We premise that a straight line inverts into a circle passing through the pole, and vice versa; that a circle inverts into a circle, the two circles being subcontrary sections of a cone of the second degree passing through the pole; and that the angles between lines and surfaces at their points of intersection are the same as the angles between the inverse lines and surfaces at the inverse points. A normal is a straight line cutting a curve or a surface at right angles; it will therefore invert into a circle through the pole cutting the inverse curve or surface at the inverse point at right angles. Such a circle we will call a normal circle. We will now prove that, if two normals at any two points of a surface intersect and be equal, the normals at the inverse points of the inverse surface also will intersect and be equal.

The object of this paper is to ascertain the relative values of the pieces on a chessboard. If a piece be placed on a square of a chessboard, the number of squares it commands depends in general on its position. If we calculate the average number of squares which any particular piece commands when placed in succession on every square of the board, it seems fair to assume that this gives a not very inexact measure of the value of the piece. For special reasons the above problem is stated in the following manner:- "A king and a piece of different colours are placed at random on two squares of a chessboard of $n^{2}$ squares; it is required the find the chance that the king is in check." The ordinary chessboard has an even number of squares; and as some of the results take different forms for odd and even values of n, the results are given merely for even values of n, and the results for the ordinary chessboard of 64 squares deduced from them.

The 1870s were an active time of academic reform in Cambridge. The old Elizabethan statutes had been partially modified by the first Victorian statutes in 1858; but a comprehensive change was made by the Universities of Oxford and Cambridge Act of 1877. Under that Act, each college framed its own statutes for submission to the Privy Council. Accordingly, Trinity proceeded with that duty in a long succession of meetings ... When decisions had been completed, the final drafting of the statutes was entrusted to three of the fellows: Prof Cayley, whose reputation as a draughtsman long survived his retirement from practice at the Bar: Rev Coutts Trotter, conspicuous for his share in the organisation of the University, especially in the domain of natural science: and H M Taylor, whose legal training proved of high value. Then, and for many years to come, Taylor had a prominent (if not foremost) part in giving effect to the necessary changes in the old system. Independent in thought, and scrupulously just, he maintained the even tenor of his views, devoted to progress yet mindful of the ancient ways, fair in constructive act, and straight in opposition. Those statutes are now under repeal; their actual initial working owed much to the prudent wisdom of a band of reformers, among whom Taylor held a not unworthy place.

My best thanks are due to Mr H M Taylor of Trinity College, Cambridge, who has been good enough to read the proofs. By his kind assistance several errors and obscurities have been eliminated, and the volume generally rendered less imperfect than it would otherwise have been.

It is again a satisfaction to me to inscribe on the fly-leaf the name of Mr H M Taylor, whose kindly encouragement first led me to write on the subject, and whose help in revision I had gratefully acknowledged on former occasions.

I wish to express the very great obligations under which I lie to my friend and former tutor Mr H M Taylor, of Trinity College, Cambridge, for his kindness in the revision of the proof-sheets. He has caused the removal of obscurities and has made many valuable suggestions of which I have continually availed myself.

Mr H M Taylor, M.A., Fellow of Trinity College, Cambridge, had read the proofs with great care: the kind assistance he has given me in this way has proved of substantial service and usefulness in correcting the sheets. I desire to recognise most gratefully my sense of the value of the work which these gentlemen [Taylor and William Burnside] have done.

It was with extreme diffidence that I accepted an invitation from the Syndics of the Cambridge University Press to undertake for them a new edition of the 'Elements of Euclid'. Though I was deeply sensible of the honour, which the invitation conferred, I could not but recognise the great responsibility, which the acceptance of it would entail.

The invitation of the Syndics was in itself, to my mind, a sign of a widely felt conviction that the editions in common use were capable of improvement. Now improvement necessitates change, and every change made in a work, which has been a text book for centuries, must run the gauntlet of severe criticism, for while some will view every alteration with aversion, others will consider that every change demands an apology for the absence of more and greater changes.

I hereby acknowledge the great help I have received in this portion of my work from friends, and especially from Dr Forsyth and from my brother Mr J H Taylor. To the latter I am indebted for the Index to Books I-VI, which I hope may prove of some assistance to persons using this edition.

It is only of late years that the University of Cambridge has taken the wise step of giving greater scope to the teaching of geometry by not insisting on the actual proofs of propositions as presented in Euclid's text. All the definitions, axioms, and postulates, together with the sequence of propositions which he adopted, are retained, but in solving them "no proof will be accepted that assumes anything not proved in preceding propositions." The work under consideration is published in the "Pitt Press Mathematical Series," and it will be found to contain some important changes, both with regard to the proofs and method of arrangement. In the first few propositions of Book III the author has introduced several proofs which seem preferable to those generally adopted, while their order of sequence has been extensively changed. The alterations in the remaining propositions of this book have not been carried to any extent, but several outlines of alternative constructions have been inserted in cases where they may be most instructive.

One Braille notation was devised by the eminent Cambridge mathematician, Henry Martyn Taylor, who was overtaken by blindness in 1894, when engaged in the preparation of an edition of Euclid for the Cambridge University Press. By means of his ingenious and well thought out Braille notation he was enabled to transcribe many advanced scientific and mathematical works, and in 1917, with the assistance of Mr Emblen, a blind member of the staff of the National Institute for the Blind, he perfected it. It was recognised as so comprehensive that it was soon adopted as the standard mathematical and chemical notation, and is universally used by English-speaking people.

Sacred to the memory of Henry Martyn Taylor, Fellow of the College for sixty-one years and at one time Lecturer and Tutor. At the age of fifty-one he lost his sight; but undiscouraged and eagerly he kindled a torch of hope for those who shared his darkness. His invention of new symbols for Braille enabled them to compensate for the loss of their eyes with their fingers and to follow mathematics more clearly than they could have done by any light.  Although blind and almost sixty years old he became Mayor of Cambridge, and he dealt with College affairs assiduously to the last. He was born on 6th June 1842 and died on 16th October 1927, being by then Senior Fellow.

Taylor was singularly modest and devoid of personal ambition. He did not seek positions of honour and responsibility, but if they came his way he applied himself conscientiously to the duties which he had undertaken. Throughout his life he was a loyal friend and a fair opponent, generous and just in his thoughts, as in his dealings. Before his blindness he had shared in the usual recreations of his time, "real" tennis, cricket, shooting, fishing, billiards, in all of which he was proficient. He was also fond of foreign travel and mountain excursions. But the privation when it came did not provoke a murmur, and he maintained the steady even temper characteristic of him. The last few years of his life were clouded by increasing infirmity, and he died on October 16, 1927. The funeral service in the Chapel of Trinity College drew together a large company of friends and former colleagues to pay the last tribute of affection and respect to a noble and lovable character.