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The gods that guard the poles have been assigned the function of assembling the separate and unifying the manifold members of the whole, while those appointed to the axes keep the circuits in everlasting revolution around and around. And if I may add my own conceit, the centers and poles of all the spheres symbolize the wry-necked gods (τῶν ίυγγικῶν θεῶν) by imitating the mysterious union and synthesis which they effect; the axes represent the connectors (συνοχάς) of all the cosmic orders ... and the very spheres are likenesses of the perfecting divinities (τῶν τελεσιουργῶν θεῶν) ... (Proclus 1992a , pp. 74–5; 1873 , pp. 90–91) Proclus's Commentary on the First Book of Euclid's Elements contains one of the most important discussions of the nature of mathematical objects and preserves a wealth of historical information about mathematics from antiquity. However, large sections of the text have been neglected by scholars since the text's first publication in the sixteenth century. Proclus expounds at length on correspondences between mathematical objects and the gods. At times he uses the language of "theurgy," the ritual practices that later Neoplatonists insisted were necessary for the human soul to ascend to the intelligibles. The contention of this article is that the Commentary, taken as a whole, concerns the practice of "inner theurgy," mental rituals used by advanced students to raise the soul beyond its assigned bounds. This practice can be traced back to Plotinus and the very beginnings of Neoplatonism; however, the particular methods presented by Proclus are his own development. This interpretation of the Commentary is supported by Proclus's works Commentary on the Parmenides and Platonic Theology. Parts of the Commentary that have been studied carefully by modern scholars, especially the theory of the mathematical phantasia or imagination, will be shown to be crucial elements in Proclus's theory and practice of inner theurgy. [ABSTRACT FROM AUTHOR]

In 1626, René Descartes and Claude Mydorge worked closely together on the problem of refraction, apparently discovering what is now known as the sine law of refraction. They constructed a plano-hyperbolic lens in order to test out the truth of this mathematical relationship. In 1637, Descartes finally published the sine method of determining refractions in his Dioptrique, which also demonstrated, on the basis of this relationship, that the hyperbola and ellipse were anaclastic lines (that is, that a lens with their profile would refract rays perfectly to a single point) without mentioning Mydorge. Mydorge himself wrote a short manuscript on the subject of refraction and lenses, which is found in a copy among Mersenne's papers. Scholars have usually assumed that this treatise was written as Mydorge and Descartes worked together in 1626. However, other evidence shows that only the first part of the treatise records the two men's early collaboration. Subsequently, Mydorge and Descartes completed their arguments separately, Descartes with the help of Isaac Beeckman. [ABSTRACT FROM AUTHOR]

Kepler’s 1604 Optics (Ad Vitellionem Paralipomena) proposed among many other things a new way of locating the place of the image under reflection or refraction. He rejected the “perspectivist” method that had been used through antiquity and the Middle Ages, whereby the image was located on the perpendicular between the object and the mirror (the “cathetus”). Kepler faulted the method for requiring a metaphysical commitment to the action of final causes in optics: the notion that the image was at that place because it was best or appropriate for it to be there, and for no other discernible reason. Kepler’s new theory relied on binocular vision and depth perception to determine the location of the image. No final causes were required, and he showed that the image would in general not be found on the cathetus. According to modern scholarship, Kepler’s theory was part of his revolutionary transformation of the science of optics, and his abandonment of perspectivist optics; as a consequence, the theory of binocular vision is also thought to be original with him. This article demonstrates that the very same theory of binocular image location was set out by Giovanni Battista Benedetti some twenty years earlier, and his writings on this subject may have been Kepler’s unacknowledged source for his own theory. Furthermore, another mathematician, Simon Stevin, developed much the same theory at the same time as Kepler and, it seems, independently of either Benedetti or Kepler. The discovery of these other binocular theories, especially Benedetti’s, requires us to recognize that Kepler’s revolution (if it can be called that) emerged out of a wider dissatisfaction with the foundations of perspectivist optics, which other lesser-known opticians resolved in much the same way that Kepler did. [ABSTRACT FROM AUTHOR]

The article discusses the rediscovery and analysis of the mathematical work of the mathematician Thomas Harriot by the researcher Jacqueline Stedall before she died.

