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From the Preface. The theory of the Ikosahedron has during the last few years obtained a place of such importance for nearly all departments of modern analysis, that it seemed expedient to publish a systematic exposition of the same. Should this prove acceptable, I propose to continue in the same course and to treat in a similar manner the subject of Elliptic Modular Functions, and the general investigations newly made of Single-valued Functions, with linear transformations into themselves. Thus a treatise of several volumes would grow, in which I should expect to promote science, at least in so far as it might introduce many to realms of modern mathematics rich in far-stretching vistas. Referring generally as to the limitations of the material, which I have observed in this publication, to the future exposition itself, I would here only draw attention to the second part, which treats of the solution of equations of the fifth degree. - It is now fully twenty-five years since Brioschi, Hermite, and Kronecker in joint labours created the modern theory of equations of the fifth degree. But though these investigations are now and again quoted, the mathematical world at large has hitherto failed to grasp their true import. By giving the first place in the following pages to the subject of the Ikosahedron, and by treating this as the true basis of the processes of solution, a view of the theory is brought forward than which a simpler and more lucid one cannot well be desired. A special difficulty, which presented itself in the execution of my plan, lay in the great variety of mathematical methods entering into the theory of the Ikosahedron. On this account it seemed advisable to take for granted no specific previous knowledge in any direction, but rather to introduce, where necessary, such explanations and references as might suffice as preliminary landmarks on the field under immediate survey. What I, however, do expect in my reader is a certain ripeness of mathematical judgment, which shall enable him to interpret concise, brief statements, so as to see in them the general principle involved in the particular case. This is the same method I have ever pursued in my more advanced lectures; indeed I have introduced into the details of these expositions the practices of my lecture-room. It is in this spirit I would have the title interpreted which I have given to my disquisition. I cannot close these short prefatory remarks without expressing my special thanks to my honoured friends Professor Lie in Christiania and Professor Gordan in Erlangen for manifold suggestions and assistance. My indebtedness to Professor Lie dates back to the years 1869-70, when we were spending the last period of our student-life in Berlin and Paris together in intimate comradeship. At that time we jointly conceived the scheme of investigating geometric or analytic forms susceptible of transformation by means of groups of changes. This purpose has been of directing influence in our subsequent labours, though these may have appeared to lie far asunder. Whilst I primarily directed my attention to groups of discrete operations, and was thus led to the investigation of regular solids and their relations to the theory of equations, Professor Lie attacked the more recondite theory of continued groups of transformations, and therewith of differential equations. It was in the autumn of 1874 that I first came into real contact with Professor Gordan. I had at that time already commenced the study of the Ikosahedron for myself (without then knowing Professor Schwarz's earlier works, to which we shall hereafter frequently have occasion to refer); but I considered my whole manner of attacking the question rather in the light of preliminary training. If now a far-reaching theory has grown from those beginnings, I attribute this result primarily to Professor Gordan....

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