This report provides a mathematically thorough review and investigation of Metric Multidimensional scaling (MDS) through the analysis of Euclidean distances in input and output spaces. By combining a geometric approach with modern linear algebra and multivariate analysis, Metric MDS is viewed as a Euclidean distance embedding transformation that converts between coordinate and coordinate-free representations of data. In this work we link Mercer kernel functions, data in infinite-dimensional Hilbert space and coordinate-free distance metrics to a finite-dimensional Euclidean representation. We further set a foundation for a principled treatment of non-linear extensions of MDS as optimization programs on kernel matrices and Euclidean distances.