The origins of Euler's early work on continued fractions

In this paper, I examine Eulerʼs early work on the elementary properties of continued fractions in the 1730s, and investigate its possible links to previous writings on continued fractions by authors such as William Brouncker. By analysing the content of Eulerʼs first paper on continued fractions, ‘De fractionibus continuis dissertatio’ (1737, published 1744) I conclude that, contrary to what one might expect, Eulerʼs work on continued fractions initially arose not from earlier writings on continued fractions, but from a wish to solve the Riccati differential equation

Effect of fenofibrate on progression of diabetic retinopathy

Background Findings from cardiovascular outcome trials suggest that fenofibrate therapy may reduce the progression of diabetic retinopathy. Methods We recruited and followed adults with nonreferable diabetic retinopathy or maculopathy using the national Diabetic Eye Screening (DES) program in Scotland. We randomly assigned participants to receive 145-mg fenofibrate tablets or placebo (taken daily or, in those with impaired renal function, on alternate days). The primary outcome was a composite of developing referable diabetic retinopathy or maculopathy (based on Scotland’s DES grading scheme) or treatment (intravitreal injection, retinal laser, vitrectomy) for retinopathy or maculopathy. Results A total of 1151 participants were randomly assigned to treatment. During a median of 4.0 years, progression to referable diabetic retinopathy or maculopathy, or treatment thereof, occurred in 131 (22.7%) of 576 participants in the fenofibrate group and 168 (29.2%) of 575 in the placebo group (hazard ratio, 0.73; 95% confidence interval [CI], 0.58 to 0.91; P=0.006). In the fenofibrate group compared with the placebo group, the frequencies for any progression of retinopathy or maculopathy were 185 (32.1%) vs. 231 (40.2%); hazard ratio, 0.74; 95% CI, 0.61 to 0.90 and for the development of macular edema were 22 (3.8%) vs. 43 (7.5%); hazard ratio, 0.50; 95% CI, 0.30 to 0.84. Seventeen (3.0%) participants assigned fenofibrate and 28 (4.9%) assigned placebo were given treatment for retinopathy (hazard ratio, 0.58; 95% CI, 0.31 to 1.06). There was no effect on visual function, quality of life, or visual acuity. Trial-averaged estimated glomerular filtration rate was 7.9 (95% CI, 6.8 to 9.1) ml/min/1.73 m2 lower in participants in the fenofibrate group compared with the placebo group. Serious adverse events occurred in 208 (36.1%) participants allocated fenofibrate and 204 (35.5%) participants allocated placebo. Conclusions Fenofibrate reduced progression of diabetic retinopathy compared with placebo among participants with early retinal changes. (Funded by the National Institute for Health and Care Research; ClinicalTrials.gov number, NCT03439345; ISRCTN number, ISRCTN15073006.

Editing and reading early modern mathematical texts in the digital age

The advent of digital technology has brought a world of new possibilities for editors of historical texts. Though much has been written about conventions for digital editing, relatively little attention has been paid to the particular question of how best to deal with texts with heavily mathematical content. This essay outlines some ways of encoding mathematics in digital form, and then discusses three recent digital editions of collections of early modern mathematical manuscripts