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FROM time to time we have noted the annual volumes of this Society from vol. ii. to vol. xi., which appeared last year. The volume before us fills up a lacuna and now makes the series complete. In the early days of the Society the publication of Proceedings was not contemplated, and when an access of members rendered publication possible, the cost of printing absorbed the major part of the funds, and each session's subscriptions have only sufficed for the current session's volume Some few years since a special appeal was made and funds sufficient to warrant publication obtained. The result is the admirable piece of geometrical work before us. For, in fact, the volume is almost entirely one man's work. The first president was Dr. J. S. Mackay, whose edition of Euclid for the Messrs. Chambers in 1884 gave ample evidence that there was an elegant and specially learned geometer in our midst. The article on “Euclid” in the Encyclop. Britannica confirmed this discovery. It has been long known that Dr. Mackay had large stores of notes, and we are glad to find that he has found an outlet for much of this interesting matter. At the second meeting of the Society the president read a paper on the triangle and its six scribed circles. A portion of this paper was given in abstract in vol. ii., and was considerably enlarged in vol. xi., under the heading “History of the Nine-point Circle.” In the long interval, with the permission of the Council, Dr. Mackay has amassed a collection of notes, divided into twenty sections, filling more than 1600 quarto pages of manuscript. A selection has been made which most nearly corresponds with what was actually communicated to the Society in 1883. The nine-point circle is accounted for above. The sections embraced in the present instalment treat of the centroid, the circumcentre, the incentre, the excentres, the orthocentre, Euler's line, relations among the radii, and area. They occupy pp. 6-128, and are accompanied by sixty-eight lithographed figures. Each property is traced back, as far as can be ascertained, to the first discoverer, the author having had the assistance of French, German, and Belgian mathematicians in addition to the aid of personal friends in Great Britain. The result is a rich repertory of almost, if not quite, all that is known on the special points indicated above.

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