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THESE Exercises are set out in; what I call The Vertical and Horizontal Arrangement. They may -be taken in the vertical order in which they stand, or the Examples ' numbered 1, 1, 1, maybe taken first, and then 2, 2, 2, and so on. Thus of the six Examples in each of the sets numbered I. to XXX. Three are on Euclid, Book I. to Prop. 33; Two are on Book I., 34 to end of Book II.; One is on Book III. to Prop. 16. Further, in each Exercise from I to XXX., Example 1. Is on Euclid, Book I. to Prop. 19. 2. Is on Euclid, Book I Prop. 20 to 20. 3. Is on Euclid, Book I. to 27 to 33. 4. Is on Euclid, Book I. to 34 to 48.5. Is on Euclid, Book II. 6. Is on Euclid, Book III. to Prop. 16. So by taking the first Example in each Exercise in succession, a graduated series of deductions from early propositions in Book I. will be obtained, and a similar result will follow from taking the second Examples in succession, and so on for the third and following Examples. Next, observe that in the Exercises numbered XXXI. to L. 1 and 2 are on Euclid, I. to Prop. 33. 3 and 4 are on Euclid, I., 84 to end of Book II. 5 and 6 are on Euclid, III., 17 to end. Lastly, in Exercise LI. the 4th and 6th Books of Euclid are brought in, and in this and the succeeding Exercises 1 and 2 are on Euclid, Books I. and II. 8 is on Book III. 4 is on Book IV. 5 and 6 are on Book VI. My intention has been to make the Examples progressive, those first given in illustration of each Book of Euclid being very easy, and those in subsequent Exercises increasing gradually in difficulty, with easy deductions interspersed among the later sets. I shall be grateful for the advice as to the transposition or omission of particular Examples. -J. HAMBLIN SMITH, CAMBRIDGE, May 29, 1896.

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