-known, this method consists of approximation of the circle with diameter 1 by inscribed and circumscribed regular polygons. Denote the circumference of the inscribed and circumscribed regular N-gon by PN and QN respectively. Then PN < ?< QN and lim N !1 PN = lim N !1 QN = ? A little trigonometry shows us that QN = N tan ? N ; PN = N sin ? N (1) from which the duplication formulae Q2N = 2PNQN PN + QN ; P2N = p PNQ2N (2) follow readily.As is well-known, Archimedes, started with the values Q6 = 2p3 and P6 = 3 and calculated Q12; P12;Q24; : : : ; Q96; P96 consecutively using the duplication formulae (2). See [A]. We also know that Ludolph van Ceulen, around 1600, continued this procedure until he obtained 35 decimal places of ?. To get an idea of the accuracy of the approximation QN to ? we use the Taylor series expansion of tan x

Changes in breaststroke swimming performances in national and international athletes competing between 1994 and 2011 -a comparison with freestyle swimming performances

BACKGROUND: The purpose of the present study was to analyse potential changes in performance of elite breaststroke swimmers competing at national and international level and to compare to elite freestyle swimming performance. METHODS: Temporal trends in performance of elite breaststroke swimmers were analysed from records of the Swiss Swimming Federation and the FINA (Fédération Internationale de Natation) World Swimming Championships during the 1994-2011 period. Swimming speeds of elite female and male breaststroke swimmers competing in 50 m, 100 m, and 200 m were examined using linear regression, non-linear regression and analysis of variance. Results of breaststroke swimmers were compared to results of freestyle swimmers. RESULTS: Swimming speed in both strokes improved significantly (p < 0.0001-0.025) over time for both sexes, with the exception of 50 m breaststroke for FINA men. Sex differences in swimming speed increased significantly over time for Swiss freestyle swimmers (p < 0.0001), but not for FINA swimmers for freestyle, while the sex difference remained stable for Swiss and FINA breaststroke swimmers. The sex differences in swimming speed decreased significantly (p < 0.0001) with increasing race distance. CONCLUSIONS: The present study showed that elite male and female swimmers competing during the 1994-2011 period at national and international level improved their swimming speed in both breaststroke and freestyle. The sex difference in freestyle swimming speed consistently increased in athletes competing at national level, whereas it remained unchanged in athletes competing at international level. Future studies should investigate temporal trends for recent time in other strokes, to determine whether this improvement is a generalized phenomenon

Women Outperform Men in Ultra-Distance Swimming - The 'Manhattan Island Marathon Swim' From 1983 to 2013

PURPOSE: Recent studies suggested that women and men's ultra-swim performances may be similar for distances of ~35 km. The present study investigated both the gender difference and the age of peak ultra-swim performance between 1983 and 2013 at the 46-km 'Manhattan Island Marathon Swim' with water temperatures <20°C. METHODS: Changes in race times and gender difference in 551 male and 237 female finishers were investigated using linear, non-linear, and hierarchical multi-level regression analyses. RESULTS: The top ten race times ever were significantly (P<0.0001) lower for women (371±11 min) than for men (424±9 min). Race times of the annual fastest and annual three fastest women and men did not differ between genders and remained stable across years. The age of the annual three fastest swimmer increased from 28±4 years (1983) to 38±6 years (2013) (r2=0.06, P=0.03) in women and from 23±4 years (1984) to 42±8 years (2013) (r2=0.19, P<0.0001) in men. CONCLUSIONS: The best women were ~12-14% faster than the best men in a 46-km open-water ultra-distance race with temperatures <20°C. The maturity of ultra-distance swimmers has changed during the last decades with the fastest swimmers becoming older across the years