’DEAR MICKLETHWAITE,—­I am back in England, and ought before this to have written to you.  I see you have just published a book with an alarming title, “A Treatise on Trilinear Co-ordinates.”  My hearty congratulations on the completion of such a labour; were you not the most disinterested of mortals, I would add a hope that it may somehow benefit you financially.  I presume there are people who purchase such works.  But of course the main point with you is to have delivered your soul on Trilinear Co-ordinates.  Shall I run down to Sheffield to see you, or is there any chance of the holidays bringing you this way?  I have found a cheap flat, poorly furnished, in Bayswater; the man who let it to me happens to be an engineer, and is absent on Italian railway work for a year or so.  My stay in London won’t, I think, be for longer than six months, but we must see each other and talk over old times,’ etc.

This he addressed to a school at Sheffield.  The answer, directed to the club, reached him in three days.

’My DEAR BARFOOT,—­I also am in London; your letter has been forwarded from the school, which I quitted last Easter.  Disinterested or not, I am happy to tell you that I have got a vastly better appointment.  Let me know when and where to meet you; or if you like, come to these lodgings of mine.  I don’t enter upon duties till end of October, and am at present revelling in mathematical freedom.  There’s a great deal to tell.—­Sincerely yours,

THOMAS MICKLETHWAITE.’

Having no occupation for his morning, Barfoot went at once to the obscure little street by Primrose Hill where his friend was lodging.  He reached the house about noon, and, as he had anticipated, found the mathematician deep in study.  Micklethwaite was a man of forty, bent in the shoulders, sallow, but not otherwise of unhealthy appearance; he had a merry countenance, a great deal of lank, disorderly hair, and a beard that reached to the middle of his waistcoat.  Everard’s acquaintance with him dated from ten years ago, when Micklethwaite had acted as his private tutor in mathematics.

The room was a musty little back-parlour on the ground floor.

‘Quiet, perfectly quiet,’ declared its occupant, ’and that’s all I care for.  Two other lodgers in the house; but they go to business every morning at half-past eight, and are in bed by ten at night.  Besides, it’s only temporary.  I have great things in view—­ portentous changes!  I’ll tell you all about it presently.’

He insisted, first of all, on hearing a full account of Barfoot’s history since they both met.  They had corresponded about twice a year, but Everard was not fond of letter-writing, and on each occasion gave only the briefest account of himself.  In listening, Micklethwaite assumed extraordinary positions, the result, presumably, of a need of physical exercise after hours spent over his work.  Now he stretched himself at full length on the edge of his chair, his arms extended above him; now he drew up his legs, fixed his feet on the chair, and locked his hands round his knees; thus perched, he swayed his body backwards and forwards, till it seemed likely that he would pitch head foremost on to the floor.  Barfoot knew these eccentricities of old, and paid no attention to them.