Some time in the late 1590s, the Welsh amateur mathematician John Bulkeley wrote to Thomas Harriot asking his opinion about the properties of a truly gargantuan (but totally imaginary) plano-spherical convex lens, 48 feet in diameter. While Bulkeley's original letter is lost, Harriot devoted several pages to the optical properties of 'Mr Bulkeley his Glasse' in his optical papers (now in British Library MS Add. 6789), paying particular attention to the place of its burning point. Harriot's calculational methods in these papers are almost unique in Harriot's optical remains, in that he uses both the sine law of refraction and interpolation from Witelo's refraction tables in order to analyze the passage of light through the glass. For this and other reasons, it is very likely that Harriot wrote his papers on Bulkeley's glass very shortly after his discovery of the law and while still working closely with Witelo's great Optics; the papers represent, perhaps, his very first application of the law. His and Bulkeley's interest in this giant glass conform to a long English tradition of curiosity about the optical and burning properties of large glasses, which grew more intense in late sixteenth-century England. In particular, Thomas Digges's bold and widely known assertions about his father's glasses that could see things several miles distant and could burn objects a half-mile or further away may have attracted Harriot and Bulkeley's skeptical attention; for Harriot's analysis of the burning distance and the intensity of Bulkeley's fantastic lens, it shows that Digges's claims could never have been true about any real lens (and this, I propose, was what Bulkeley had asked about in his original letter to Harriot). There was also a deeper, mathematical relevance to the problem that may have caught Harriot's attention. His most recent source on refraction-Giambattista della Porta's De refractione of 1593-identified a mathematical flaw in Witelo's cursory suggestion about the optics of a lens (the only place that lenses appear, however fleetingly, in the writings of the thirteenth-century Perspectivist authors). In his early notes on optics, in a copy of Witelo's optics, Harriot highlighted Witelo's remarks on the lens and della Porta's criticism (which he found unsatisfactory). The most significant problem with Witelo's theorem would disappear as the radius of curvature of the lens approached infinity. Bulkeley's gigantic glass, then, may have provided Harriot an opportunity to test out Witelo's claims about a plano-spherical glass, at a time when he was still intensely concerned with the problems and methods of the Perspectivist school. [ABSTRACT FROM AUTHOR]

The article focuses on Peter Ramus' histories of science. Ramus concluded that mathematics was the most natural and well organized of arts. Ramus's last work on mathematics was published in two versions, which contained the three books of the Prooemium plus another twenty-eight books of criticism of Euclid. The impact of Ramus's history can be seen in the mathematical lecture of Henry Savile delivered at Oxford in 1570 which concerns with mathematical astronomy and consist in part of a brilliant exposition of the mathematics of both Ptolemy's Almagest and Copernicus's De revolutionibus.

The article presents an introduction on the histories of science in early modern Europe. The papers here argue that there were indeed histories of mathematics before Jean-Etienne Montucla which are worthy of scholarly attention and that the dismissal of Renaissance histories of science as nothing more than a cloud of fine adjectives and metaphors is a fallacy. Three of the papers in this collection address histories of mathematics, which were written from the late fifteenth century to the early seventeenth century. It also provides a new approach that considers the relationship which humanist culture had with the emerging new sciences.

Objective To develop and validate two automatic methods for the detection of burst and interburst periods in preterm eight-channel electroencephalographs (EEG). To perform a detailed analysis of interobserver agreement on burst and interburst periods and use this as a benchmark for the performance of the automatic methods. To examine mathematical features of the EEG signal and their potential correlation with gestational age. Methods Multi-channel EEG from 36 infants, born at less than 30 weeks gestation was utilised, with a 10 min artifact-free epoch selected for each subject. Three independent expert observers annotated all EEG activity bursts in the dataset. Two automatic algorithms for burst/interburst detection were applied to the EEG data and their performances were analysed and compared with interobserver agreement. A total of 12 mathematical features of the EEG signal were calculated and correlated with gestational age. Results The mean interobserver agreement was found to be 77% while mean algorithm/observer agreement was 81%. Six of the mathematical features calculated (spectral entropy, Higuchi fractal dimension, spectral edge frequency, variance, extrema median and Hilberts transform amplitude) were found to have significant correlation with gestational age. Conclusions Automatic detection of burst/interburst periods has been performed in multi-channel EEG of 36 preterm infants. The algorithm agreement with expert observers is found to be on a par with interobserver agreement. Mathematical features of EEG have been calculated which show significant correlation with gestational age. Significance Automatic analysis of preterm multi-channel EEG is possible. The methods described here have the potential to be incorporated into a fully automatic system to quantitatively assess brain maturity from preterm EEG. [ABSTRACT FROM AUTHOR]

Aim: To develop a method that segments preterm EEG into bursts and inter-bursts by extracting and combining multiple EEG features. Methods: Two EEG experts annotated bursts in individual EEG channels for 36 preterm infants with gestational age < 30 weeks. The feature set included spectral, amplitude, and frequency-weighted energy features. Using a consensus annotation, feature selection removed redundant features and a support vector machine combined features. Area under the receiver operator characteristic (AUC) and Cohen’s kappa ( κ ) evaluated performance within a cross-validation procedure. Results: The proposed channel-independent method improves AUC by 4–5% over existing methods ( p < 0.001, n = 36 ), with median (95% confidence interval) AUC of 0.989 (0.973–0.997) and sensitivity–specificity of 95.8–94.4%. Agreement rates between the detector and experts’ annotations, κ = 0.72 (0.36–0.83) and κ = 0.65 (0.32–0.81), are comparable to inter-rater agreement, κ = 0.60 (0.21–0.74). Conclusions: Automating the visual identification of bursts in preterm EEG is achievable with a high level of accuracy. Multiple features, combined using a data-driven approach, improves on existing single-feature methods. [ABSTRACT FROM AUTHOR]