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Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid’s reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory.(More)
This paper presents and compares FG and Eu, two recent formalizations of Euclid’s diagrammatic arguments in the Elements. The analysis of FG, developed by the mathematician Nathaniel Miller, and that of Eu, developed by the author, both exploit the fact that Euclid’s diagrammatic inferences depend only on the topology of the diagram. In both systems, the(More)
In his writings on Hilbert’s foundational work, Paul Bernays sometimes contrasts Euclid’s geometrical method in the Elements with Hilbert’s in Foundations of Geometry, identifying two features that distinguished the latter from the former (notably, in [6] and in the introduction to [3]). First, Hilbert’s theory is abstract. Though the primitives of the(More)
Fifty-four cases of congenital dacryocele from several medical centers were reviewed retrospectively. There was strong female preponderance (73%) and unilateral involvement (88%). Lacrimal sac contents could be expressed by local massage through the puncta in 21% of cases. Probing and irrigation were done under general (27.8%) or local (55.6%) anesthesia,(More)
The case of a 4 1/2-year-old boy with congenital vertical ocular motor apraxia who was otherwise developmentally and neurologically normal is reported. The presence of perinatal hypoxia in this patient may have been etiologic. While the presence of a supranuclear vertical ocular motor abnormality usually suggests a serious, acquired neurologic or systemic(More)
Six cases of classic achromatopsia are presented. The methods of practical clinical diagnosis are discussed, including paradoxical pupillary constriction in darkness, the easily performed Sloan achromatopsia test, and electrophysiologic studies which are useful in young children. The visual and cosmetic benefits of heavily tinted contact lenses in such(More)