Again and again the attempt has been made to detect harmonic ratios in the measurement of Athenian architecture, but ever without reward. Mr Hay has, however, made the discovery, and to an extent of which no one had previously dreamt. IV. In 8vo, 100 Plates, price 6s., FIRST PRINCIPLES OF SYMMETRICAL BEAUTY. _From the Spectator._ This is a grammar of pure form, in which the elements of symmetrical, as distinguished from picturesque beauty, are demonstrated, by reducing the outlines or planes of curvilinear and rectilinear forms to their origin in the principles of geometrical proportion. In thus analysing symmetry of outline in natural and artificial objects, Mr Hay determines the fixed principles of beauty in positive shape, and shews how beautiful forms may be reproduced and infinitely varied with mathematical precision. Hitherto the originating and copying of beautiful forms have been alike empirical; the production of a new design for a vase or a jug has been a matter of chance between the eye and the hand; and the copying of a Greek moulding or ornament, a merely mechanical process. By a series of problems, Mr Hay places both the invention and imitation of beautiful forms on a sure basis of science, giving to the fancy of original minds a clue to the evolving of new and elegant shapes, in which the infinite resources of nature are made subservient to the uses of art. The volume is illustrated by one hundred diagrams beautifully executed, that serve to explain the text, and suggest new ideas of beauty of contour in common objects. To designers of pottery, hardware, and architectural ornaments, this work is particularly valuable; but artists of every kind, and workmen of intelligence, will find it of great utility. _From the Athenæum._ The volume before us is the seventh of Mr Hay’s works. It is the most practical and systematic, and likely to be one of the most useful. It is, in short, a grammar of form, or a spelling-book of beauty. This is beginning at the right end of the matter; and the necessity for this kind of knowledge will inevitably, though gradually, be felt. The work will, therefore, be ultimately appreciated and adopted as an introduction to the study of beautiful forms. The third part of the work treats of the Greek oval or composite ellipse, as Mr Hay calls it. It is an ellipse of three foci, and gives practical forms for vases and architectural mouldings, which are curious to the architect, and will be very useful both to the potter, the moulder, and the pattern-drawer. A fourth part contains applications of this to practice. Of the details worked out with so much judgment and ingenuity by Mr Hay, we should in vain attempt to communicate just notions without the engravings of which his book is full. We must, therefore, refer to the work itself. The forms there given are full of beauty, and so far tend to prove the system. V. In 8vo, 14 Coloured Diagrams, Second Edition, price 15s.,